MATH 38Mathematical Analysis III

I. F. Evidente

IMSP (UPLB)

Unit 3

Differentiation of Functions of Two or More Variables1 Limits and Continuity2 Partial Derivatives, Differentiability3 Higher Order Derivatives, The Chain Rule4 Local Linearity and Differentials5 Relative and Absolute Extrema of Functions of Two Variables

Outline

1 Functions of Two or More VariablesBasic IdeasGraphs of Functions of Two or More Variables

2 Limits of Functions of Two or More VariablesLimits Along PathsIntuitive Notion of LimitsBasic Limit Theorems

Outline

1 Functions of Two or More VariablesBasic IdeasGraphs of Functions of Two or More Variables

2 Limits of Functions of Two or More VariablesLimits Along PathsIntuitive Notion of LimitsBasic Limit Theorems

RecallRR=R2 = {(x, y) | x R and y R}Geometrically, R2 is associated with two-dimensional Euclidean space(Euclidean 2-space)

DefinitionA function of two variables f (x, y) is a rule that assigns a unique realnumber to each point (x, y) in some subset D of R2.

Examples

Example1 f (x, y)= 63x2y2 f (x, y)= 4x2+ y23 f (x, y)=

9x2 y2

Evaluation

Example1 f (x, y)= 63x2y .

1 Find f (0,2)2 Find f (x+x, y +y)

2 f (x, y)= 4x2+ y2.1 Find f (

p2,1).

2 Find f (x+x, y)

RemarkThe set D is called the domain of f , which we denote by dom f .

If no set D is specified, then D is the natural domain of f : thelargest subset of R2 containing all points (x, y) for which f (x, y) isdefined.The set of all possible values of f (x, y) is called the range of f and isdenoted by ran f .

RemarkThe set D is called the domain of f , which we denote by dom f .If no set D is specified, then D is the natural domain of f :

thelargest subset of R2 containing all points (x, y) for which f (x, y) isdefined.The set of all possible values of f (x, y) is called the range of f and isdenoted by ran f .

RemarkThe set D is called the domain of f , which we denote by dom f .If no set D is specified, then D is the natural domain of f : thelargest subset of R2 containing all points (x, y) for which f (x, y) isdefined.

The set of all possible values of f (x, y) is called the range of f and isdenoted by ran f .

RemarkThe set D is called the domain of f , which we denote by dom f .If no set D is specified, then D is the natural domain of f : thelargest subset of R2 containing all points (x, y) for which f (x, y) isdefined.The set of all possible values of f (x, y) is called the range of f and isdenoted by ran f .

Examples

ExampleFind and sketch the domain of the following:

1 f (x, y)= x2x+ y

2 f (x, y)= 4x2+ y23 f (x, y)=

9x2 y2

RemarkThe graph of f (x, y) is the surface in R3 (Euclidean 3-space) satisfyingthe equation z = f (x, y).

ExampleDetermine the graph of the following:

1 f (x, y)= 63x2y2 f (x, y)= 4x2+ y23 f (x, y)=

9x2 y2

Examplef (x, y)= 63x2y

Examplef (x, y)= 4x2+ y2

Example

f (x, y)=9x2 y2

RemarkLet pi be a plane and a surface in R3. Let C be the curve of intersectionof pi and . We call C the trace of on the plane pi.

Example1 Determine the trace of the graph of f (x, y)= 4x2+ y2 on the plane

z = 4, the yz-plane and the xy-plane.2 Determine the trace of the graph of f (x, y)=

9x2 y2 on the

plane z = 0, the xz-plane and the xy-plane.

RemarkLet S be a set of points in R3. The projection of S on the xy-plane isobtained by "dropping" all points of S orthogonally on the xy-plane. Wedenote the projection of S on xy-plane by Projxy (S).

Example

Find Projxy (S) for the following:1 S = {(1,1,1), (1,0,2), (0,1,1)}2 S is the graph of f (x, y)=

9x2 y2

3 S is the graph of f (x, y)= 4x2+ y2

RemarkSuppose is the graph of f . Geometrically:

The domain of f is Projxy ()

The range of f is the "projection" of onto the z-axis.

RemarkSuppose is the graph of f . Geometrically:

The domain of f is Projxy ()The range of f is the "projection" of onto the z-axis.

Example

Use the graph of f to determine its domain and range:1 f (x, y)= 4x2+ y22 f (x, y)=

9x2 y2

Examplef (x, y)= 4x2+ y2

Example

f (x, y)=9x2 y2

Contour MapsLet f (x, y) be a function of two variables with graph given by z = f (x, y).

Let Ck be the trace of the graph of f (x, y) on the plane z = kThe level curve of height k is Projxy (Ck )A set of level curves for various values of k is called a contour mapof fWe get an idea about the graph of f from its contour map.

Raise the level curve of height k to z = k.The "steepness" of the graph can be gleaned from the contour map.

Contour MapsLet f (x, y) be a function of two variables with graph given by z = f (x, y).

Let Ck be the trace of the graph of f (x, y) on the plane z = k

The level curve of height k is Projxy (Ck )A set of level curves for various values of k is called a contour mapof fWe get an idea about the graph of f from its contour map.

Raise the level curve of height k to z = k.The "steepness" of the graph can be gleaned from the contour map.

Contour MapsLet f (x, y) be a function of two variables with graph given by z = f (x, y).

Let Ck be the trace of the graph of f (x, y) on the plane z = kThe level curve of height k is Projxy (Ck )

A set of level curves for various values of k is called a contour mapof fWe get an idea about the graph of f from its contour map.

Raise the level curve of height k to z = k.The "steepness" of the graph can be gleaned from the contour map.

Contour MapsLet f (x, y) be a function of two variables with graph given by z = f (x, y).

Let Ck be the trace of the graph of f (x, y) on the plane z = kThe level curve of height k is Projxy (Ck )A set of level curves for various values of k is called a contour mapof f

We get an idea about the graph of f from its contour map.Raise the level curve of height k to z = k.The "steepness" of the graph can be gleaned from the contour map.

Contour MapsLet f (x, y) be a function of two variables with graph given by z = f (x, y).

Let Ck be the trace of the graph of f (x, y) on the plane z = kThe level curve of height k is Projxy (Ck )A set of level curves for various values of k is called a contour mapof fWe get an idea about the graph of f from its contour map.

Raise the level curve of height k to z = k.The "steepness" of the graph can be gleaned from the contour map.

Example

Draw a contour map for:1 f (x, y)= 4x2+