Experimental Notes on Elementary Differential Geometry · Euclidean geometry can therefore be...

Experimental Notes on Elementary DifferentialGeometry

Deane Yang

Last updated: 2020-03-20 22:02 ET

Contents

1 Introduction 9

2 Flat geometry 112.1 Axiomatic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Axioms of the line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Axiomatic plane geometry . . . . . . . . . . . . . . . . . . . . . . . . 132.1.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Flat geometry using coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 Cartesian line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.2 Cartesian plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.3 Length and distance . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.4 Angle via dot product . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Abstract vector space with an inner product . . . . . . . . . . . . . . . . . . 162.4 Euclidean geometry on an affine space . . . . . . . . . . . . . . . . . . . . . 17

3 Vector Spaces and Linear Maps 193.1 Definition of a vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Examples of vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Linearly independent vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 Basis and dimension of a vector space . . . . . . . . . . . . . . . . . . . . . . 213.5 Linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.5.1 Isomorphism of an m-dimensional vector space with Rm . . . . . . . 233.5.2 The space of linear maps . . . . . . . . . . . . . . . . . . . . . . . . . 233.5.3 Subspaces and quotient spaces . . . . . . . . . . . . . . . . . . . . . 233.5.4 Kernel and image of a linear map . . . . . . . . . . . . . . . . . . . . 24

3.6 Normal forms of a linear transformation . . . . . . . . . . . . . . . . . . . . 253.7 Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Affine Spaces and Maps 274.1 1-dimensional spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 The real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.2 Abstract 1-dimensional vector space . . . . . . . . . . . . . . . . . . 274.1.3 The affine line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 2-dimensional spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2.1 R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3

4 CONTENTS

4.2.2 2-dimensional vector space . . . . . . . . . . . . . . . . . . . . . . . . 284.2.3 2-dimensional affine space . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Affine spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4 Affine maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4.1 Affine maps of Cartesian spaces . . . . . . . . . . . . . . . . . . . . . 314.4.2 Abstract affine maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4.3 Affine independence and basis . . . . . . . . . . . . . . . . . . . . . . 314.4.4 The space of affine maps . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Concrete Category Theory 335.1 Definition of a category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2.2 Finite dimensional real vector spaces . . . . . . . . . . . . . . . . . . 345.2.3 Finite dimensional affine spaces . . . . . . . . . . . . . . . . . . . . . 345.2.4 C1 manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.3 Definition of Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.4 Examples of functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.4.1 Forgetful functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.4.2 Dualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6 Euclidean space 376.1 Cartesian space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.1.1 Length and angle using the dot product . . . . . . . . . . . . . . . . 376.2 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.3 Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.4 Orthogonal maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.5 Orthonormal vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.6 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.7 Isometries of Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . 42

7 Curves in Affine Space 437.0.1 Curves in Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.0.2 Curves in affine space . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7.1 Velocity and acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.2 Curves in Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7.2.1 Velocity, speed, and length . . . . . . . . . . . . . . . . . . . . . . . . 447.2.2 Acceleration and curvature . . . . . . . . . . . . . . . . . . . . . . . 457.2.3 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.2.4 Frenet-Serret frame in Euclidean 2-space . . . . . . . . . . . . . . . . 467.2.5 Frenet-Serret frame in Euclidean 3-space . . . . . . . . . . . . . . . . 47

CONTENTS 5

8 Surfaces in affine 3-space 498.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

9 Surfaces in Cartesian 3-space 519.1 Informal but flawed descriptions of a surface . . . . . . . . . . . . . . . . . . 519.2 2-planes in 3-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529.3 Local descriptions of a surface . . . . . . . . . . . . . . . . . . . . . . . . . . 539.4 Equivalence of definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549.5 Coordinate charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559.6 Tangent space at a point on a surface . . . . . . . . . . . . . . . . . . . . . . 55

10 Calculus on Affine Space 5710.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.2 Functions and maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.3 Dual vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.4 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

10.4.1 Directional derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 5810.4.2 Differential of a function . . . . . . . . . . . . . . . . . . . . . . . . . 58

10.5 Differential 1-form on an open subset of Am . . . . . . . . . . . . . . . . . . 5910.6 Line integral of a 1-form along a curve in Am . . . . . . . . . . . . . . . . . 5910.7 Fundamental example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

11 Differential of a map 6111.1 Characterization of a linear map . . . . . . . . . . . . . . . . . . . . . . . . . 61

11.1.1 Differential of a map . . . . . . . . . . . . . . . . . . . . . . . . . . . 6111.1.2 Pushforward of a tangent vector . . . . . . . . . . . . . . . . . . . . . 6111.1.3 Pullback of a cotangent vector . . . . . . . . . . . . . . . . . . . . . . 62

11.2 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

12 Surfaces in affine 3-space 6312.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.2 Surfaces in Cartesian 3-space . . . . . . . . . . . . . . . . . . . . . . . . . . 63

12.2.1 Informal but flawed descriptions of a surface . . . . . . . . . . . . . . 6312.2.2 2-planes in 3-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6412.2.3 Local descriptions of a surface . . . . . . . . . . . . . . . . . . . . . . 6512.2.4 Equivalence of definitions . . . . . . . . . . . . . . . . . . . . . . . . 6612.2.5 Coordinate charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.2.6 Tangent space at a point on a surface . . . . . . . . . . . . . . . . . . 67

13 Surfaces in Euclidean 3-space 6913.1 First fundamental form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.2 Change of notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7013.3 Gauss map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.4 Geometry of a curve on a surface . . . . . . . . . . . . . . . . . . . . . . . . 7213.5 Second fundamental form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 CONTENTS

A Notation 79A.1 Cartesian spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79A.2 Abstract vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79A.3 Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80A.4 Abstract affine space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

B Normal forms of maps 83B.1 Normal form of a linear map . . . . . . . . . . . . . . . . . . . . . . . . . . . 83B.2 Characterization of a linear subspace . . . . . . . . . . . . . . . . . . . . . . 83B.3 Linear subspace as a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84B.4 Topology of the space of linear maps . . . . . . . . . . . . . . . . . . . . . . 86B.5 Local versus global behavior of a 1-dimensional nonlinear map . . . . . . . . 87B.6 Normal form of C1 maps with maximal rank . . . . . . . . . . . . . . . . . . 87B.7 Contraction map lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88B.8 Inverse function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89B.9 Normal forms of nonlinear maps . . . . . . . . . . . . . . . . . . . . . . . . . 90

C The differential of a function 91C.1 The dual vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91C.2 Differentiability of a function . . . . . . . . . . . . . . . . . . . . . . . . . . 92C.3 Directional derivative of a function . . . . . . . . . . . . . . . . . . . . . . . 92C.4 C1 functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92C.5 Differential 1-form on an open subset of Am . . . . . . . . . . . . . . . . . . 92C.6 Line integral of a 1-form along a curve in Am . . . . . . . . . . . . . . . . . 93C.7 Fundamental example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

D Differential of a map 95D.1 Characterization of a linear map . . . . . . . . . . . . . . . . . . . . . . . . . 95D.2 Norm of a linear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95D.3 Differential of a map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95D.4 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

E Topology 97E.1 Topology of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

E.1.1 Dot product and norm of a vector . . . . . . . . . . . . . . . . . . . 97E.1.2 Distance between two points . . . . . . . . . . . . . . . . . . . . . . . 97E.1.3 Open and closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 98E.1.4 Convergent and Cauchy sequences . . . . . . . . . . . . . . . . . . . 98E.1.5 Compact sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

E.2 Topology of affine space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

F Maps 101F.1 Maps on Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

F.1.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101F.1.2 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

CONTENTS 7

F.1.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102F.2 Maps on affine space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102F.3 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103F.4 Partition of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

8 CONTENTS

Chapter 1

Introduction

The goal of these notes is to provide an introduction to differential geometry, first by studyinggeometric properties of curves and surfaces in Euclidean 3-space. Guided by what we learnthere, we develop the modern abstract theory of differential geometry.

The approach taken here is radically different from previous approaches. Instead ofworking initially with Rn exclusively and introducing the modern abstract perspective later,I have chosen to introduce and use abstraction from the start. The philosophy here is to useabstraction whenever possible, and switch back to using coordinates on Rn only as needed.

Why do it this way, especially since abstraction can be difficult to understand? First,I believe that, although coordinates are at first sight easier to understand, they interferewith developing a clear conceptual understanding of the ideas. My view is that the abstractapproach reflects more closely the geometric intuition driving the subject. Second, althoughcoordinates are essential for proving some of the fundamental technical theorems, such asthe implicit function theorem, most of the lemmas needed for laying the foundations of thesubject are more easily proved using the abstract framework than using coordinates.

Last, the abstract perspective of mathematics is extremely useful, if not essential, forlearning and doing advanced mathematics, even applied mathematics. Since it is so difficultto learn, it requires for most of us several attempts before we get it. The sooner you startstruggling with it, the better.

Warning: All functions and maps are assumed to be continuous, C1, or C2 (we’ll discusswhat this means later), depending on the context. Many of the results presented do not holdif the functions or maps are assumed to be only differentiable.

9

10 CHAPTER 1. INTRODUCTION

Chapter 2

Flat geometry

Flat plane geometry was developed by Euclid using an axiomatic approach. He formulated aset of definitions and axioms (assumptions) and derived geometric theorems from them usingonly deductive logic.[2] Later, Descartes observed that Euclidean plane can be representedusing coordinates, and the theorems of Euclidean geometry proved using basic algebra. Sometime after that, it was recognized that Descartes’ approach can be done using abstractalgebra.

But what is geometry? What is a geometric space, and when is a statement about thespace called geometric? An initial answer comes from physics. When we study physics, wehave to make measurements, which in turn require using units. For example, for length wecan use inches, centimeters, or other units. However, we believe that the laws of physicsshould not depend on the units used. This means that a law of physics should hold, nomatter what units are used. Ideally, one should be able to state the laws of physics withoutusing any units at all. This i fact is always possible by expressing them using unitless ratios.

In particular, in Newtonian physics it is assumed that empty space satisfies the propertiesof Euclidean 3-space. Therefore, any law of physics has to be logically consistent with theaxioms and theorems of Euclidean geometry. On the other hand, spatial measurementsrequire units, which in higher dimensions, includes a way to measure directions. The mostconvenient way to do this is to use Cartesian coordinates. However, the statements of physicallaws and properties should not depend on where the origin is placed and what directions thecoordinate axes point in.

Euclidean geometry can therefore be defined in one of two equivalent ways. It is the studyof rigorous logical consequences of the Euclidean axioms. Or it is the study of theorems aboutEuclidean space, where the theorems and proofs might be stated using Cartesian coordinatesbut remain valid if the coordinates are changed by either shifting the origin to a differentpoint in space or rotating the coordinate axes.

After the introduction of Cartesian coordinates, Euclid’s original axiomatic approach hasbeen rarely used, because the proofs are often quite subtle and difficult to find. On the otherhand, although proofs using Cartesian coordinates are usually more straightforward, onemust also verify that the theorem proved remains valid under a change of coordinates. Themodern abstract algebraic approach allows theorems to be stated without any reference tocoordinates and sometimes short straightforward proofs.

In this chapter we briefly review the three different approaches. More details can be

11

12 CHAPTER 2. FLAT GEOMETRY

found in the book of Stillwell [1].

2.1 Axiomatic geometryThere are several different formulations of the axioms for Euclidean plane geometry. In all ofthem one starts with points, lines, and circles. Euclid himself first defined what are known asstraightedge and compass constructions and then additional axioms. Others who formulatedEuclidean geometry in terms of independent aioms include Hilbert, Birkhoff, and Tarski.Prsented here are Hilbert’s axioms.

2.1.1 Axioms of the lineWe begin with axioms that specify the properties of a line and points on the line. Note thatthese axioms never refer to an origin, the length of a line segment, or numbers at all.

Order.

Concept of order. Given any three distinct points lying on a line, exactly one lies betweenthe other two.

Existence of betweenness. Given two distinct points on a line, there exists a point on theline between them.

Linear congruence.

Definition of oriented line segment. Given any two points A and B, the oriented line seg-ment AB is defined to be the set containing A, B, and all points between A and B, alongwith the designation that A is the start and B is the end of AB. In particular, if A 6= B,then AB 6= BA.

• Definition of ray. Given two distinct points A and B, the ray −→AB is defined to be theunion of overlineAB and the set of all points for which B lies between A and the point inquestion.

• Directions of rays. Two rays point in the same direction, if one contains the other.

• Directions of oriented line segments. Two oriented line segments AB and CD point inthe same direction, if the rays −→AB and −−→CD point in the same direction

Congruence of oriented line segments. There is an equivalence relation called congruence onthe set of oriented line segments and denoted by ∼=.

Existence of congruent line segments. Given a line segment AB and a point C, there existsa unique point D such that AB ∼= CD.

Concatenation of line segments. Given line segments AB and CD, let AB ∨CD denote theline segment AE, where BE ∼= CD.

2.1. AXIOMATIC GEOMETRY 13

Multiple of a line segment. Given a line segment AB, let 1AB = AB. If n ∈ Z+, let

(n+ 1)AB = nAB ∨ AB.

Additivity of congruence. Given points A,B,C,D,E, F,G,H,

AB ∼= EF and CD ∼= GH =⇒ AB ∨ CD ∼= EF ∨GH.

Topology of a line

Relative length. A line segment AB is longer than a line segment CD, if D lies between C

and E, where CE is the line segment in the ray −−→CD that is congruent to AB.

Archimedean axiom. For any line segments AB and CD, there exists n ∈ Z+ such that nABis longer than CD.

Completeness. Let S, T be nonempty subsets of a line ℓ, such that S ∪ T = ℓ and no pointin one subset lies between two points in the other. Then there exists a unique point P suchthat for any points A ∈ S\{P} and B ∈ T\{P}, P lies between A and B.

2.1.2 Axiomatic plane geometryIncidence.

Unique line containing two points. For any two distinct points A and B, there exists aunique line ←→AB passing though them.

Line has at least two points. Every line contains at least two distinct points.

Definition of collinearity. A set of points is collinear, if it is contained in a line.

Dimension is greater than 1. There exist three points that are not collinear.

Parallel axiom. For each line ℓ and point P not on ℓ, there exists a unique line through Pthat does not intersect ℓ.

Pasch’s axiom. Suppose the points A, B, C are not collinear and ℓ is a line that does notpass through any of these points. If ℓ contains a point between A and B, then it also containseither a point between A and C or a point between B and C, but not both.

Angles.

Concept of angle. Any two distinct rays −→BA and −−→BC defines an angle ∠ABC.

Concept of congruence. There is an equivalence relation called congruence on the set of an-gles and denoted ∼=.

Existence of congruent angles. Given any angle ∠ABC and a ray −−→ED, there exists a uniqueray −→EF such that ∠DEF ∼= ∠ABC.


Definition of triangle. A triangle consists of non-collinear points A, B, C and is denoted by4ABC.

Congruence of triangles. Two triangles4ABC and4DEF are congruent,4ABC ∼= 4DEF ,if

AB ∼= DE, BC ∼= EF, CA ∼= FD,

and∠ABC ∼= ∠DEF, ∠BCA ∼= ∠EFD, ∠CAB ∼= ∠FDE.

SAS. If the triangles 4ABC and 4DEF satisfy

AB ∼= DE, BC ∼= EF , and ∠ABC ∼= ∠DEF,

then they are congruent.

Circles

Definition of a circle. A circle centered at A is a set of points such that for any two pointsB and C lying in the circle, AB ∼= AC.

Interior of a circle. A point C lies inside a circle centered at A, if there exists a point B inthe circle such that C lies between A and B. A point lies outside a circle if it does not lieon or inside the circle.

Intersection of circles. If one circle contains both points inside and outside another circle,then the two circles intersect.

2.1.3 ConsequencesLength and angles

Areas of triangles and parallelograms

Pythagorean theorem

Arithmetic from geometry

2.2 Flat geometry using coordinates2.2.1 Cartesian lineGiven a line ℓ, choose a point O ∈ ℓ, call it the origin, and label it by 0. Choose anotherpoint E 6= O on the line and label it by 1. There exists a unique point F 6= O such thatOE ∼= EF . Label F by 2. Continuing by induction, each nonnegative integer labels a uniquepoint.

There exists a unique point E ′ 6= E such that E ′O ∼= OE. Label it by −1. Continuingby induction, each integer labels a unique point.

2.2. FLAT GEOMETRY USING COORDINATES 15

Each unit segment can be bisected. Each resulting smaller segment can also be bisected.Continuing by induction, each fraction whose denominator is a power of 2 labels a uniquepoint.

By the completeness axiom, every real number labels a unique point, and every point islabeled by a unique real number. we call this the real line and denote it by R.

Given any two points a, b ∈ R, the line segment ab is congruent to a line segment 0c,where c > 0, if and only if c = |b− a|. We can therefore define the length of a line segmentab to be |b− a| and the distance between a and b to be |b− a|.

2.2.2 Cartesian planeThe Cartesian plane can be defined from the Euclidean plane as follows. Start with a pointO on the plane. Choose a line passing through O and call it the x-axis. Choose a secondline orthogonal to the x-axis and call it the y-axis. Choose a point E1 6= O on the x-axis anda point E2 6= O on the y-axis such that OE1

∼= OE2. By the construction of the Cartesianline, each axis is a real line, where E1 and E2 are labeled by 1.

For each point P in the plane, it follows by the parallel axiom that there exists a uniqueline passing through P and parallel to the y-axis. That line must intersect the x-axis. Leta denote the real number labeling the point of intersection. Similarly, there is a unique linethrough P that is parallel to the x-axis. Let b label the intersection of that line with they-axis. Label P by the ordered pair (a, b). Let R2 denote the set of all possible ordered pairs(a, b), where a, b ∈ R.

Conversely, given an ordered pair (a, b), there exist a unique line parallel to the y-axisand passing through a on the x-axis and a second unique line parallel to the x-axis andpassing through b on the y-axis. These two lines must intersect, and the point of intersectionis the point labeled by (a, b).

2.2.3 Length and distanceGiven points P1 = (a1, b1) and P2 = (a2, b2), there exists a unique c ≥ 0 such that P1P2

∼=0c. We define the length of P1P2 and the distance between P1 and P2 to be c. If we letQ = (a1, b2), then 4P1QP2 is a right triangle. By the Pythagorean theorem,

(ℓ(P1P2))2 = (ℓ(P1Q))

2 + (ℓ(QP2))2.

This is equivalent to(d(P1, P2))

2 = (a2 − a1)2 + (b2 − b1)2.

2.2.4 Angle via dot productRecall that, if P0 = (a0, b0), P1 = (a1, b1), and P2 = (a2, b2), then

(P1 − P0) · (P2 − P0) = (a1 − a0)(a2 − a0) + (b1 − b0)(b2 − b0).

By the law of cosines,(P1 − P0) · (P2 − P0) = d1d2 cos θ,

https://en.wikipedia.org/wiki/Law_of_cosines


, where

d1 =√

(a1 − a0)2 + (b1 − b0)2

d2 =√

(a2 − a0)2 + (b2 − b0)2

θ = ∠P1P0P2.

2.3 Abstract vector space with an inner productLet Vm be an abstract m-dimensional vector space. A function

V × V → R(v1, v2) 7→ 〈v1, v2〉

is called an inner product, if the following hold for any v, v1, v2 ∈ V and c ∈ R:

〈v2, v1〉 = 〈v1, v2〉 (symmetry)〈v, v1 + v2〉 = 〈v, v1〉+ 〈v, v2〉〈v1, cv2〉 = c〈v1, v2〉

}(bilinearity)

〈v, v〉 ≥ 0

〈v, v〉 = 0 ⇐⇒ v = 0

}(positive definiteness).

We can then define the length of a vector to be

|v| =√〈v, v〉

and the angle θ between two nonzero vectors v1 and v2 by

cos θ =〈v1, v2〉|v1||v2|

The basic example of an abstract m-dimensional vector space is Rm, and the basic exampleof an inner product is the dot product,

(v11, . . . , vm2 ) · (v12, . . . , vm2 ) = v11v

12 + · · ·+ vm1 v

m2

We will see that this is essentially the only possible example. Nevertheless, it is often usefulto forget about the Cartesian coordinates and use only the abstract definitions.

It is now straightforward to show that if you start with the Euclidean axioms, choose apoint and call it the origin, then you can define vector addition and scalar multiplication onthe Euclidean space, where the Euclidean axioms imply the properties listed above and thedefinitions of length and distance above are consistent with the definitions implied by theaxioms.

Conversely, it is also straightforward to show that the properties of a vector space withan inner product imply the Euclidean axioms.

2.4. EUCLIDEAN GEOMETRY ON AN AFFINE SPACE 17

2.4 Euclidean geometry on an affine spaceThe vector space approach to Euclidean geometry requires choosing a special point in spaceand designating it as the origin. This can be avoided using affine geometry. In this approach,there is a space of points (which are not the same as vectors), called affine space, and anassociated vector space, which we will call the tangent space, with an inner product. Adetailed presentation is given in Chapter 4. Again, it can be shown that the Euclideanaxioms are equivalent to the definition of an affine space with an inner product on its tangentspace.

Chapter 3

Vector Spaces and Linear Maps

This is a review of concepts and definitions that you should have already learned in linearalgebra.

3.1 Definition of a vector spaceA vector space can be defined with respect to any field F such as the rationals Q, reals R, orcomplex numbers C. We, however, will restrict our attention to vector spaces with respectto the reals.

Definition 3.1. A vector space over R that has the operations of addition,

v1, v2 ∈ V 7→ v1 + v2 ∈ V,

and scalar multiplicationr ∈ R, v ∈ V 7→ rv ∈ V,

which satisfy the following properties:

(v1 + v2) + v3 = v1 + (v2 + v3), ∀v1, v2, v3 ∈ V (3.1)v1 + v2 = v2 + v1, ∀v1, v2 ∈ V (3.2)

∀ v ∈ V, ∃ −v ∈ V such that v + (−v) = 0 (3.3)(r1r2)v = r1(r2v), ∀r1, r2 ∈ R, v ∈ V (3.4)

r(v1 + v2) = rv1 + rv2, ∀r ∈ R, v1, v2 ∈ V (3.5)∃ 0 ∈ V such that 0 = 0v, ∀v ∈ V (3.6)

1v = v, ∀v ∈ V. (3.7)

If we denote −v = (−1)v, then, by the properties above,

v + (−v) = 1v + (−1)v = (1 + (−1))v = 0v = 0.

For convenience we will denote 0 by simply 0. It will always be clear, from the context,whether 0 represents the scalar 0 or the vector 0.

19

20 CHAPTER 3. VECTOR SPACES AND LINEAR MAPS

3.2 Examples of vector spacesThe most basic example of a vector space is Rm. However, depending on the context, Rm

can play different roles, sometimes as a vector space, sometimes as an affine space (definedlater), and sometimes as a dual vector space. It is useful to distinguish clearly between thesedifferent types of Rm. We therefore shall use the following notation.

Let−→Rm = {v = 〈v1, . . . , vm〉, where v1, . . . , vm ∈ R}.

We shall also sometimes write a vector v as a column vector,

v = 〈v1, . . . , vm〉 =

v1

...vm

.Other examples of vector spaces are

1. The set of all solutions to a system of homogeneous linear equations.

2. Polynomials with coefficients in R.

3. Continuous functions f : R→ R

3.3 Linearly independent vectorsDefinition 3.2. A finite set of vectors, {v1, . . . , vk} ⊂ V is linearly independent, if, for anyscalars a1, . . . , ak ∈ R,

a1v1 + · · ·+ akvk = 0 =⇒ a1 = · · · = ak = 0.

Definition 3.3. Given a subset S ⊂ V, the span of S in V is defined to be

[S] = {a1v1 + · · · akvk : ∀a1, . . . , ak ∈ R, v1, . . . , vk ∈ S, k > 0}.

Note that the set S need not be finite.

Lemma 3.4. Any nonempty finite set S ⊂ V, where S 6= {0}, contains a linearly independentsubset T such that [T ] = [S].

Lemma 3.5. If S = {v1, . . . , vk} ⊂ V is linearly independent, then, for any v ∈ [S], thereexists a unique set of scalars a1, . . . , ak ∈ R such that

v = a1v1 + · · · akvk.

3.4. BASIS AND DIMENSION OF A VECTOR SPACE 21

3.4 Basis and dimension of a vector spaceDefinition 3.6. A vector space V is finite dimensional, if there exists a finite set S ={v1, . . . , vN} ⊂ V such that [S] = V .

Definition 3.7. An ordered list of vectors, E = (e1, . . . , em), is a basis of V, if the vectorsare linearly independent and span V, i.e., [E] = V.

Lemma 3.8. Any finite dimensional vector space has at least one basis.

Lemma 3.9. If (e1, . . . , em) and (f1, . . . , fn) are both bases of V, then m = n.

Proof. Since e1, . . . , em is a basis, there is a unique matrix A such that, for each 1 ≤ i ≤ n,

fi = A1i e1 + · · ·+ Am

i em.

On the other hand, since f1, . . . , fn is a basis, there exists a unique matrix B such that, foreach 1 ≤ p ≤ m,

ep = B1pf1 + · · ·+Bn

p fn.

Therefore, for each 1 ≤ p ≤ m,

ep =n∑

i=1

Bipfi

=n∑

i=1

Bip

m∑q=1

Aqi eq

=m∑q=1

(n∑

i=1

BipA

qi

)eq

It follows thatn∑

i=1

BipA

qi =

{1 if p = q

0 if p 6= q,(3.8)

The same calculation above with the two bases switched givesm∑p=1

ApiB

jp =

{1 if i = j

0 if i 6= j.(3.9)

Therefore, first by (3.8) and then by (3.9),

m =m∑p=1

n∑i=1

BipA

pi =

n∑i=1

m∑p=1

ApiB

ip = n.

Definition 3.10. The dimension of a vector space V is m, if it has a basis with m elements.The vector space V = {0} is said to be 0-dimensional.


Lemma 3.11. If E = (e1, . . . , em) is a basis of Vm, then, for each v ∈ Vm, there is a uniqueA = 〈a1, . . . , am〉 ∈

−→Rm such that

v = eiai.

It is convenient here to write everything using formal matrix notation. First, we view abasis as a row matrix of vectors (which is why the indices are subscripts):

E = (e1, . . . , em) =[e1 e2 · · · em

],

and an element A ∈ −→Rm as a column matrix (which is why the indices are superscripts):

A =

a1

...am

.The vector v can now be written as

v = eiai = EA.

This notation will simplify many of the formulas later.

3.5 Linear mapsA linear map from a vector space V to a vector space W (which could be V itself) thatpreserves addition and scaling. In particular, it is a map denoted

L : V→Wv 7→ L(v),

where, for any v1, v2, v ∈ V and a ∈ R,L(v1 + v2) = Lv1 + Lv2

L(av) = a(Lv).

Example. Let V =−→Rm, W =

−→R n, and L is defined using an n-by-m matrix A,

L :−→Rm →

−→R n

v = 〈v1, . . . , vm〉 =

v1

...vm

7→ Av =

A1i v

i

...An

i vi

= 〈A1i v

i, . . . , Ani v

i〉.

Example. Let P be the set of polynomials with coefficients in R. Differentiation defines alinear map L : P → P , where, for each p ∈ P ,

L(p) = p′.

Definition 3.12. A linear map L : V→W is injective or 1-1, if, for any v1, v2 ∈ V,Lv1 = Lv2 ⇐⇒ v1 = v2.

It is surjective or onto, if, for any w ∈ W, there exists v ∈ V such that Lv = w. It is anisomorphism if it is both injective and surjective.

3.5. LINEAR MAPS 23

3.5.1 Isomorphism of an m-dimensional vector space with Rm

Lemma 3.13. Given a basis E = (e1, . . . , em) of Vm, the map

IE :−→Rm → Vm

〈v1, . . . , vm〉 7→ v1e1 + · · ·+ vmem,

is a linear isomorphism.

3.5.2 The space of linear mapsLet Hom(Vm,Wn) denote the space of all linear maps from Vm to Wn. We can define thesum of two linear maps L1, L2 ∈ Hom(Vm,Wn) to be M : Vm →Wn, where

M(v) = L1(v) + L2(v), ∀ v ∈ Vm.

It is straightforward to verify that M is linear and therefore an element of Hom(Vm,Wn).We shall denote it by L1 + L2. Similar, the scalar multiple of L ∈ Hom(Vm,Wn) by r ∈ Ris the map

(rL)(v) = r(L(v)), ∀ v ∈ Vm.

It is straightforward to show that using these definitions, Hom(V m,Wn) is a vector space.Let M(n,m) = Hom(Rn,Rm) denote the set of n-by-m matrices.Given a basis E = (e1, . . . , em) of Vm and a basis F = (f1, . . . , fn) of Wn, then for any

linear map L : Vm →Wn, there is a sequence of maps

−→Rm IE−→ Vm L−→Wn I−1

F−−→−→R n.

This defines a map

IE,F : Hom(Vm,Wn) 'M(n,m)

L 7→ I−1F ◦ L ◦ IE

Lemma 3.14. IE,F is a linear isomorphism, and therefore Hom(Vm,Wn) is an mn-dimensionalvector space.

3.5.3 Subspaces and quotient spacesDefinition 3.15. A set S ⊂ V is a subspace of V, if it is closed under addition and scalar mul-tiplication. As a consequence, using the same addition and scalar multiplication operationsas V, the subspace S is itself a vector space.

Example. The setS = {(x, y) : x+ y = 0} ⊂

−→R 2

is a subspace.


Example. The setT = {(x, y) : x+ y = 1} ⊂

−→R 2

is not a subspace. However, you can define different addition and scalar operations for whichS is a vector space, namely

(x1, y1) + (x2, y2) = (1, 0) + [(x1, y1)− (1, 0)] + [(x2, y2)− (1, 0)]

= (x1 + x2 − 1, y1 + y2)

a · (x, y) = (1, 0) + a[(x, y)− (1, 0)]

= (1 + a(x− 1), ay).

If S ⊂ V is a linear subspace and v ∈ V, let

v + S = {v + s : s ∈ S}.

If the subspace S is clearly indicated, then we can also write [v] = v + S. Note that

[v] = [w] ⇐⇒ w − v ∈ S.

Definition 3.16. LetV/S = {v + S : v ∈ V}.

There is a natural short exact sequence (see below)

0→ S → V→ V/S → 0 (3.10)

Definition 3.17. A chain of two maps

Af−→ B

g−→ C

is exact, if im f = ker g. A longer chain of maps is exact, if any two consecutive maps areexact.

The exactness of the sequence (3.10) means that the inclusion map S ⊂ V is injective,the kernel of the map V→ V/S is S, and the map V→ V/S is surjective.

Lemma 3.18. Given a subspace S ⊂ V,

dimV = dimS + dimV/S.

3.5.4 Kernel and image of a linear mapDefinition 3.19. Given a linear map L : V→W, the kernel, image, and cokernel of L aredefined to be

kerL = {v ∈ V : L(v) = 0} ⊂ VimL = {L(v) : v ∈ V} ⊂W

cokerL = W/ imL.

3.6. NORMAL FORMS OF A LINEAR TRANSFORMATION 25

Lemma 3.20.

kerL is a subspace of VimL is a subspace of WimL ' V/ kerL

dimkerL+ dim imL = dimVdim imL+ dim cokerL = dimW.

Definition 3.21. The rank of L is defined to be

rankL = dim imL.

Therefore, the identities above can also be written as

dimkerL+ rankL = dimVrankL+ dim cokerL = dimW.

3.6 Normal forms of a linear transformationLemma 3.22. Let L : V → W be a linear map, and F = (f1, . . . , fn) a basis of Wn.There exists a basis E = (e1, . . . , em) and a reordering of the basis F such that the matrixM = IE,F (L) satisfies the following:

M ij =

{δij if 1 ≤ i, j ≤ k

0 if k + 1 ≤ i ≤ n and 1 ≤ j ≤ m,(3.11)

where k = rankL.

Proof. There exists a reordering of F such that ([fk+1], . . . , [fn]) is a basis of cokerL. Foreach 1 ≤ i ≤ k, there exists Aµ

i , k+1 ≤ µ ≤ n, such that the vectors fi+Aµi fµ are a basis of

imL. Therefore, there exist e1, . . . , ek such that L(ei) = fi + Aµi fµ. Let (ek+1, . . . , em) be a

basis of kerL. Note that (e1, . . . , em) is a basis of V. The matrix M = IE,F (L) now satisfies(3.11)

Corollary 3.23. For any n-by-m matrix M , there exists an invertible m-by-m matrix Qsuch that MQ is in row-echelong form.

Lemma 3.24. For any linear map L : Vm → Wn, there exist bases E = (e1, . . . , em) of Vand F = (f1, . . . , fn) of W such that the entries of the matrix M = IE,F (L) = I−1

F ◦ L ◦ IEare

M ij =

{δij , if 1 ≤ i, j ≤ rankM

0 = , otherwise.


3.7 Change of basisLet E and F be bases of a vector space Vm. Therefore, there exists a matrix

R =

R11 · · · R1m... ...

Rm1 · · · Rmm

such that, for each 1 ≤ j ≤ m,

fj = eiRij.

Equivalently, if E and F are treated as row matrices of vectors, then

F = ER.

Using this notation, it is easy to derive the change of basis formulas for a vector. For eachv ∈ Vm, there exist coefficients A,B ∈ −→Rm, viewed as column vectors, such that

v = EA = FB = FRB.

Since the coefficient matrices A and B are uniquely determined by the bases and the vectorv, it follows that

A = RB.

Chapter 4

Affine Spaces and Maps

4.1 1-dimensional spaces4.1.1 The real lineThe real line is the 1-dimensional vector space but labeled by real numbers with respect toa chosen set of units.

−4 −3 −1 0 2 3 4q p•

4.1.2 Abstract 1-dimensional vector spaceAn abstract 1-dimensional vector space L can be thought of as the real line with all of thenumbers, except 0, erased. It is, in particular, the set of all oriented line segments in aEuclidean line that start at zero. The concepts of congruence and rescaling of vectors canstill be defined geometrically. They can in turn be used to define concepts of vector additionand rescaling of a vector that satisfy properties listed in Definition 3.1.

0

A vector v looks like this in L:

0•

v

4.1.3 The affine lineThe affine line A can be viewed as an abstract 1-dimensional vector space with the originerased. More precisely, it is the Euclidean line that satisfies the axioms in Subection 2.1.1.Associated with each point p ∈ A is the vector space of all oriented line segments that startat p. We shall denote this vector space by Lp.

27

28 CHAPTER 4. AFFINE SPACES AND MAPS

Given two points p, q ∈ A, there is a vector v ∈ Lp from p to q

•q p

We denote v = q − p ∈ Lp. Given any two points p, q ∈ A, there is a natural linearisomorphism Ipq : Lp → Lq, where, given any vector (i.e., oriented line segment) v ∈ Lp,Ipq(v) ∈ Lq is the unique vector (i.e., oriented line segment) that is congruent to v. SinceIpr = Ipq ◦ Iqr and Ipp is the identity map, there is an abstract vector space L and linearisomorphisms Ip : L→ Lp such that, for any p, q ∈ A, Ipq = Iq ◦ I−1

p .

4.2 2-dimensional spaces

4.2.1 R2

−4 −2 2 4

−4

−2

2

4

•A

•B

•C

4.2.2 2-dimensional vector space

B − A

C − A = (B − A) + (C −B)

C −B

0

4.3. AFFINE SPACES 29

4.2.3 2-dimensional affine space

•A

•B

•C

4.3 Affine spaces4.3.1 Definition

•p

•q = p+ v

v = q − p

Definition 4.1. An affine space associated with an m-dimensional vector space Vm is a setAm with operations denoted, for any p, q ∈ Am and v ∈ Vm,

q − p ∈ Vm

p+ v ∈ A,

which satisfy the following properties:q − p = 0 ⇐⇒ q = p

p+ v = q ⇐⇒ v = q − pp+ (v1 + v2) = (p+ v1) + v2.

We will often call Vm the tangent space of Am. As we will see, it can also be viewed asthe set of all possible velocity vectors of parameterized curves in Am.

In particular, given any p ∈ Am, the mapIp : Vm → Am

v 7→ p+ v

is bijective, and its inverse map isI−1p : Am → Vm

q 7→ q − p.

For each p, q ∈ Am, the vector Ip(q) is called the position vector of q relative to p.Definition 4.2. The dimension of an affine space Am is defined to be the dimension of thevector space associated to it.


4.3.2 ExamplesThe fundamental example of an affine space is Cartesian space Rm. To distinguish betweenvectors and points in Rm, we will use the following notation for the Cartesian space of points:

Rm = {p = (p1, . . . , pm) : p1, . . . , pm ∈ R}.

Since this is an affine space, the rules of algebra on Rm are different from the usual onesfor Rm. They are instead the following: You cannot add two points p1, p2 ∈ Rm, and youcannot rescale a point by a factor c ∈ R. You can, however, do the following: Given

p = (p1, . . . , pm) ∈ Rm

p1 = (p11, . . . , pm1 ) ∈ Rm

p2 = (p12, . . . , pm2 ) ∈ Rm

v = 〈v1, . . . , vm〉 ∈−→Rm

the following algebraic operations are allowed:

p2 − p1 = 〈p12 − p11, . . . , pm2 − pm1 〉 ∈−→Rm

p+ v = (p1 + v1, . . . , pm + vm) ∈ Rm.

With this notation, Rm is an affine space with tangent space −→Rm.A similar example is a vector space Vm with itself as its tangent space.Another example is

Hm = {xm+1 = 1} ⊂ Rm+1,

with tangent spaceT = {xm+1 = 0} ⊂

−→Rm+1.

More generally, given a1, . . . , am+1, b ∈ R,

Am = {a1x1 + · · · am+1xm+1 = b} ⊂ Rm+1.

is an affine space with tangent space

Vm = {a1v1 + · · · am+1vm+1 = 0} ⊂

−→Rm+1.

The abstract version of this is the following: Given an abstract vector space Wm+1 and anonzero linear function ℓ : Wm+1 → R, the space

Am = {w ∈Wm+1 : ℓ(w) = 1}

is affine with tangent space

Vm = {v ∈Wm+1 : ℓ(v) = 0}.

4.4. AFFINE MAPS 31

4.4 Affine maps4.4.1 Affine maps of Cartesian spacesA map F : Rm → Rn is called affine, if there is a n-by-m matrix L :

−→Rm →

−→R n and b ∈ Rn

such that, for any p ∈ Rm,F (p) = L(p− 0) + b.

Note that, for any v ∈ −→Rm,L(v) = F (0 + v)− F (0).

More generally, for any p ∈ Rm and v in −→Rm,

L(v) = F (p+ v)− F (p)

Equivalently, for any p, q ∈ Rm,

F (q) = F (p) + L(q − p).

In particular, if m = n = 1, then M : R→ R looks like

F (x) = ax+ b, ∀ x ∈ R,

where a and b are constants.

4.4.2 Abstract affine mapsDefinition 4.3. Let Am and Bn be affine spaces with corresponding tangent spaces Vm andWn. A map M : Am → Bn is affine, if there is a linear map L : Vm →Wn such that, for anyp, q ∈ Am,

M(q) =M(p) + L(q − p) (4.1)

This implies that, given p ∈ Am, any affine map M : Am → Bn is uniquely determined bythe linear map L : Vm →Wn and M(p) ∈Wn and vice versa.

A affine map is a affine isomorphism, if it is both injective and surjective. An affine mapM : Am → R is called an affine function.

4.4.3 Affine independence and basisDefinition 4.4. A set of points p0, . . . , pk ∈ Am is in general position or, equivalently, areaffinely independent, if the points p1 − p0, . . . , pk − p0 ∈ Vm are linearly independent.

Lemma 4.5. A set of points p0, . . . , pk ∈ Am are in general position if and only if, for each0 ≤ i ≤ k, the set

{p0 − pi, . . . , pk − pi} ⊂ Vm

consists of the zero vector and a linearly independent subset of Vm.


Definition 4.6. An ordered list of points, (p0, . . . , pm), is an affine basis of Am, if the vectors(p1 − p0, . . . , pm − p0) is a basis of Vm.

Lemma 4.7. Given an affine basis P = (p0, . . . , pn) ⊂ Am, the map

IP : Rm → Am

(a1, . . . , am) 7→ p0 + a1(p1 − p0) + · · ·+ am(pm − p0).

is an affine isomorphism.

Lemma 4.8. Given an affine isomorphism I : Am → Bm, an ordered set (p0, . . . , pm) is anaffine basis of Am if and only if (I(p0), . . . , I(pm)) is an affine basis of Bm.

Example. An affine basis of R3 is given by the points

(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1).

4.4.4 The space of affine mapsLet Homaff(Am,Bn) denote the space of all affine maps from Am to Bn.

Lemma 4.9. Homaff(Am,Bn) is an affine space with tangent space Homaff(Vm,Wn).

Equation (4.1) implies that the map

Ip : Hom(Vm,Wn)× Bn → Homaff(Am,Bn)

(L, b) 7→M,

where, for each q ∈ Am,M(q) = L(q − p) + b,

is an bijection.Given affine bases P = (p0, . . . , pm) of Am and Q = (q0, . . . , qn) of Bn, let

IP,Q : Homaff(Rm,Rn)→ Hom(Am,Bn),

denote the map, where for each M ∈ Hom(Rm,Rn),

IP,Q(M) = IQ ◦M ◦ I−1P .

Lemma 4.10. The map IP,Q is an affine isomorphism.

Lemma 4.11. dimHom(Am,Bn) = (m+ 1)n.

Chapter 5

Concrete Category Theory

The concept of a category is implicit in much of the mathematics you have already learned.The idea is that, within a specific area such as linear algebra, you often restrict your attentionto a certain class of sets with given properties, such as vector spaces, and a correspondingclass of maps, such as linear transformations.

Another example is the class of open sets in Rn for any n > 0 and the correspondingclass of smooth maps from an open set in Rn to Rm for any m,n > 0.

A functor is in a loose sense a map from one category to another. We explain moreprecisely below.

5.1 Definition of a categoryA category C consists of the following:

• A collection S of things we call objects

• A collectionM of things we call morphisms or arrows. Associated with any morphismare two objects, one we call the source and the other the target. Any morphism canbe written as f : A→ B, where F is the name of the morphism, A is the source, andB is the target.

Moreover, the following properties must hold:

• (Composition) Any two morphisms f : A → B, g : B → C (where the target of f isthe source of g) uniquely determine a morphism with source A and target C, whichwe denote by g ◦ f : A→ C.

• (Associativity) Given morphisms f : A→ B, g : B → C, and h : C → D,

(h ◦ g) ◦ f = h ◦ (g ◦ f) : A→ D.

• (Identity) For each object A, there is a morphism IA such that, for any morphismf : A→ B, f ◦ IA = f and, for any morphism g : B → A, IA ◦ g = g.

33

34 CHAPTER 5. CONCRETE CATEGORY THEORY

Here, we will restrict to the following types of categories: Objects are sets that satisfy acommon set of properties. In this situation an object is often called a space. A morphism isa map between two spaces, where the source is the domain and the target is the range, thatsatisfies a specified set of properties. The map g ◦ f is the composition of the two maps, andIA : A→ A is simply the identity map.

5.2 Examples5.2.1 SetsLet Set denote the category, where the spaces are sets and the maps are maps from a set toanother.

5.2.2 Finite dimensional real vector spacesLet Lin denote the category, where the spaces are finite dimensional real vector spaces andthe maps are linear maps betwee two vector spaces.

5.2.3 Finite dimensional affine spacesLet Aff denote the category, where the spaces are finite dimensional real affine spaces andthe maps are affine maps between two affine spaces.

5.2.4 C1 manifoldsLet Man denote the category, where the spaces are open subsets of a finite-dimensional affinespace, and the maps are C1 maps from one space to another. Later, this category will beextended to a larger collection of spaces and maps.

5.3 Definition of FunctorA covariant functor F : C1 → C2 consists of maps

FS : S1 → S2FM :M1 →M2,

such that the following hold:

• For each map f : X → Y in M1, the corresponding map is

FM(f) : FS(X)→ FS(Y ).

• Given maps f : X → Y and g : Y → Z,

FM(g ◦ f) = FM(g) ◦ FM(f)

5.4. EXAMPLES OF FUNCTORS 35

• For each space X,FM(idX) = idFS(X).

A contravariant functor F : C1 → C2 reverses the arrows. It consists of maps

FS : S1 → S2FM :M1 →M2,

such that the following hold:

• For each map f : X → Y in M1, the corresponding map is

FM(f) : FS(Y )→ FS(X).

• Given maps f : X → Y and g : Y → Z,

FM(g ◦ f) = FM(f) ◦ FM(g)

• For each space X,FM(idX) = idFS(X).

5.4 Examples of functors5.4.1 Forgetful functorsGiven any of the categories C above, there is a forgetful functor F : C → Set , since eachspace in C is a space in Set and each map in C is a map in Set .

Another forgetful functor F : Lin → Aff forgets the origin of each vector space. In otherwords, any vector space is itself an affine space and any linear map is an affine map.

There is also the forgetful functor F : Aff → Man that forgets the affine structure butremembers the topological structure of each affine space. Each affine space is itself an openspace of itself, and each map in Aff is a C1 map.

5.4.2 DualizationThere is a contravariant functor Dual : Lin → Lin, where, for each vector space V ,Dual(V ) = V ∗ and, given A : V → W , Dual(A) = At : W ∗ → V ∗.

36 CHAPTER 5. CONCRETE CATEGORY THEORY

Chapter 6

Euclidean space

The concepts of a vector space and an affine space provide an abstract algebraic formulationof the properties of points and lines in Euclidean space. In this chapter we introduce anabstract version of the dot product, which is then used to define the length of a vector andthe angle between two vectors in an abstract vector space. This in turn can be used to definethe distance between two points in an affine space and the angle defined by three points.

6.1 Cartesian space

6.1.1 Length and angle using the dot productRecall that the dot product of two vectors in −→Rm is defined to be

〈v11, . . . , vm1 〉 · 〈v12, . . . , vm2 ) = v11v12 + · · ·+ vm1 v

m2 . (6.1)

The length or magnitude of a vector is

|v| =√v · v.

Recall that, if the angle between two nonzero vectors v1 and v2 is θ, where 0 ≤ θ ≤ π, then

cos θ =v1 · v2|v1||v2|

.

In particular, the two vectors are orthogonal, if

v1 · v2 = 0,

and, if this holds,

|v1|2 + |v2|2 = |v2 − v1|2,

which is the Pythagorean theorem.

37

38 CHAPTER 6. EUCLIDEAN SPACE

The dot product also has the following fundamental properties: For each v1, v2, v3 ∈−→Rm

and a ∈ R,

v1 · v2 = v2 · v1(v1 + v2) · v3 = v1 · z + v2 · v3

(av1) · v2 = a(v1 · v2)v1 · v1 ≥ 0

v1 · v1 = 0 ⇐⇒ v1 = 0.

6.2 Inner productIt turns out that almost all geometric theorems about Euclidean space can be proved usingonly the properties of the dot product, and its definition (6.1) is rarely needed. Due to this,it turns out to be useful to define an abstract concept of the dot product, which is usuallycalled the inner product.Definition 6.1. An inner product on a vector space V is a real-valued function of twovectors, which we denote by

v1, v2 ∈ V 7→ v1 · v2 ∈ R,that satisfies satisfies the following properties for any v1, v2, v3, v ∈ V and a ∈ R:

v1 · v2〉 = v2 · v1(v1 + v2) · v3 = v1 · v3 + v2 · v3

(av1) · v2 = a(v1 · v2)v · v ≥ 0

v · v = 0 ⇐⇒ v = 0.

Given an inner product on V, we define the norm or magnitude of a vector v ∈ V to be

|v| =√v · v. (6.2)

An inner product space is a vector space with an inner product. Cartesian space Rm

with the dot product is an inner product space. There are, however, an infinite number ofdifferent possible inner products on a vector space V.Lemma 6.2. Given a basis (e1, . . . , em) of a vector space Vm, an inner product on Vm isuniquely determined by the symmetric matrix

A =

A11 . . . A1m... . . .

...Am1 . . . Amm

,where Aij = ei · ej. In particular, given vectors v = viei and w = wiei,

v · w =[v1 · · · vm

]A

w1

...wm

. (6.3)

6.3. NORM 39

Given a symmetric matrix A, (6.3) does not necessarily define an inner product. Simplecounterexamples for m = 2 include

A =

[0 00 0

]and A =

[1 00 −1

].

Definition 6.3. A symmetric m-by-m matrix A is positive definite, if (6.3) defines an innerproduct.

Lemma 6.4. A symmetric 2-by-2 matrix

A =

[a11 a12a21 a22

],

where a12 = a21, defines an inner product if and only if

a11, a22 > 0 and a11a22 − a212 > 0.

6.3 NormThe norm defined, using an inner product, by (6.2), satisfies the following basic properties:For any v, v1, v2 ∈ V,

|v| ≥ 0

|v| = 0 ⇐⇒ v = 0

|cv| = c|v||v1 + v2| ≤ |v1|+ |v2|.

There are, however, other norms that can be defined on a vector space and satisfy theproperties above. Examples of norms on −→Rm are:

|〈v1, . . . , vm〉|1 = |v1|+ · · ·+ |vm|

|〈v1, . . . , vm〉|p =

(m∑i=1

|vi|p)1/p

, where 1 ≤ p <∞

|〈v1, . . . , vm〉|∞ = max(|v1|, . . . , |vm|).

6.4 Orthogonal mapsDefinition 6.5. An orthogonal map is a linear map F : Vm → Wn, where Vm and Wn areinner product spaces, such that

F (v1) · F (v2) = v1 · v2, ∀ v1, v2 ∈ Vm.

If Wn = Vm, then F is also called an orthogonal transformation.

Let O(Vm,Wn) denote the set of all orthogonal maps from Vm to Wn and O(Vm) =O(Vm,Vm).


6.5 Orthonormal vectorsLet Vm be m-dimensional inner product space.

Definition 6.6. A set of vectors,

E = {e1, . . . , ek} ⊂ Vm

is orthonormal, if for any 1 ≤ i, j ≤ k,

ei · ej =

{1 if i = j

0 if i 6= j.

Note that if v ∈ [E], then

v =k∑

i=1

aiei,

where, for each 1 ≤ i ≤ k,ai = v · ei.

Lemma 6.7. A set of orthonormal vectors is linearly independent.

For each orthonormal set E, define

[E]⊥ = {v ∈ Vm : v · e = 0, ∀e ∈ [E]}

and the linear maps

πE : Vm → [E]

v 7→i=k∑i=1

(v · ei)ei

π⊥E : Vm → Vm

v 7→ v − πE(v).

Lemma 6.8. For each v ∈ Vm,

v = v1 + v2, where v1 ∈ [E] and v2 ∈ [E]⊥

if and only ifv1 = πE(v) and v2 = π⊥

E(v).

Lemma 6.9. A finite-dimensional inner product space has at least one orthornomal basis.

Lemma 6.10. If E = (e1, . . . , em) is an orthonormal basis of Vm, then F = (f1, . . . , fm) isalso an orthonormal basis if and only if

fj = eiMij , ∀ 1 ≤ j ≤ m,

6.6. METRIC SPACES 41

whereM i

jMik = δjk, ∀ 1 ≤ j, k ≤ m,

or, equivalently, if and only ifF = EM,

whereM tM = I. (6.4)

Let O(m) denote the set of all m-by-m matrices satisfying (6.4). It is a group undermatrix multiplication.

Lemma 6.11. Given an orthonormal basis E of an inner product space Vm, there is a grouphomomorphism

O(m)→ O(Vm)

M 7→ A,

where, for each v = eiai,

A(eiai) = ejM

ji a

i.

6.6 Metric spacesRecall that the distance between two points p1, p2 ∈ Rm is given by

d(p1, p2) = |p2 − p1| =√

(p2 − p1) · (p2 − p1).

and satisfies the following properties for any p1, p2, p3 ∈ Rm:

d(p1, p2) = d(p2, p1)

d(p1, p2) ≥ 0

d(p1, p2) = 0 ⇐⇒ p2 = p1

d(p1, p3) ≤ d(p1, p2) + d(p2, p3).

(6.5)

The leads to the following abstract definition:

Definition 6.12. A metric space is a set X and a function d : X×X → [0,∞) that satisfies,for every p1, p2, p3 ∈ X, the properties given by (6.5).

If X and Y are both metric spaces, then a map F : X → Y is called an isometry, if forany x1, x2 ∈ X,

dY (F (x1), F (x2)) = dX(x1, x2).

If A is an affine space with tangent space V and | · | is a norm on V, then A with thedistance function

d(p1, p2) = |p2 − p1|

is a metric space.


6.7 Isometries of Euclidean spaceDefinition 6.13. An m-dimensional Euclidean space is an m-dimensional affine space whosetangent space is an inner product space.

Lemma 6.14. Let Am and Bn be Euclidean spaces. A map F : Am → Bn is an isometry,if and only if there exists L ∈ O(Vm,Wn) such that the following holds: For any p ∈ Am,there exists q ∈ Bn such that, for any a ∈ Am,

F (a) = L(a− p) + q.

Definition 6.15. An isometry F : Am → Am is called a rigid motion.

Lemma 6.16. The set G of all possible rigid motions of m-dimensional Euclidean space isa group, because it satisfies the following properties:

1. If F1, F2 ∈ G, then F2 ◦ F1 ∈ G.

2. The identity map I : Am → Am is in G.

3. Any rigid motion F : Am → Am has an inverse map F−1 : Am → Am that is also arigid motion.

Chapter 7

Curves in Affine Space

7.0.1 Curves in Rm

A parameterized curve in Rm is a map of the form

c : I → Rm

t 7→ c(t) = (c1(t), . . . , cm(t)),

where I ⊂ R is an interval and each ci is a real-valued function on I.

Definition 7.1. The curve c is defined to be continuous, if each of the functions c1, . . . , cmare continuous.

The curve c is defined to be C1, if the functions c1, . . . , cm and their derivatives c1, . . . , cmare continuous.

The curve c is C2, if, in addition, the second derivatives c1, . . . , cm are continuous.

Since, given s, t ∈ I with s 6= t,

c(s)− c(t)s− t

∈−→Rm,

it follows that if c is C1, then for each t ∈ I,

c(t) = lims→t

c(s)− c(t)s− t

= lims→t

⟨c1(s)− c1(t)

s− t, . . . ,

cm(s)− c1(t)s− t

⟩= 〈c1(t), . . . , cm(t)〉 ∈

−→Rm.

Therefore, the derivative c is a map from I to −→Rm. Similarly, if c is C2, then c is alao a mapfrom I to −→Rm.

43

44 CHAPTER 7. CURVES IN AFFINE SPACE

7.0.2 Curves in affine spaceA parameterized curve in Am is a map of the form

c : I → Am,

where I ⊂ R is an interval. The curve c is defined to be continuous, if, for any p ∈ Am, thecurve I−1

p ◦ c : R → Rm is continuous. The curve c is C1 if I−1p ◦ c is C1 is C1 and C2 if

I−1p ◦ c is C2.

Lemma 7.2. Given a curve c : I → Am and pq ∈ Am, the curve I−1p ◦ c : I → Rm is

continuous if and only if I−1q ◦ c is continuous. Similar statements hold for C1 and C2.

7.1 Velocity and accelerationDefinition 7.3. The velocity of a C1 curve c : I → Am at time t0 ∈ I is defined to be thederivative of c at time t0,

v(t) = c(t) = lims→t

c(s)− c(t)s− t

∈ Vm.

Since c′(t) ∈ Vm is tangent to c(t) ∈ Am, the vector space Vm is called the tangent spaceof Am.

If c is C2, then the acceleration is the second derivative of c,

a(t) = c(t) = lims→t

c′(s)− c′(t)s− t

.

7.2 Curves in Euclidean spaceLet Em be m-dimensional Euclidean space with tangent space Em.

7.2.1 Velocity, speed, and lengthGiven a C1 curve c : [t0, t1] → Em with velocity v : I → Em, its speed is defined to beσ = |v| : I → [0,∞). The distance along the curve, i.e., the length of the curve, betweentwo points c(t0) and c(t1) is therefore

ℓ =

∫ t=t1

t=t0

σ(t) dt.

In particular, we can define the arclength function to be

[t0, t1]→ [0, ℓ]

t 7→ s(t) =

∫ τ=t

τ=t0

σ(τ) dτ.

7.2. CURVES IN EUCLIDEAN SPACE 45

If the velocity is never zero, then σ is continuous and, by the fundamental theorem of calculus,s′ = σ. Moreover, s is strictly increasing, and therefore, by the chain rule, has a C1 inversefunction t(s). The arclength parameterization of c is defined to be

c : [0, ℓ]→ Em

s 7→ c(t(s)).

Since the speed of c is 1, it is called a unit speed curve.A simple fact we will use is the following: Given any C1 function f : [t0, t1]→ R,

d

ds(f(t(s))) =

f ′(t(s))

s′(t).

7.2.2 Acceleration and curvatureThe velocity of a C2 curve c : I → Em can be written as

v(t) = σ(t)u(t), where u =v

|v|is a unit vector for each t ∈ I.

Therefore, the acceleration of c is

a = v = σ′u+ σu′.

The first term represents the acceleration of speed along the curve, while the second repre-sents the acceleration due to the change in direction of the curve.

We now use the following lemma:

Lemma 7.4. If u : I → Tm is C1 and u(t) is a unit vector for every t ∈ I, then u′(t) isorthogonal to u(t), for every t ∈ I.

Proof.0 =

d

dt(u · u) = 2u · u′.

Here, this implies that the acceleration due to change in direction is orthogonal to thetangent direction.

7.2.3 CurvatureGeometric properties of the curve should not depend on the parameterization. We thereforewant properties that do not depend on the speed or acceleration along the curve. It sufficesto study only unit speed curves. Notice that this restricts us to curves that have at leastone parameterization with nonzero velocity everywhere along the curve.

The other thing to note is that a curve is straight if u, as defined above, never changesdirection. Moreover, the greater the magnitude of u′ is, the more sharply curved the curveis. It therefore makes sense to define the curvature at each point c(t) of a unit speed C2

curve to beκ(t) = |a(t)| = |u′(t)|.


7.2.4 Frenet-Serret frame in Euclidean 2-spaceLet c : [0, T ]→ E2 be a C2 curve with nonzero velocity ·c and speed σ = | · c|. Let e1 be theunit vector pointing in the same direction as c, i.e.,

c = |c|e1 = σe1.

There exists, for each t ∈ [0, T ], a unit vector e2(t), that is a C1 function of t, such that

E(t) =[e1(t) e2(t)

](7.1)

is an orthonormal basis of E2. By Lemma 7.4, e1 · e′1 = 0. This implies that e′1(t) is a scalarmultiple of e2. It follows that there is a function κ such that, for each t ∈ [0, T ],

e′1(t) = σ(t)κ(t)e2(t).

Again, since e′2 is orthogonal to e2, it is a scalar multiple of e1. On other hand, since e1·e2 = 0,

0 =d

dt(e1 · e2) = e′1 · e2 + e1 · e′2 = σκ+ e1 · e′2,

and thereforee1 · e′2 = −σκ.

It follows thate′2 = −σκe1.

The 1-parameter family E(t) of orthonormal bases of E2 is called the Frenet-Serret frame ofc. We have shownthat it satisfies the equations[

e′1(t) e′2(t)]= σ

[κe2(t) −κe1(t)

]= σ

[e1(t) e2(t)

] [ 0 −κ(t)κ(t) 0

].

(7.2)

This is known as the Frenet-Serret equations in Euclidean 2-space.If we however, assume that c : [0, T ] → E2 is a C2 unit speed curve such that a Frenet-

Serret frame satisfying the Frenet-Serret equations(7.2), then the curvature function uniquelydetermines the shape but not the position of the curve.

Theorem 7.5. Given a continuous function κ : [0, T ] → E2, p0 ∈ E2, and e0 ∈ E2, thereexists a unique curve c : [0, T ]→ E2 and moving frame

E(t) =[e1(t) e2(t)

], 0 ≤ t ≤ T,

such that c = e1 and E satisfies (7.2).

Corollary 7.6. Given a continuous function κ : [0, T ]→ R, if a curve c1 with moving frameE1 and another curve c2 with moving frame E2 both satisfy c = e1 and (7.2), then thereexists a rigid motion R : E2 → E2 such that c2 = R ◦ c1.

7.2. CURVES IN EUCLIDEAN SPACE 47

7.2.5 Frenet-Serret frame in Euclidean 3-spaceThe following lemma will be useful:

Lemma 7.7. If v1, v2 : I → Em are C1 vector-valued functions satisfying

v1 · v2 = 0,

thenv′1 · v2 + v1 · v′2 = 0,

which is equivalent tov2 · v′1 = −v1 · v′2. (7.3)

Let c : I → E3 be a C2 curve with nonzero velocity ·c, speed σ = |c speed, and

e1 =c

|c|.

Also, assume that e1 is nonzero and let e2 be the unit vector pointing in the direction of e1.It follows that there is a nonnegative function κ such that

e1 = σκe2. (7.4)

There now exists a C1 vector-valued function e3 : I → E3 such that, for each t ∈ I,

E(t) =[e1(t) e2(t) e3(t)

]is an orthonormal basis of E3. Since e3 · e3 = 0, e3 is a linear combination of e1 and e2. ByLemma 7.7 and (7.4),

e1 · e3 = −e3 · e1 = 0.

It follows that e3 is a scalar multiple of e2. In other words, there is a function τ such that

e3 = −στe2.

From the equations above and Lemma 7.7,

e1 · e′2 = −e2 · e′1 = −κ and e3 · e′2 = −e2 · e′3 = τ,

and thereforee2 = −κe1 + τe3.

Everything above can be summarized as following: Let c : I → E3 be a C2 curve, whereits velocity v(t) and accelerations a(t) are linearly independent for each t ∈ I. Then thereexists, for each t ∈ I, an orthonormal frame E(t) such that[

e1(t) e2(t) e3(t)]= σ

[κe2(t) −κe1(t) + τe3(t) −τe1(t).

]= σ

[e1(t) e2(t) e3(t)

] 0 −κ 0κ 0 −τ0 τ 0

(7.5)

This frame is called the Frenet-Serret frame.

Chapter 8

Surfaces in affine 3-space

8.1 PreliminariesA set O ⊂ Rm is called open, if, for any p ∈ O, there is an r > 0 such that

B(p, r) ⊂ O.

Think of an open set as one that does not contain any point on its boundary. A set D ⊂ Rm

is called an open domain, if it is an open set. Given a point p ∈ Rm, an open neighborhoodof p is an open set that contains p.

49

50 CHAPTER 8. SURFACES IN AFFINE 3-SPACE

Chapter 9

Surfaces in Cartesian 3-space

9.1 Informal but flawed descriptions of a surface1. Graph: Given an open domain D ⊂ R2 and a C1 function h : D → R, let

S = {z = h(x, y)} = {(x, y, h(x, y)) : (x, y) ∈ D}.

Example:S = {z = 1√

1− x2 − y2.

2. Level set: Given an open set O ⊂ R3 and a C1 function f : O → R, let

S = {f(x, y, z) = 0} = {(x, y, z) ∈ O : f(x, y, z) = 0}.

Examples:

S = {x2 + y2 + z2 − 1 = 0}S = {x2 + y2 − z2 = 0}.

3. Parameterized surface: Given an open domain D ⊂ R2 and a C1 map Φ : D → R3,

S = Φ(D) = {(Φ(u, v) : (u, v) ∈ D} = {(x(u, v), y(u, v), z(u, v) : (u, v) ∈ D},

whereΦ(u, v) = (x(u, v), y(u, v), z(u, v)).

Example:S = {(r cos θ, r sin θ, r)}.

But these descriptions are FLAWED. And are they equivalent? How can these descrip-tions be made abstract for a surface in A3?

51

52 CHAPTER 9. SURFACES IN CARTESIAN 3-SPACE

9.2 2-planes in 3-spaceThe simplest surface in R3 is a 2-plane. It can be described as follows:

1. Graph: Given an affine function

h(x, y) = ax+ by + c,

letS = {z = h(x, y)} = {z = ax+ by + c}.

Examples:

S = {z = 0}S = {z = x+ y + 1}.

The only shortcoming is that vertical planes are not graphs. This can be resolvedby allowing x or y be the output variable and the other two variables be the inputvariables.

2. Level set. Given an affine function

f(x, y, z) = αx+ βy + δz + γ,

let

S = {f(x, y, z) = 0} = {αx+ βy + δz + γ = 0} =

[α β δ] xy

z

+ γ = 0

.

This, however, is not always a surface, because, if α = β = δ = 0, then S is not a2-plane. An extra assumption is needed, namely 〈α, β, γ〉 6= 0. We can write this asfollows: First, recall that the differential of f is defined to be

df = ∂xf dx+ ∂yf dy + ∂zf dz.

In order for S to be a plane and therefore a surface, we need to assume that

df 6= 0

3. Parameterized surface. Given an affine map Φ : R2 → R3 given by

Φ(u, v) = (x0, y0, z0) + A〈u, v〉.

where (x0, y0, z0) ∈ R3 and

A =

a11 a12a21 a22a31 a32,

,

9.3. LOCAL DESCRIPTIONS OF A SURFACE 53

let

S = {Φ(u, v) : 〈u, v〉}= {(x0, y0, z0) + A〈u, v〉, 〈u, v〉 ∈ R2}= {(x0 + a11u+ a12v, y0 + a21u+ a22v, z0 + a31u+ a32v)}.

This, however, does not always work. If the rank of A is not maximal, i.e., 2, then Sis either a point or a line. We must therefore assume that the rank of A is maximal.Equivalently, we must assume kerA = {0}.

The assumptions needed for the function f and map Φ are nondegeneracy conditions.Notice that they can be expressed in terms of the differentials of f and Φ. In particular, theset S = {f = 0} is a surface, if

df 6= 0.

The set S = Φ(R2) is a surface, if the Jacobian of Φ,

∂Φ =

∂ux ∂vx∂uy ∂vy∂uz ∂vz

has maximal rank.

9.3 Local descriptions of a surface1. Graph: Given an open domain D ⊂ R2 and C1 function h : D → R, the set

S = {z = h(x, y)}

is a surface. It suffices to do this locally.

Definition 9.1. A set S ⊂ R3 is a surface, if for any p0 = (x0, y0, z0) ∈ R3, thereexists an open neighborhood O ⊂ R3 of p0, an open domain D ⊂ R2, and a C1

function f : D → R such that

S ∩O = {z = f(x, y)}.

2. Level set: We start with a global definition. If

S = {f(x, y, z) = 0},

where O ⊂ R3 is open and f : O → R is C1, then S is a surface.The equivalent local definition is the following:

Definition 9.2. A set S ⊂ R3 is a surface, if for each p0 = (x0, y0, z0) ∈ S, there existsan open neighborhood O of p0 and a C1 function f : O → R such that

S ∩O = {f(x, y, z) = 0} and df(p0) 6= 0,


The idea is to approximate f by its linear approximation at p0. For (x, y, z) closeenough to (x0, y0, z0),

f(x, y, z) ' f(x0, y0, z0) + (x− x0)∂xf(p0) + (y − y0)∂yf(p0) + (z − z0)∂zf(p0).

3. Parameterized surface: We just give the local definition:

Definition 9.3. A set S ⊂ R3 is a surface, if for each p0 ∈ S, there exists an opendomain D ⊂ R2, an open neighborhood O ⊂ R3 of p0, and a C1 map Φ : D → R3 suchthat

S ∩O = {Φ(u, v) : (u, v) ∈ D}

and the Jacobian matrix of Φ at (u0, v0), where Φ(u0, v0) = p0, has maximal rank. Inother words,

∂Φ(u0, v0) =

∂ux(u0, v0) ∂vx(u0, v0)∂uy(u0, v0) ∂vy(u0, v0)∂uz(u0, v0) ∂vz(u0, v0)

has rank 2.

9.4 Equivalence of definitionsIf S is locally a graph, then, near each p0 ∈ S,

S = {z = h(x, y)} = {f(x, y, z) = 0} = {(x, y, z) = Φ(u, v)},

where

f(x, y, z) = z − h(x, y)Φ(u, v) = (u, v, h(u, v)).

Therefore, S is locally both a level set and a parameterized surface.The converse statements require the implicit and inverse function theorems. We state

the exactly versions we need here.

Theorem 9.4. (Implicit function theorem) Given an open set O ⊂ R3, a point p0 ∈ O, andC1 function f : O → R such that

f(p0) = 0 and ∂zf(p0) 6= 0,

there exists an open neighborhood N ⊂ R3 of p0, an open domain D ⊂ R2, and a C1 functionh : D → R such that, for any p ∈ N ,

{f(x, y, z) = 0 : (x, y, z) ∈ N} = {(x, y, h(x, y)) : (x, y) ∈ D}.

9.5. COORDINATE CHARTS 55

Theorem 9.5. (Inverse function theorem) Given an open domain D ⊂ R2, a point (u0, v0) ∈D, and a C1 map Φ : D → R3, let

Φ : D → R2

(u, v) 7→ (x(u, v), y(u, v)).

If the Jacobian matrix ∂Φ(u0, v0) has maximal rank (i.e., is invertible), then there exists anopen neighborhood N ⊂ D of (u0, v0) and a C1 map Ψ : Φ(N) → R2, such that, for any(x, y) ∈ Φ(N),

Φ(Ψ(x, y)) = (x, y).

9.5 Coordinate chartsA C1 coordinate chart of a surface S ⊂ R3 consists of an open domain D ⊂ R2, a C1 map.

9.6 Tangent space at a point on a surfaceThe tangent space at a point p ∈ S will be denoted TpS. It consists of all possible velocitiesat p of curves that lie in the surface and pass through p. In other words,

TpS = {v ∈ R3 : there exists a C1 curve c : I → S such that c(0) = p and c(0) = v}.

Suppose D is an open neighborhood of (0, 0) ∈ R2 and Φ : D → S is a coordinate chart,where Φ(0) = p0 ∈ S. Then, given any C1 curve c : (−T, T ) → D, where c(0) = (0, 0),c = Φ ◦ c : (−T, T )→ S is a C1 curve in S. Moreover, since

c(t) = (x(u(t), v(t)), y(u(t), v(t)), z(u(t), v(t))),

The velocity of c at p0 is

c(0) =

⟨∂x

∂uu+

∂x

∂vv,∂y

∂uu+

∂y

∂vv,∂z

∂uu+

∂z

∂vv

⟩

=

∂x

∂u

∂x

∂v∂y

∂u

∂y

∂v∂z

∂u

∂z

∂v

[uv

]

= ∂Φ(p0)〈u, v〉

where 〈u, v〉 = c′(0). Therefore, the Jacobian of Φ defines at each p0 ∈ S, a linear map

∂Φ(p0) : R2 → Tp0S

〈u, v〉 7→ ∂Φ(p0)〈u, v〉(9.1)


Since ∂Φ(p0) has maximal rank, this map is injective.Conversely, since Φ is an injective map, given any C1 curve c : (−T, T ) → S, there is a

corresponding curve c = Φ−1 ◦ c : (−T, T ) → D. One can use the inverse function theoremto prove that c is C1. The argument above shows that c(0) ∈ TpS lies in the image of thelinear map ∂Φ(p0). This implies that the map (12.1) is a linear isomorphism.

The tangent space Tp can also described using the definition of the surface S as a levelset. Suppose

S = {p ∈ O : f(p) = 0},

where O is an open neighborhood of p0 in R3 and f is a C1 function on O such that df 6= 0.Given any C1 curve c : (−T, T )→ S ∩O such that c(0) = p0 and c(0) = 〈x, y, z〉,

0 =d

dtf(c(t)) =

d

dt(f(x(t), y(t), z(t))) = x∂xf + y∂yf + z∂zf = ∇f · c. (9.2)

It follows thatTpS ⊂ {v ∈ R3 : v · ∇f(p)〉 = 0}.

However, since both sides are 2-dimensional linear subpsaces, they must be the same. There-fore, they are in fact equal.

Chapter 10

Calculus on Affine Space

10.1 TopologyGiven an affine space Am, there is an affine isomorphism Φ : Rm → Am. The topology onAm is assumed to be the one on Rm pushed forward to Am. Specifics follow.

Recall first the following: Given p0 ∈ Rm and r > 0, define the open ball of radius rcentered at p0 to be

B(p0, r) = {p ∈ Rm : |p− p0| < r}.

A set O ⊂ Rm is open, if and only if, for each p0 ∈ O, there exists r > 0 such thatB(p0, r) ⊂ O.

10.2 Functions and mapsGiven an open set O ⊂ Am, a function f : O → R is Ck, where k ≥ 0, if the functionf ◦ Φ : Φ−1(O)→ R is Ck.

More generally, given affine spaces Am, Bn and an open set O ⊂ Am, a map F : O → Bn

is Ck, if Ψ−1 ◦ F ◦ Φ : Φ−1(O)→ Rn is Ck, where Ψ : Rn → Bn is an affine isomorphism.

10.3 Dual vector spaceDefinition 10.1. Given a vector space V, its dual vector space V∗ is the set of all linearfunctions of V,

V∗ = {θ : V → R : θ is linear}.

For convenience, we shall use the following angle bracket notation: Given θ ∈ V∗ andv ∈ V,

〈θ, v〉 = 〈v, θ〉 = θ(v).

Lemma 10.2. Given any linear map L : V→W, there is a unique linear map Lt : W∗ → V∗,such that, for any v ∈ V and η ∈W∗,

〈η, L(v)〉 = 〈L∗(η), v〉.

57

58 CHAPTER 10. CALCULUS ON AFFINE SPACE

Lemma 10.3. There is a natural isomorphism (V∗)∗ = V.

Given a basis (e1, . . . , em) of V, there is a natural basis (θ1, . . . , θm) of V∗, where, for each1 ≤ i, j ≤ m,

〈θi, ej〉 = δij.

In particular, if v = aiei, then, for each 1 ≤ j ≤ m,

〈θj, v〉 = aj.

10.4 DerivativesLet O ⊂ Am be open.

10.4.1 Directional derivativeGiven a C1 function f : O → R and a tangent vector v ∈ Am, its directional derivative atp ∈ O with velocity v is defined to be

dvf(p) =d

dt

∣∣∣∣t=0

f(c(t)) = limt→0

f(c(t))− f(c(0))t

, (10.1)

where c : (−T, T )→ Am is a C1 curve such that c(0) = p and c(0) = v.

Lemma 10.4. If c1 and c2 are curves that satisfy the assumptions above, then they give thesame value for dvf(p) in (10.1).

10.4.2 Differential of a functionGiven a C1 function f : O → R and p ∈ O, the function

df(p) : A→ Rv 7→ dvf(p)

is a linear function of v ∈ A. Therefore, df(p) ∈ A∗. We can therefore, define the differentialof a C1 function f : O → R to be the map

df : O → Ap 7→ df(p),

where, for each v ∈ A,〈df(p), v〉 = dvf(p),

as defined above.

10.5. DIFFERENTIAL 1-FORM ON AN OPEN SUBSET OF AM 59

10.5 Differential 1-form on an open subset of Am

Definition 10.5. Given an open set O ⊂ Am, a differential 1-form is a map ω : O → V∗.

Example. The differential of a function is a 1-form.

10.6 Line integral of a 1-form along a curve in Am

Definition 10.6. If O ⊂ Am is open, c : [0, T ] → O is a C1 curve, and ω : O → (Vm)∗ a1-form, then the line integral of ω along c is defined to be∫

c

ω =

∫ T

0

〈ω(c(t)), c′(t)〉 dt.

Observe that the integrand on the right is a real-valued function of t ∈ [0, T ] and thereforethe integral is standard single variable integral.

By using the same proof of the chain rule for functions of a single variable, the followingholds:

Lemma 10.7. If c : I → Am is a C1 curve and ϕ : I ′ → I is a C1 function on the intervalI ′, then

(c ◦ ϕ)′(t) = c′(ϕ(t))ϕ′(t).

The same proof for the change of variables for a single variable integral now implies:

Theorem 10.8. If O ⊂ Am is open, c : [0, T ] → O is a C1 curve, and ω : O → (Vm)∗ a1-form, then for any C1 function ϕ : [0, T ]→ [0, T ] such that ϕ(0) = 0 and ϕ(T ) = T ,∫

c

ω =

∫c

ω,

where c = c ◦ ϕ.

The fundamental theorem of calculus implies the following:

Theorem 10.9. If c : [0, T ]→ Am is a C1 curve and f : O → R a C1 function on an openset O ⊂ Am, then ∫

c

df = f(c(T ))− f(c(0)).

In particular, the line integral of df from p to q does not depend on which C1 curve isused to connect p to q.

60 CHAPTER 10. CALCULUS ON AFFINE SPACE

10.7 Fundamental exampleRecall that, associated with an affine basis (p0, . . . , pm) of Am is a basis E = (e− 1, . . . , em)of Vm, where

e1 = p1 − p0, . . . , em = pm − p0.

On the other hand, recall that I−1E (p) = (x1, . . . , xm), wherefor each 1 ≤ i ≤ m, t

xi : Am → Rp0 + v 7→ vi.

The differential of xi is defined to be dxi ∈ (Vm)∗, where

〈dxi(p), v〉 = d

dt

∣∣∣∣t=0

xi(p+ tv) = vi.

Equivalently,〈dxi, ej〉 = δij.

and, therefore, (dx1, . . . , dxm) is the basis of (Vm)∗ dual to the basis E = (e1, . . . , em) of Vm.Given a C1 function f : O → R, note that, if p0 ∈ O and IE(x

1, . . . , xn) = p, then

∂i((f ◦ IE)(x1, . . . , xn)) =d

dt

∣∣∣∣t=0

f(p+ tei)

= 〈df(p), ei〉.

‘Therefore,df(p) = 〈df(p), ei〉dxi.

This leads to the following notation: Given an affine basis P = (p0, . . . , pm) of Am,instead of denoting the corresponding basis of Vm by E = (e1, . . . , em), we denote it by∂ = (∂1, . . . , ∂m), where, for each i = 1, . . . ,m,

∂i = pi − p0.

As noted above, its dual basis is dx = (dx1, . . . , dxm). Given a C1 function f : O → R, wedenote

∂if(p) = 〈df(p), ∂i〉.

Therefore, omitting the p for brevity,

df = ∂if dxi.

Chapter 11

Differential of a map

11.1 Characterization of a linear mapLemma 11.1. Given a map F : Vm →Wn, the following are equivalent:

• F is linear.

• ℓ ◦ F : Vm → R is linear for any ℓ ∈ (Wn)∗.

• Given a basis ℓ1, . . . , ℓn of Wn, ℓi ◦ F is linear for each i = 1, . . . , n.

11.1.1 Differential of a mapGiven affine spaces A, B and an open set O ⊂ A, let F : O → B be a C1 map. Given p ∈ O,we can define a linear map

∂pF : A→ B

as follows: Given v ∈ A, let c : (−T, T )→ A be a C1 curve such that c(0) = p and c(0) = v.Then F ◦ c : (−T, T ) → B is also a C1 curve. We then define ∂pF (v) to be the velocity ofF ◦ c at t = 0,

∂pF (v) =d

dt

∣∣∣∣t=0

F (c(t)).

This linear map is also known as the Jacobian of F . If A = Rm and B = Rn, then ∂Fp is the n-by-m matrix of partial derivatives of F (x1, . . . , xm) = (F 1(x1, . . . , xm), . . . , F n(x1, . . . , xm)).

11.1.2 Pushforward of a tangent vectorGiven p ∈ O and v ∈ A, we will also write

F∗v = ∂pF (v)

and call it the pushforward of v by F .

61

62 CHAPTER 11. DIFFERENTIAL OF A MAP

11.1.3 Pullback of a cotangent vectorGiven p ∈ O and θ ∈ A∗, we will write

F ∗θ = (∂pF )t(θ).

11.2 Chain RuleLemma 11.2. If F : Am → Bn and G : Bn → C are C1 maps, then so is G ◦ F , and

d(G ◦ F )(p) = dG(F (p)) ◦ dF (p).

Chapter 12

Surfaces in affine 3-space

12.1 PreliminariesA set O ⊂ Rm is called open, if, for any p ∈ O, there is an r > 0 such that

B(p, r) ⊂ O.

Think of an open set as one that does not contain any point on its boundary. A set D ⊂ Rm

is called an open domain, if it is an open set. Given a point p ∈ Rm, an open neighborhoodof p is an open set that contains p.

12.2 Surfaces in Cartesian 3-space12.2.1 Informal but flawed descriptions of a surface

1. Graph: Given an open domain D ⊂ R2 and a C1 function h : D → R, let

S = {z = h(x, y)} = {(x, y, h(x, y)) : (x, y) ∈ D}.

Example:S = {z = 1√

1− x2 − y2.

2. Level set: Given an open set O ⊂ R3 and a C1 function f : O → R, let

S = {f(x, y, z) = 0} = {(x, y, z) ∈ O : f(x, y, z) = 0}.

Examples:

S = {x2 + y2 + z2 − 1 = 0}S = {x2 + y2 − z2 = 0}.

63


3. Parameterized surface: Given an open domain D ⊂ R2 and a C1 map Φ : D → R3,

S = Φ(D) = {(Φ(u, v) : (u, v) ∈ D} = {(x(u, v), y(u, v), z(u, v) : (u, v) ∈ D},

whereΦ(u, v) = (x(u, v), y(u, v), z(u, v)).

Example:S = {(r cos θ, r sin θ, r)}.

But these descriptions are FLAWED. And are they equivalent? How can these descrip-tions be made abstract for a surface in A3?

12.2.2 2-planes in 3-spaceThe simplest surface in R3 is a 2-plane. It can be described as follows:

1. Graph: Given an affine function

h(x, y) = ax+ by + c,

letS = {z = h(x, y)} = {z = ax+ by + c}.

Examples:S = {z = 0}S = {z = x+ y + 1}.

The only shortcoming is that vertical planes are not graphs. This can be resolvedby allowing x or y be the output variable and the other two variables be the inputvariables.

2. Level set. Given an affine function

f(x, y, z) = αx+ βy + δz + γ,

let

S = {f(x, y, z) = 0} = {αx+ βy + δz + γ = 0} =

[α β δ] xy

z

+ γ = 0

.

This, however, is not always a surface, because, if α = β = δ = 0, then S is not a2-plane. An extra assumption is needed, namely 〈α, β, γ〉 6= 0. We can write this asfollows: First, recall that the differential of f is defined to be

df = ∂xf dx+ ∂yf dy + ∂zf dz.

In order for S to be a plane and therefore a surface, we need to assume that

df 6= 0

12.2. SURFACES IN CARTESIAN 3-SPACE 65

3. Parameterized surface. Given an affine map Φ : R2 → R3 given by

Φ(u, v) = (x0, y0, z0) + A〈u, v〉.

where (x0, y0, z0) ∈ R3 and

A =

a11 a12a21 a22a31 a32,

,let

S = {Φ(u, v) : 〈u, v〉}= {(x0, y0, z0) + A〈u, v〉, 〈u, v〉 ∈ R2}= {(x0 + a11u+ a12v, y0 + a21u+ a22v, z0 + a31u+ a32v)}.

This, however, does not always work. If the rank of A is not maximal, i.e., 2, then Sis either a point or a line. We must therefore assume that the rank of A is maximal.Equivalently, we must assume kerA = {0}.

The assumptions needed for the function f and map Φ are nondegeneracy conditions.Notice that they can be expressed in terms of the differentials of f and Φ. In particular, theset S = {f = 0} is a surface, if

df 6= 0.

The set S = Φ(R2) is a surface, if the Jacobian of Φ,

∂Φ =

∂ux ∂vx∂uy ∂vy∂uz ∂vz

has maximal rank.

12.2.3 Local descriptions of a surface1. Graph: Given an open domain D ⊂ R2 and C1 function h : D → R, the set

S = {z = h(x, y)}

is a surface. It suffices to do this locally.Definition 12.1. A set S ⊂ R3 is a surface, if for any p0 = (x0, y0, z0) ∈ R3, thereexists an open neighborhood O ⊂ R3 of p0, an open domain D ⊂ R2, and a C1 functionf : D → R such that

S ∩O = {z = f(x, y)}.

2. Level set: We start with a global definition. If

S = {f(x, y, z) = 0},

where O ⊂ R3 is open and f : O → R is C1, then S is a surface.The equivalent local definition is the following:


Definition 12.2. A set S ⊂ R3 is a surface, if for each p0 = (x0, y0, z0) ∈ S, thereexists an open neighborhood O of p0 and a C1 function f : O → R such that

S ∩O = {f(x, y, z) = 0} and df(p0) 6= 0,

The idea is to approximate f by its linear approximation at p0. For (x, y, z) closeenough to (x0, y0, z0),

f(x, y, z) ' f(x0, y0, z0) + (x− x0)∂xf(p0) + (y − y0)∂yf(p0) + (z − z0)∂zf(p0).

3. Parameterized surface: We just give the local definition:

Definition 12.3. A set S ⊂ R3 is a surface, if for each p0 ∈ S, there exists an opendomain D ⊂ R2, an open neighborhood O ⊂ R3 of p0, and a C1 map Φ : D → R3 suchthat

S ∩O = {Φ(u, v) : (u, v) ∈ D}

and the Jacobian matrix of Φ at (u0, v0), where Φ(u0, v0) = p0, has maximal rank. Inother words,

∂Φ(u0, v0) =

∂ux(u0, v0) ∂vx(u0, v0)∂uy(u0, v0) ∂vy(u0, v0)∂uz(u0, v0) ∂vz(u0, v0)

has rank 2.

12.2.4 Equivalence of definitionsIf S is locally a graph, then, near each p0 ∈ S,

S = {z = h(x, y)} = {f(x, y, z) = 0} = {(x, y, z) = Φ(u, v)},

where

f(x, y, z) = z − h(x, y)Φ(u, v) = (u, v, h(u, v)).

Therefore, S is locally both a level set and a parameterized surface.The converse statements require the implicit and inverse function theorems. We state

the exactly versions we need here.

Theorem 12.4. (Implicit function theorem) Given an open set O ⊂ R3, a point p0 ∈ O,and C1 function f : O → R such that

f(p0) = 0 and ∂zf(p0) 6= 0,

there exists an open neighborhood N ⊂ R3 of p0, an open domain D ⊂ R2, and a C1 functionh : D → R such that, for any p ∈ N ,

{f(x, y, z) = 0 : (x, y, z) ∈ N} = {(x, y, h(x, y)) : (x, y) ∈ D}.

12.2. SURFACES IN CARTESIAN 3-SPACE 67

Theorem 12.5. (Inverse function theorem) Given an open domain D ⊂ R2, a point(u0, v0) ∈ D, and a C1 map Φ : D → R3, let

Φ : D → R2

(u, v) 7→ (x(u, v), y(u, v)).

If the Jacobian matrix ∂Φ(u0, v0) has maximal rank (i.e., is invertible), then there exists anopen neighborhood N ⊂ D of (u0, v0) and a C1 map Ψ : Φ(N) → R2, such that, for any(x, y) ∈ Φ(N),

Φ(Ψ(x, y)) = (x, y).

12.2.5 Coordinate chartsA C1 coordinate chart of a surface S ⊂ R3 consists of an open domain D ⊂ R2, a C1 map.

12.2.6 Tangent space at a point on a surfaceThe tangent space at a point p ∈ S will be denoted TpS. It consists of all possible velocitiesat p of curves that lie in the surface and pass through p. In other words,

TpS = {v ∈ R3 : there exists a C1 curve c : I → S such that c(0) = p and c(0) = v}.

Suppose D is an open neighborhood of (0, 0) ∈ R2 and Φ : D → S is a coordinate chart,where Φ(0) = p0 ∈ S. Then, given any C1 curve c : (−T, T ) → D, where c(0) = (0, 0),c = Φ ◦ c : (−T, T )→ S is a C1 curve in S. Moreover, since

c(t) = (x(u(t), v(t)), y(u(t), v(t)), z(u(t), v(t))),

The velocity of c at p0 is

c(0) =

⟨∂x

∂uu+

∂x

∂vv,∂y

∂uu+

∂y

∂vv,∂z

∂uu+

∂z

∂vv

⟩

=

∂x

∂u

∂x

∂v∂y

∂u

∂y

∂v∂z

∂u

∂z

∂v

[uv

]

= ∂Φ(p0)〈u, v〉

where 〈u, v〉 = c′(0). Therefore, the Jacobian of Φ defines at each p0 ∈ S, a linear map

∂Φ(p0) : R2 → Tp0S

〈u, v〉 7→ ∂Φ(p0)〈u, v〉(12.1)

Since ∂Φ(p0) has maximal rank, this map is injective.


Conversely, since Φ is an injective map, given any C1 curve c : (−T, T ) → S, there is acorresponding curve c = Φ−1 ◦ c : (−T, T ) → D. One can use the inverse function theoremto prove that c is C1. The argument above shows that c(0) ∈ TpS lies in the image of thelinear map ∂Φ(p0). This implies that the map (12.1) is a linear isomorphism.

The tangent space Tp can also described using the definition of the surface S as a levelset. Suppose

S = {p ∈ O : f(p) = 0},

where O is an open neighborhood of p0 in R3 and f is a C1 function on O such that df 6= 0.Given any C1 curve c : (−T, T )→ S ∩O such that c(0) = p0 and c(0) = 〈x, y, z〉,

0 =d

dtf(c(t)) =

d

dt(f(x(t), y(t), z(t))) = x∂xf + y∂yf + z∂zf = ∇f · c. (12.2)

It follows thatTpS ⊂ {v ∈ R3 : v · ∇f(p)〉 = 0}.

However, since both sides are 2-dimensional linear subpsaces, they must be the same. There-fore, they are in fact equal.

Chapter 13

Surfaces in Euclidean 3-space

In the previous chapter, we never used any geometric concepts such as distance, length, orangle. In this chapter, we use them to study the geometry of a surface in Euclidean 3-space.Specifically, we use here the standard dot product on R3.

13.1 First fundamental form

The first fundamental form is simply the dot product restricted to the subspace TpS ⊂ R3.We want, however, to understand what it looks like using a coordinate chart Φ : D → R3.In particular, we want to “pull back” the dot product to the vector space R2, viewed asvelocities of curves in D.

First, recall that the dot product of any two vectors 〈x1, y1, z1〉 and 〈x2, y2, z2〉 is

〈x1, y1, z1〉 · 〈x2, y2, z2〉 = x1x2 + y1y2 + z1z2.

Therefore, given any two c1, c2 : (−T, T ) → D such that c1(0) = c2(0) = p0 ∈ S, the dot

69

70 CHAPTER 13. SURFACES IN EUCLIDEAN 3-SPACE

product of the velocities c1 = (Φ ◦ c1)′(0) and c2 = (Φ ◦ c2)′(0) at p0 is given by

c1(0) · c2(0) = (Φ ◦ c1)′(0) · (Φ ◦ c2)′(0)= (∂Φc′1(0)) · (∂Φc′2(0))

=

(∂x

∂uu1 +

∂x

∂vv1

)(∂x

∂uu2 +

∂x

∂vv2

)+

(∂y

∂uu1 +

∂y

∂vv1

)(∂y

∂uu2 +

∂y

∂vv2

)+

(∂z

∂uu1 +

∂z

∂vv1)(

∂z

∂uu2 +

∂z

∂vv2

)=

(∂x

∂u

∂x

∂u+∂y

∂u

∂y

∂u+∂z

∂u

∂z

∂u

)u1u2

+

(∂x

∂u

∂x

∂v+∂x

∂v

∂x

∂u+∂y

∂u

∂y

∂v+∂y

∂v

∂y

∂u+∂z

∂u

∂z

∂v+∂z

∂v

∂z

∂u

)(u1v2 + u2v1)

+

(∂x

∂v

∂x

∂v+∂y

∂v

∂y

∂v+∂z

∂v

∂z

∂v

)(v)2

=[u1 v1

] ∂x∂u ∂y

∂u

∂z

∂u∂x

∂v

∂y

∂v

∂z

∂v

∂x

∂u

∂x

∂v∂y

∂u

∂y

∂v∂z

∂u

∂z

∂v

[u2v2

]

= 〈u1, v1〉 · (∂Φ)T (∂Φ)〈u2, v2〉,

where (∂Φ)T denotes the transpose of the matrix ∂Φ.Therefore, if, for each p ∈ S, we define the symmetric 2-by-2 matrix

G(p) = (∂Φ(p))T (∂Φ(p)),

and, given any tangent vectors 〈u1, v1〉, 〈u2, v2〉 ∈ TpS, define

g(p)(〈u1, v1〉, 〈u2, v2〉) = 〈u1, v1〉 ·G〈u2, v2〉,

then,〈(∂Φ(p))〈u1, v1〉 · 〈(∂Φ(p))〈u2, v2〉 = g(p)(〈u1, v1〉, 〈u2, v2〉).

13.2 Change of notationThe calculation above is a mess, using several different variables. We introduce new notationthat is easier to write and has the advantage that it makes generalization to higher dimensionseasy.

From now on, a point in R2 will be denoted x = (x1, x2), and a vector in R2 by x =

〈x1, x2〉. A point in R3 will be denoted y = (y1, y2, y3), and a vector in R3 by 〈y1, y2, y3〉. Acoordinate chart will be denoted y(x) = (y1(x1, x2), y2(x1, x2), y3(x1, x2)).

If we fix x2, then the map t 7→ y(t, x2) defines a curve in S. This defines a family ofcurves on the coordinate chart. There is a second family of curves defined by fixing x1 and

13.2. CHANGE OF NOTATION 71

using x2 as the parameter of a curve. The velocities of these curves are the partial derivativesof y,

∂y

∂x1and ∂y

∂x2,

which are the columns of the Jacobian

∂y

∂x=

∂y1

∂x1∂y1

∂x2

∂y2

∂x1∂y2

∂x2

∂y3

∂x1∂y3

∂x2

.

That the Jacobian has rank 2 implies that(∂y

∂x1,∂y

∂x2

)is, for each p ∈ S, a basis of TpS. Morevoer, the dot product of

v1 = v11∂y

∂x1+ v21

∂y

∂x2

v2 = v12∂y

∂x1+ v22

∂y

∂x2,

is given by

v1 · v2 =(v11∂y

∂x1+ v21

∂y

∂x2

)·(v12∂y

∂x1+ v22

∂y

∂x2

)= gij(p)v

i1v2v

j,

where, for each 1 ≤ i, j ≤ 2,

gij =∂y

∂xi· ∂y∂xj

. (13.1)

It follows that the length of any tangent vector or curve at p ∈ S and the angle of anypair of vectors in TpS can be determined using the coordinates (x1, x2) and the symmetricpositive definite matrix G = [gij(p)].

More generally, an inner product on TpS, for each p ∈ S, is called a Riemannian metric.Given a Riemannian metric g, p ∈ S, and a basis e1, . . . , em ∈ TpS, there is a positive definitematrix G(p) = [gij(p)], where

gij(p) = ei · ej.

Such a Riemannian metric need not be the same as the one given by (13.1). Moreover, givena Riemannian metric g, there can be more than one embedding that satisfies (13.1). Forthese reasons, we call a Riemannian metric an intrinsic geometric structure on S.


13.3 Gauss map

If a surface S ⊂ R is defined as the level set of a C2 function f , then the Gauss map isdefined to be

η : S → R3

p 7→ ∇f(p)|∇f(p)|

,

For each p ∈ S, the vector η(p) is a unit vector orthogonal to TpS. It is unique up to sign.

13.4 Geometry of a curve on a surfaceRecall that the curvature of a curve in E2 and the curvature and torsion of a unit speed curvein E3 uniquely determine the shape of the curve. Moreover, the definition of these functionsdoes not depend on how the curve is described (i.e., the parameterization of the curve orthe coordinates in Euclidean space). The analogous geometric invariants for a surface inE3 are called the first and second fundamental forms. The first fundamental form plays arole similar to the unit speed parameterization of a curve. The second fundamental form isanalogous to the curvature and torsion of a curve. We will see later that the first and secondfundamental forms uniquely determine the shape of the surface.

Before describing the second fundamental form of a surface S, we note that the firstfundamental form at each point p ∈ S is simply the dot product on E3 but restricted tovectors tangent to S at p. Its

Given a C2 surface S ⊂ R3, a point p ∈ S, and a unit tangent vector u ∈ TpS, considera C2 unit speed curve c : I → S such that c(0) = p and c(0) = u. We can define anorthonormal frame (e1, e2, e3) along c, where

e1(t) = c(t)

e3(t) = η(c(t)).

This is not the Frenet-Serret frame, and therefore we have to recompute the derivatives ofe1, e2, e3 with respect to t.

First, since e1 is unit, it follows by Lemma 7.4 that e1 is orthogonal to e1 and, therefore,there are functions κ2 and κ3 such that

c(t) = e1(t) = κ2(t)e2(t)− κ3(t)e3(t).

Since e2 is tangent to S and e3 is normal to S, we call κ2 the tangential curvature andκ3 the normal curvature to the curve c, relative to the surface S. The tangential curvaturemeasures how quickly e1 is twisting in the tangent plane, and the normal curvature measureshow quickly e1 is twisting out of the tangent plane.

13.5. SECOND FUNDAMENTAL FORM 73

If S is the level set of a C2 function f , then, differentiating (12.2), we get

0 =d2

dt2f(x(t), y(t), z(t))

=d

dt(〈∂xf, ∂yf, ∂zf〉 · 〈x, y, z〉

=[x y z

] ∂2xxf ∂2xyf ∂2xzf∂2yxf ∂2yyf ∂2yzf∂2zxf ∂2zyf ∂2zzf

xyz

+ 〈∂xf, ∂yf, ∂zf〉 · 〈x, y, z〉

= c · (∂2f)c+∇f · c= ·c · (∂2f)c− |∇f |e3 · (κ2e2 + κ3e3)

= ·c · (∂2f)c− |∇f |e3

(13.2)

where

∂2f =

∂2xxf ∂2xyf ∂2xzf∂2yxf ∂2yyf ∂2yzf∂2zxf ∂2zyf ∂2zzf

is the Hessian of f . Therefore, setting t = 0, we get

κ3(0) = −u ·H(p)u,

whereH =

∂2f

|∇f |is a 2-by-2 symmetric matrix. This shows that the normal curvature at each point of thecurve depends only on the direction of the curve at that point. It does not depend on theshape of the curve at all.

13.5 Second fundamental formWe can also show that, for any v ∈ TpS, v ·Hv depends only on the surface S and not on thefunction f as follows: If S is also the level set of another C2 function g, then, by Lemma F.8,there exists a C2 function ϕ such that g = ϕf . Flipping the sign of g, if necessary, we canassume that, along S, ∇g points in the same direction as∇f . That implies ϕ > 0. Therefore,along S,

∂2g

|∇g|=

∂2g

|∇g|

=∂2(ϕf)

|∇(ϕf)|

=ϕ∂2f +∇ϕ⊗∇f +∇f ⊗∇ϕ+ f∂2ϕ

|ϕ∇f + f∇ϕ|

=ϕ∂2f +∇ϕ⊗∇f +∇f ⊗∇ϕ

ϕ|∇f |,


where we use the following notation: Given functions f and g,

∇f ⊗∇g =

∂xf∂xg ∂xf∂yg ∂xf∂zg∂yf∂xg ∂yf∂yg ∂yf∂zg∂zf∂xg ∂zf∂yg ∂zf∂zg

.Therefore, if p ∈ S and v ∈ TpS, then f(p) = 0 and v · ∇f = 0. It follows that

v · (∂2g)v|∇g|

=v · (ϕ∂2f +∇ϕ⊗∇f +∇f ⊗∇ϕ)v

ϕ|∇f |

=v · ∂2fv + 2(v · ∇ϕ)(v · ∇f)

|∇f |

We can therefore define, for each p, a bilinear function

H(p) : TpS × TpS → R

(v, w) 7→ v · (∂2f)w|∇f |

.

This is called the second fundamental form of the surface S at p ∈ S. It is a geometricinvariant of S that measures how curved the surface is. In particular, if c is a unit speedcurve on S, then its normal curvature at a point c(t) is

κ3 = −v ·Hv,

where v = c(t). This implies that how quickly the direction of c twists out of the tangentplane of S depends only on the geometry of S and the direction of c.

Given any orthonormal basis of TpS, H(p) can be written as a symmetric 2-by-2 ma-trix, which always has 2 real eigenvalues. These are called the principal curvatures. Theyrepresent the maximum and minimum possible values of the normal curvature at a point.

Example. The sphere of radius r centered at the origin is the level set f = r2 for thefunction

f(x, y, z) = x2 + y2 + z2.

The gradient of f is∇f = 2〈x, y, z〉,

and therefore

|∇f | = 2√x2 + y2 + z2

= 2r

∂2f

|∇f |=

1

2r

2 0 00 2 00 0 2

=

1

rI,


where I is the 3-by-3 identity matrix. Since

v ·(1

rI

)v =|v|2

r=

1

r,

for any unit vector v, it follows that the principal curvatures are both r−1.

Example. Consider the 1–sheeted hyperboloid

S = {(x, y, z) ∈ R3 : x2 + y2 − z2 = a2}.

This can be written in cylindrical coordinates as

S = {(r cos θ, r sin θ, z) : r2 − z2 = a2},

where r2 = x2 + y2. It is the level set f = a2 for the function f(x, y, z) = x2 + y2 − z2.Therefore,

∇f = 2〈x, y,−z〉 = 2〈r cos θ, r sin θ,−z〉|∇f | = 2

√x2 + y2 + z2 = 2

√r2 + z2 = 2

√a2 + 2z2

∂2f

|∇f |=

1√a2 + 2z2

1 0 00 1 00 0 −1

.One orthonormal basis of TpS is

e1 = 〈− sin θ, cos θ, 0〉, e2 =〈z cos θ, z sin θ, r〉√

r2 + z2=〈z cos θ, z sin θ, r〉√

a2 + 2z2

Therefore, the second fundamental form of S is given by

H(e1, e1) =1√

a2 + 2z2

[− sin θ cos θ 0

] 1 0 00 1 00 0 −1

− sin θcos θ0

=

1√a2 + 2z2

w

H(e1, e2) = H(e2, e1) =1

a2 + 2z2[z cos θ z sin θ r

] 1 0 00 1 00 0 −1

− sin θcos θ0

= 0

H(e2, e2) =1

(a2 + z2)3/2[z cos θ z sin θ r

] 1 0 00 1 00 0 −1

z cos θz sin θr

=

z2 − r2

(a2 + 2z2)3/2

=−a2

(a2 + 2z2)3/2.


It follows that the principal curvatures are

1√a2 + 2z2

and −a2

(a2 + 2z2)3/2.

Example. More generally, consider a surface of revolution

S = {(r(z) cos θ, r(z) sin θ, z)},

where r : R → (0,∞) is the given radial profile function. This is the level set f = 0 of thefunction

f(x, y, z) =1

2(x2 + y2 − (r(z))2).

Therefore,

∂f = 〈x, y,−rrz〉|∂f | =

√x2 + y2 + (rrz)2

= r√

1 + r2z

∂2f =

1 0 00 1 00 0 −rrzz − (rz)

2

∂2f

|∂f |=

1

r√

1 + r2z

1 0 00 1 00 0 −rrzz − (rz)

2

=

1

r√

1 + r2z

1 0 00 1 00 0 −rrzz − (rz)

2

.An orthonormal basis of S is given by

e1 =〈−y, x, 0〉√x2 + y2

=〈−y, x, 0〉

r

e2 =〈rzx, rzy, r〉r√

1 + r2z.


Therefore, the second fundamental form of S is given by

H(e1, e1) =1

r3√

1 + r2z

[−y x 0

] 1 0 00 1 00 0 −rrzz − (rz)

2

−yx0

=

1

r√

1 + r2z

H(e1, e2) = H(e2, e1) =1

r3(1 + r2z)

[rzx rzy r

] 1 0 00 1 00 0 −rrzz − (rz)

2

−yx0

= 0

H(e2, e2) =1

r3(1 + r2z)3/2

[rzx rzy r

] 1 0 00 1 00 0 −rrzz − (rz)

2

rzxrzyr

=

(rrz)2 − r2(rrzz + (rz)

2)

r3(1 + r2z)3/2

=−rzz

(1 + r2z)3/2.

Appendix A

Notation

A.1 Cartesian spacesThe m-dimensional Cartesian affine space will be denoted Rm, and a point in it by

x = (x1, . . . , xm) =

x1

...xm

∈ Rm.

Its tangent space will be denoted Rm = T0Rm, and a vector in it by

v = vi∂i = 〈v1, . . . , vm〉 = [v1, . . . , vm]t =

v1

...vm

∈ Rm.

In particular, the standard basis of Rm is [∂1, . . . , ∂m], where, for each 1 ≤ i ≤ m,

∂i = [0, . . . , 1, . . . , 0]t, where the 1 is in the i-th slot..

The cotangent space (dual to Rm) will be denoted Rm = T ∗0Rm, and a covector in it by

θ = bi dxi = 〈b1, . . . , bm〉∗ = [b1, . . . , bm] =

[b1 · · · bm

]∈ Rm.

In particular, the basis dual to [∂1, . . . , ∂m] is denoted

[dx1, . . . , dxm]t =

dx1

...dxm

.A.2 Abstract vector spaceA basis of an m-dimensional abstract vector space will be denoted

E = [e1, . . . , em] =[e1 · · · em

],

79

80 APPENDIX A. NOTATION

and a vector with respect to this basis as

v = eiai =

[e1 · · · em

] a1

...am

.The dual basis to E is denoted

Θ = [θ1, . . . , θm]t =

θ1

...θm

,and a covector with respect to this basis by

ω = biθi =

[b1 · · · bm

] θ1

...θm

.A.3 ManifoldCoordinates on an m-dimensional manifold M is a C1 map

x = (x1, . . . , xm) :M → Rm

that is injective and

[dx1, . . . , dxm]t =

dx1

...dxm

is a basis of T ∗pM , for each p ∈M,

where, for each 1 ≤ i ≤ m, dxi is the differential of the C1 function xi :M → R.The dual basis at each p ∈M is a basis of the tangent space TpM and is denoted by

[∂1, . . . , ∂m].

This can also be defined equivalently as follows: The map x :M → Rm is a diffeomorphismonto its image, and therefore, the inverse map

x−1 : x(M)→M

is a C1 diffeomorphism. For each p ∈ M , there are m coordinate curves passing throughit, where each coordinate curve is defined by holding all but one of the coordinates fixedand letting the remaining coordinate be the curve parameter. The velocity vectors of thesecurves at p are the tangent vectors ∂1, . . . , ∂m.

Note that not every manifold has coordinates. However, a manifold is always the unionof manifolds that have coordinates.

A.4. ABSTRACT AFFINE SPACE 81

A.4 Abstract affine spaceAssociated to each affine space Am is its tangent space Amm. Recall that each affine basisP = (p0, . . . , pm) of Am defines a map

IP : Rm → Am

[a1, . . . , am]t 7→ p0 + aivi,

wherevi = pi − p0, for each 1 ≤ i ≤ m.

The map

xP : Am → Rm

p 7→ 0 + I−1P (p)

defines coordinates on Am, known as affine coordinates with respect to the afffine basis P .

82 APPENDIX A. NOTATION

Appendix B

Normal forms of maps

B.1 Normal form of a linear mapLemma B.1. Let L : Vm → Wn be a linear map. There exist invertible linear mapsA : Rm → Vm and B : Rn →Wn such that

B−1LA =

[Ir 0r,m−r

0n−r,r 0n−r,m−r

],

where Ir is the r-by-r identity matrix and 0k,l is the k-by-l zero matrix.

It follows that the maximum possible rank of a linear map L : Vm → Wn is min(m,n).If m ≥ n and L : Vm →Wn has maximum rank, then the rank is n,

L = B[In 0n,m−n

]A−1,

and therefore L is surjective.If m ≤ n and L : Vm →Wn has maximum rank, then the rank is m,

L = B

[Im

0n−m,m

]A−1,

and therefore L is injective.It follows that if m = n and L : Vm →Wn has maximum rank, then

L = B[Im]A−1

is bijective.

B.2 Characterization of a linear subspaceLemma B.2. Given 0 ≤ m ≤ n, an n-dimensional vector space Wn, and S ⊂ Rn, thefollowing are equivalent:

1. S is an m-dimensional linear subspace of Wn.

83

84 APPENDIX B. NORMAL FORMS OF MAPS

2. There exists a linear map L : Rm →Wn with rank m such that

imL = S.

3. There exists a linear map M : Wn → Rn−m with rank n−m such that

kerM = S.

The most basic example is

S = {(x1, . . . , xm, 0, . . . , 0) : (x1, . . . , xm) ∈ Rm} ⊂ W = Rn,

with the linear maps

Ln,m : Rm → Rn

(x1, . . . , xm) 7→ (x1, . . . , xm, 0, . . . , 0)

and

Mn,m : Rn → Rn−m

(x1, . . . , xn) 7→ (xn−m+1, . . . , xn).

Lemma B.3. If S ⊂ Wn is an m-dimensional subspace, then there exists a basis E of Wn

such that the linear maps L = IELn,m and M =Mn,mI−1E satisfy the properties in Lemma B.2.

B.3 Linear subspace as a graphA 1-dimensional linear subspace of R2 is given by an equation of the form

ax+ by = 0.

When can this be written as a graph of a function, y = f(x)? The line cannot be vertical.Equivalently, b must be nonzero, and therefore we can solve for y,

y = −abx.

A 2-dimensional linear subspace of R3 is given by an equation of the form

ax+ by + cz = 0.

Similarly, this is the graph of a function, if c 6= 0 and therefore we can solve for zm

z = −acx− b

cy.

A 2-dimensional linear subspace of R4 is given by two linear equations of the form

a11x1 + a12x

2 + a13x3 + a14x

4 = 0

a21x1 + a22x

2 + a23x3 + a24x

4 = 0

B.3. LINEAR SUBSPACE AS A GRAPH 85

When can we solve for (x3, x4) in terms of (x1, x2)? First, note that the system can berewritten in the form [

A′ A′′] [x′x′′

]= 0, (B.1)

where

A′ =

[a11 a12a21 a22

]A′′ =

[a31 a32a41 a42

]x′ =

[x1

x2

]x′′ =

[x3

x4

].

We can therefore solve for x′ = (x3, x4) if and only if the matrix A′′ is invertible. If so, wecan multiply (B.1) on the left by (A′′)−1 to get an equivalent system of the form[

(A′′)−1A′ I] [x′x′′

]= 0,

which expands to

b11x1 + b12x

2 + x3 = 0

b21x1 + b22x

2 + x4 = 0,

where [b11 b12b21 b22

]= (A′′)−1.

Recall that Hom(Rm,Rn) is the space of n-by-m matrices. In general, an m-dimensionalsubspace of Rn, where m < n, is given by a linear system

Lx =[L′ L′′] [x′

x′′

],= 0, (B.2)

where x′ ∈ Rm, x′′ ∈ Rn−m, L′ ∈ Hom(Rm,Rn−m), and L′′ ∈ Hom(Rn−m,Rn−m). Thissystem can be rewritten equivalently as

x′′ = Ax′,

if and only if L′′ is invertible. Multiplying (B.2) on the left by (L′′)−1, we get an equivalentsystem [

(L′′)−1L′ I] [x′x′′

]= 0,

and therefore, we can solve for x′′ in terms of x′:

x′′ = −(L′′)−1L′x′.


B.4 Topology of the space of linear mapsRecall that, if we choose a basis for a vector space Vm and one for another vector space Wn,then there is a linear isomorphism

Hom(Vm,Wn) ' Hom(Rm,Rn).

There is a natural inner product defined on Hom(Rm,Rn) given by

〈A,B〉 =m∑i=1

n∑j=1

AjiB

ji ,

which defines a norm, which in turn defines a distance function. We can therefore defineopen sets, convergence and limits of sequences, and continuity of functions and maps usingthe distance function. These definitions can then be transferred to the more abstract spaceHom(Vm,Wn).

Let GL(Vm) ⊂ Hom(Vm,Vm) denote the space of invertible linear maps. For conveniencewe denote Hom(Rm,Rm) by gl(m) and GL(Rm) by GL(m).

Recall that there is a norm defined on Hom(Rm,Rm).

Theorem B.4. The following hold:

1. GL(V m) is an open subset of Hom(Vm,Vm).

2. The map F : GL(m)→ GL(m), where F (L)→ L−1 is C∞.

3. The differential of F is given by

dF (L)L = −L−1LL−1.

Proof. The determinant of a matrix is a continuous function of the matrix. It suffices toprove this for Vm = Rm. First, we show that there exists δ > 0 such that, for any C ∈ gl(m),

|C| < δ =⇒ I + C ∈ GL(m).

From this it follows that, given A ∈ GL(m), there exists δA > 0 such thatm for any B ∈gl(m),

|B| < δA =⇒ A+ b ∈ GL(m).

This shows that GL(m) is an open subset of gl(m).Recall that the inverse of A ∈ GL(m) is given by

A =1

detAAc,

where Ac is the cofactor matrix of A. Since both detA and the components of Ac arepolynomial function of the components of A and detA 6= 0 for every A ∈ GL(m), it followsthat

B.5. LOCAL VERSUS GLOBAL BEHAVIOR OF A 1-DIMENSIONAL NONLINEAR MAP87

B.5 Local versus global behavior of a 1-dimensionalnonlinear map

Compare the following maps F : R1 → R1:F0(x) = x

F1(x) = x3

F2(x) = x2

F3(x) = sin x

The maps F0 and F1 have inverse mapsF−10 (y) = y

F−11 (y) = y1/3,

but F1 is not C1 at x = 0. The map F2 and F3 have inverse maps, only if their domainsare suitably restricted: If F2 is F2 restricted to [0,∞), then it has the inverse map F−1

2 :[0,∞→ [0,∞), given by

F−12 (y) =

√y,

which is C1 if restricted to (0,∞). If F3 is F3 restricted to [−π2, π2], then it has the inverse

map F3 = arcsin, which is C1 if restricted to (−π2, π2).

We shall focus on determining whether a map is locally invertible and the inverse is C1.In other words, given x0 in the domain of a map F , is there an open set O containing x0such that F restricted to O has a C1 inverse map.

B.6 Normal form of C1 maps with maximal rankDefinition B.5. If O ⊂ Vm and F : O → Wn is a C1 map, then F has maximal rank atp ∈ O, if its differential at p ∈ O, dF (p) : Vm →Wn has maximal rank.

A C1 map F : O ⊂ Vm has maximal rank on O, if it has maximal rank at every p ∈ O.Definition B.6. If m ≥ n and F has maximal rank, then F is a submersion of O.

If m ≤ n and F has maximal rank, then F is an immersion of O.If m = n and F has maximal rank, then F is a local diffeomorphism of O.

Definition B.7. Let O ⊂ Rm be open. A map F : O → Rm is a C1 diffeomorphism, ifF (O) is open, F is C1, and there exists a C1 map G : F (O)→ O such that

G(F (x)) = x, ∀ x ∈ OF (G(y)) = y, ∀ y ∈ F (O).

In particular, if the map G exists, it is unique and the maps F and G are bijective.If F : O → Rm is a C1 map, it is a local diffeomorphism at x0 ∈ O, if there exists an

open neighborhood O′ ⊂ O of x0 such that F restricted to O′ is a C1 diffeomorphism.By the chain rule,

Lemma B.8. If F : O → Rm is a C1 diffeomorphism at x0 ∈ O, then the differentialdF (x) : Rm → Rm is invertible for any x ∈ O.


B.7 Contraction map lemmaWe state this for an arbitrary vector space with a norm with respect to which the space isa complete topological space. In other words, any Cauchy sequence has a limit.

Lemma B.9. Let V be a complete normed vector space and O ⊂ V an open subset. IfF : O → O is a continuous map such that there exists 0 ≤ c < 1 such that for anyx0, x1 ∈ O,

|F (x1)− F (x0)| ≤ c|x1 − x0|,

then there exists a unique x ∈ O such that F (x) = x.

Proof. Given x0 ∈ O, define a sequence as follows: For each k ≥ 0, let

xk+1 = F (xk).

It follows that, for each k ≥ 1,

|xk+1 − xk| = |F (xk)− F (xk−1)| ≤ c|xk − xk−1|.

From this, it follows by induction that, for any 0 ≤ k ≤ l,

|xl+1 − xl| ≤ cl−k|xk+1 − xk|.

Therefore,

|xl − xk| ≤ |xl − xl−1|+ · · ·+ |xk+1 − xk|

(cl−k + · · ·+ 1)|xk+1 − xk| ≤1− cl−k+1

1− c|xk+1 − xk|

≤ ck|x1 − x0|.

For ϵ < 0, let N > 0 satisfycN |x1 − x0| = ϵ.

Then for any N ≤ k ≤ l,

|xl − xk| ≤ ck|x1 − x0| ≤ cN |x1 − x0| ≤ ϵ.

It follows that (x1, · · · ) is a Cauchy sequence and has a limit x ∈ O. By the continuity of F

x = limk→∞

xk+1 = limk→∞

F (xk) = F (x).

Finally, if F (x1) = x1 and F (x2) = x2, then

|x2 − x1| = |F (x2)− F (x1)| ≤ c|x2 − x1|

which implies that(1− c)|x2 − x1| ≤ 0,

and therefore x2 = x1.

B.8. INVERSE FUNCTION THEOREM 89

B.8 Inverse function theoremThe following is a version of the mean value theorem.

Lemma B.10. If F : O → Rn is a C1 map and x0, x1 ∈ O are such that the line segmentjoining them also lies in O, then

F (x1)− F (x0) =∫ 1

0

dF ′((1− t)x0 + tx1)(x1 − x0) dt.

Proof.

F (x1)− F (x0) =∫ 1

0

d

dtF ((1− t)x0 + t1) dt

=

∫ 1

0

dF ′((1− t)x0 + tx1)(x1 − x0) dt.

The following is a converse to Lemma B.8:

Lemma B.11. Given A ∈ GL(m) and v ∈ Rm,

|Av| ≤ |A||v|.

Theorem B.12. If F : O → Rm is a C1 map and x0 ∈ O, then if dF (x0) : Rm → Rm isan invertible linear map, then there exists an open neighborhood O′ ⊂ O of x0 such that Frestricted to O′ is a C1 diffeomorphism. In particular, there F (O′) is open and there existsa C1 map G : F (O′)→ O such that

F (G(y)) = y, ∀ y ∈ F (O′) and G(F (x)) = x, ∀ x ∈ O′.

Proof. Let L : Rm → Rm be the inverse linear map to dF (x0) and

O′ = B(x0, δ),

whereδ =

1

2|L|−1.

Given y ∈ F (O′), ifΦ(x) = x− L(y − F (x)),

then y = F (x) if and only if Φ(x) = x, so it suffices to show that Φ : O′ → O′ is a contractionmapping. First, note that, for any x1, x2 ∈ O′,

Φ(x2)− Φ(x1) =

∫ 1

0

dF ((1− t)x2 + tx1)(x2 − x1) dt

x2 − x1 = LdF (x0)(x2 − x1)


and therefore

Φ(x2)− Φ(x1) = LdF (x0)(x2 − x1) + L−1(F (x2)− F (x1))

= L

∫ 1

0

(dF (x0)− dF ((1− t)x2 + tx1))(x2 − x1) dt

It follows that

|Φ(x2)− Φ(x1)| ≤ |L|1

2|L|−1|x2 − x1|

≤ 1

2|x2 − x1|.

B.9 Normal forms of nonlinear mapsDefinition B.13. Given an open set O ⊂ Rm and m ≤ n, a C1 map F : O → Rn is animmersion, if for any x ∈ O, the differential dF (x) : Rm → Rn is injective. This is equivalentto saying that m ≤ n and the rank of dF (x) is m.

If F is also injective, then it is an embedding.

Theorem B.14. If F : O → Rn is a C1 immersion, then for each x ∈ O, there exists an openneighborhood O′ ⊂ O of x and diffeomorphisms A : A−1(O′) → O′ and B : B−1(F (O′)) →F (O′) such that

F = B ◦ Ln,m ◦ A−1

Definition B.15. Given an open set O ⊂ Rn, a C1 map F : O → Rn−m is a submersion, iffor any x ∈ O, the differential dF (x) : Rn → Rn−m has rank n −m. This is equivalent todF (x) being surjective.

Theorem B.16. If F : O → Rn−m is a C1 submersion, then for each x ∈ O, thereexists an open neighborhood O′ ⊂ O of x and diffeomorphisms A : A−1(O′) → O′ andB : B−1(F (O′))→ F (O′) such that

F = B ◦Mn,m ◦ A−1

Appendix C

The differential of a function

C.1 The dual vector spaceDefinition C.1. The dual vector space of a vector space Vm is defined to be the space ofall linear functions of Vm,

(Vm)∗ = {ℓ : Vm → R : ℓ is linear}.

Observe that (Vm)∗ is indeed a vector space. If v1, . . . , vm is a basis of Vm, defineℓ1, . . . , ℓm ∈ V ∗ as follows:

ℓi(a1v1 + · · · amvm) = ai.

Equivalently,

ℓi(vj) = δji , ∀1 ≤ i, j ≤ m.

Lemma C.2. ℓ1, . . . , ℓm is a basis of (Vm)∗ and therefore dim(Vm)∗ = dimVm.

Lemma C.3. There is a natural isomorphism

Vm → (Vm)∗∗

that maps v ∈ Vm to the linear function on (Vm)∗ given by

ℓ 7→ ℓ(v).

Therefore, for any v ∈ Vm and ℓ ∈ (Vm)∗,

v(ℓ) = ℓ(v).

where on the left v is treated as an element of (Vm)∗∗ and on the right as an element of V .Because of this symmetry, we will denote this by

〈ℓ, v〉 = 〈v, ℓ〉 = ℓ(v).

91

92 APPENDIX C. THE DIFFERENTIAL OF A FUNCTION

C.2 Differentiability of a functionDefinition C.4. Let O ⊂ Am be open. A function f : O → R is differentiable at p ∈ O, ifthere exists a linear function ℓ : Vm → R (which depends on f and p) such that

limt→0

f(p+ tv)− f(p)− t〈ℓ, v〉t

= 0.

Equivalently, there exists a real-valued function ϵf (p, v, t) such that

f(p+ tv) = f(p) + t〈ℓ, v〉+ tϵ(p, v, t)

limt→0

ϵ(p, tv) = 0.

C.3 Directional derivative of a functionDefinition C.5. Given f : O → R, p ∈ O, and v ∈ Vm, the directional derivative of f at pin the direction v is, if it exists, defined to be

dvf(p) =d

dt

∣∣∣∣t=0

f(p+ tv) = limt→0

f(p+ tv)− f(p)t

.

In this definition the function v 7→ dvf(p) is not assumed to be linear. In fact,

Lemma C.6. The function f : O → R is differentiable at p ∈ O if and only if the functionv 7→ dvf(p) exists for all v ∈ Vm and is linear.

C.4 C1 functionsDefinition C.7. A function f : O → R is C1 on O, if, for each v ∈ Vm,

dvf : O → Rp 7→ dvf(p)

is a continuous function on O.

Lemma C.8. A C1 function on O ⊂ Am is differentiable on O.

Since, if f is C1, v 7→ dvf(p) ∈ R is linear, we can define df(p) ∈ (Vm)∗ to be the linearfunction, where

〈df(p), v〉 = dvf(p).

This in turn defines a map df : O → (Vm)∗, which is called the differential of f .

C.5 Differential 1-form on an open subset of Am

Definition C.9. Given an open set O ⊂ Am, a differential 1-form is a map ω : O → V∗.

Example. The differential of a function is a 1-form.

C.6. LINE INTEGRAL OF A 1-FORM ALONG A CURVE IN AM 93

C.6 Line integral of a 1-form along a curve in Am

Definition C.10. If O ⊂ Am is open, c : [0, T ] → O is a C1 curve, and ω : O → (Vm)∗ a1-form, then the line integral of ω along c is defined to be∫

c

ω =

∫ T

0

〈ω(c(t)), c′(t)〉 dt.

Observe that the integrand on the right is a real-valued function of t ∈ [0, T ] and thereforethe integral is standard single variable integral.

By using the same proof of the chain rule for functions of a single variable, the followingholds:

Lemma C.11. If c : I → Am is a C1 curve and ϕ : I ′ → I is a C1 function on the intervalI ′, then

(c ◦ ϕ)′(t) = c′(ϕ(t))ϕ′(t).

The same proof for the change of variables for a single variable integral now implies:

Theorem C.12. If O ⊂ Am is open, c : [0, T ] → O is a C1 curve, and ω : O → (Vm)∗ a1-form, then for any C1 function ϕ : [0, T ]→ [0, T ] such that ϕ(0) = 0 and ϕ(T ) = T ,∫

c

ω =

∫c

ω,

where c = c ◦ ϕ.

The fundamental theorem of calculus implies the following:

Theorem C.13. If c : [0, T ]→ Am is a C1 curve and f : O → R a C1 function on an openset O ⊂ Am, then ∫

c

df = f(c(T ))− f(c(0)).

In particular, the line integral of df from p to q does not depend on which C1 curve isused to connect p to q.

C.7 Fundamental exampleRecall that, associated with an affine basis (p0, . . . , pm) of Am is a basis E = (e− 1, . . . , em)of Vm, where

e1 = p1 − p0, . . . , em = pm − p0.

On the other hand, recall that I−1E (p) = (x1, . . . , xm), wherefor each 1 ≤ i ≤ m, t

xi : Am → Rp0 + v 7→ vi.

94 APPENDIX C. THE DIFFERENTIAL OF A FUNCTION

The differential of xi is defined to be dxi ∈ (Vm)∗, where

〈dxi(p), v〉 = d

dt

∣∣∣∣t=0

xi(p+ tv) = vi.

Equivalently,〈dxi, ej〉 = δij.

and, therefore, (dx1, . . . , dxm) is the basis of (Vm)∗ dual to the basis E = (e1, . . . , em) of Vm.Given a C1 function f : O → R, note that, if p0 ∈ O and IE(x

1, . . . , xn) = p, then

∂i((f ◦ IE)(x1, . . . , xn)) =d

dt

∣∣∣∣t=0

f(p+ tei)

= 〈df(p), ei〉.

‘Therefore,df(p) = 〈df(p), ei〉dxi.

This leads to the following notation: Given an affine basis P = (p0, . . . , pm) of Am,instead of denoting the corresponding basis of Vm by E = (e1, . . . , em), we denote it by∂ = (∂1, . . . , ∂m), where, for each i = 1, . . . ,m,

∂i = pi − p0.

As noted above, its dual basis is dx = (dx1, . . . , dxm). Given a C1 function f : O → R, wedenote

∂if(p) = 〈df(p), ∂i〉.

Therefore, omitting the p for brevity,

df = ∂if dxi.

Appendix D

Differential of a map

D.1 Characterization of a linear mapLemma D.1. Given a map F : Vm →Wn, the following are equivalent:

• F is linear.

• ℓ ◦ F : Vm → R is linear for any ℓ ∈ (Wn)∗.

• Given a basis ℓ1, . . . , ℓn of Wn, ℓi ◦ F is linear for each i = 1, . . . , n.

D.2 Norm of a linear mapRecall that given bases E of Vm and F of Wn, then there is a linear isomorphism IE,F :M(n,m)→ Hom(Vm,Wn). Moreover, we can define a dot product on M(n,m) given by

A ·B =m∑i=1

n∑j=1

AijB

ij

with the corresponding norm given by |A|2 = A · A. This is the same norm as if we simplyflattened each matrix into an mn-dimensional vector.

D.3 Differential of a mapIf Am and Bn are affine spaces with tangent spaces Vm and Wn respectively, O is an opensubset of Am, then a map F : O → Bn is differentiable at p, if there exists a linear mapL : Vm →Wn such that, for any v ∈ Vm,

limt→0

F (p+ tv)− F (p)− tL(v)t

= 0.

If F is differentiable at p, then the map L : Vm →Wn is called the differential of F at p anddenoted dFp. If F is differentiable for every p ∈ O and the map

dF : O → Hom(Vm,Wn)

is continuous, then the map F is C1.

95

96 APPENDIX D. DIFFERENTIAL OF A MAP

D.4 Chain RuleLemma D.2. If F : Am → Bn and G : Bn → C are C1 maps, then so is G ◦ F , and

d(G ◦ F )(p) = dG(F (p)) ◦ dF (p).

Appendix E

Topology

E.1 Topology of Rn

The topology of Rn is described using the distance function and the concepts of open andclosed sets. This is used to define the convergence of sequences and functions, which in turnare used to define differentiation and integration.

E.1.1 Dot product and norm of a vectorRecall that the dot product of v = (v1, . . . , vn), w = (w1, . . . , wn) ∈ Rn is defined to be

v · w = v1w1 + · · ·+ vnwn

and the norm of v ∈ Rn is |v| ≥ 0, where

|v|2 = v · v.

Lemma E.1. For any vectors v, w ∈ Rn,

|v − w|2 ≤ |v|2 + |w|2,

with equality holding if and only if w = tv, for some t ≥ 0.

E.1.2 Distance between two pointsRecall that the distance between two points p, q ∈ Rn is defined to be d(p, q) ≥ 0, where

d(p, q)2 = |p− q|2 = (p− q) · (p− q).

The fundamental properties of the (or any) distance function are the following: For anypoints p, q, r ∈ Rn,

d(p, q) = d(q, p)

d(p, q) ≥ 0

d(p, q) = 0 ⇐⇒ p = q

d(p, r) ≤ d(p, q) + d(q, r),

97

98 APPENDIX E. TOPOLOGY

Denote the ball of radius r > 0 centered at p ∈ Rn by

B(p, r) = {x ∈ Rn : d(p, x) < r}

and the closed ball byB(p, r) = {x ∈ Rn : d(p, x) ≤ r}

E.1.3 Open and closed setsDefinition E.2. A set O ⊂ Rn is open, if for any p ∈ O, there exists r > 0 such thatB(p, r) ⊂ O. A set C ⊂ Rn is closed, if Rn\C is open.

E.1.4 Convergent and Cauchy sequencesDefinition E.3. A sequence is a map s : Z+ → Rn.

Definition E.4. A sequence s1, . . . converges to the limit s∞ ∈ Rn, if for any ϵ > 0, thereexists N(ϵ) such that

d(si, s∞) < ϵ, ∀ i ≥ N(ϵ).

Definition E.5. A sequence is a Cauchy sequence, if for any ϵ > 0, there exists N(ϵ) suchthat

d(si, sj) < ϵ, ∀ i, j ≥ N(ϵ).

Lemma E.6. A sequence converges to a limit if and only if it is Cauchy.

E.1.5 Compact setsDefinition E.7. Let S ⊂ Rn.

• S is bounded, if there exists R > 0 such that S ⊂ B(0, R).

• S is sequentially compact, if any sequence has a convergent subsequence.

• S is compact, if, given any infinite collection of open sets O1, . . . such that

S ⊂⋃i

Oi,

there exists a finite subcollection O1, . . . , ON such that

S ⊂ O1 ∪ · · ·ON .

Lemma E.8. The following are equvialent for a set S ⊂ Rn:

1. S is compact.

2. S is sequentially compact.

3. S is closed and bounded.

E.2. TOPOLOGY OF AFFINE SPACE 99

E.2 Topology of affine spaceIt is less convenient to define directly the topology of affine space, because there is no naturaldefinition of distance.

Instead, we do it indirectly by the topology of Rm. If E = (e0, . . . , em) is an affine basisof Am, then we can define a distance function dE on Am using the distance function on Rm

as follows: For each p, q ∈ Am,

dE(p, q) = d(IE(p), IE(q)).

Although each affine basis defines a distance function, they are all equivalent in the followingsense:

Lemma E.9. If E and F are both affine bases of Am, then there exists a, b > 0 such thatfor any p, q ∈ Am,

adE(p, q) ≤ dF (p, q) ≤ bdE(p, q).

This implies that a set in Am is open with respect to the distance function dE if and onlyif it is open with respect to dF . It also implies the following:

Lemma E.10. A sequence p1, p2, · · · ∈ Am converges with respect to the distance functiondE if and only if it converges with respect to the distance function dF , and the limit is thesame in both cases.

We will follow the following principle in what follows: When working on affine space,do as much as possible abstractly without relying an affine basis or even an origin (whichdefines an isomorphism of the affine space with its vector space). Even analysis argumentsthat use the topology of affine space but do not require any explicit use of the distancefunction can be done abstractly. However, when an analysis argument requires the use of adistance function, use the distance function associated with an affine basis as defined above.

100 APPENDIX E. TOPOLOGY

Appendix F

Maps

F.1 Maps on Rm

F.1.1 LimitsDefinition F.1. Given an open set O ⊂ Rm, a map f : O → Rn, x0 ∈ O, and L ∈ Rn,

limx→x0

f(x) = L,

if for any ϵ > 0, there exists δ > 0 such that

d(x0, x) < δ =⇒ d(f(x0), f(x)).

If n = 1, we call f a map.

F.1.2 BoundednessA function f : D → R is bounded from above, if there exists U ∈ R such that

f(x) ≤ U, ∀x ∈ D.

Any U satisfying this is an upper bound. A least upper bound is an upper bound U0 ∈ R suchthat U0 ≤ U for any upper bound U .

Lemma F.2. If a function f : D → R is bounded from above, then there exists a least upperbound of f on D.

A function f : D → R is bounded from below, if there exists L ∈ R such that

L ≤ f(x), ∀x ∈ D.

Any U satisfying this is a lower bound. A greatest lower bound is an upper bound L0 ∈ Rsuch that L0 ≥ L for any lower bound L.

A function is bounded, if it is bounded from both above and below.The completeness of the real line implies the following:

101

102 APPENDIX F. MAPS

Lemma F.3. Any function bounded from above has a unique least upper bound. Any functionbounded from below has a unique greatest lower bound.

We can therefore denote the least upper bound of f by

supx∈D

f(x),

and its greatest lower bound byinfx∈D

f(x),

F.1.3 ContinuityGiven an open set O ⊂ Rm, a map f : O → Rn is continuous at x0 ∈ O, if

limx→x0

f(x) = f(x0).

A map f : O → Rn is continuous at x0 ∈ O, if

limx→x0

f(x) = f(x0).

A map f : O → Rn is continuous, if it is continuous at every x0 ∈ O.

Lemma F.4. A map f : O → Rn is continuous if and only if, for any open set S ⊂ Rn,f−1(S) is an open subset of O.

Lemma F.5. If D ⊂ Rm is compact and f : D → Rn is continuous, then it is bounded.Moreover, there exist p, q ∈ D such that

f(p) = infx∈D

f(x) and f(q) = supx∈D

f(x).

F.2 Maps on affine spaceLet P be an affine basis of the affine space Am, IP : Rm → Am be the affine isomorphismdefined by P . Let Q be an affine basis of the affine space Bn and IQ : Rn → Bn be the affineisomorphism defined by Q. Let O an open subset of Am.

Definition F.6. Given a map f : O → Bn, p0 ∈ O, and b ∈ Bn,

limp→p0

f(p) = b

if and only iflimx→x0

fP,Q(x) = bQ,

where x0 = I−1P (p0) and fP,Q = I−1

Q ◦ f ◦ IP (x), and bQ = I−1Q (b).

Definition F.7. A map f : O → Bn is continuous if I−1Q ◦f ◦IP : I−1

E (O)→ Rn is continuous.

F.3. LEMMAS 103

F.3 LemmasLemma F.8. If O ⊂ Am is open, f, g : O → R are C1 functions such that, for each p ∈ O,

f(p) = 0 =⇒ ∂f(p) 6= 0 and g(p) = 0,

then there exists a C1 function ϕ : O → R such that g = ϕf . If f and g are C2, then so is ϕ.

F.4 Partition of unityLet M be an m-dimensional C2 manifold. Therefore, there exists a countable locally finitecovering of M by coordinate charts xi : Oi → B(0, 1) ⊂ Rm. In particular, each xi is bijectiveand, for any i and j, xj ◦ x−1

i : xi(Oi ∩Oj)→ xj(Oi ∩Oj) is a C2 diffeomorphism.Let ϕ : B(0, 1)→ [0,∞) be given by

ϕ(x) =1 + cos(π|x|)

2.

Let ϕi :M → [0,∞) be the C2 function given by

ψi(p) =

{ϕ(xi(p)) , if p ∈ Oi

0 = , otherwise

Observe thatψ =

∑i

ψi

is strictly positive and therefore, the functions

χi =ψi

ψ

are well-defined C2 functions on M such that∑i

χi = 1.

104 APPENDIX F. MAPS

Bibliography

[1] John Stillwell, The four pillars of geometry, Undergraduate Texts in Mathematics,Springer, New York, 2005.

[2] Wikipedia, Euclidean geometry, 2019.

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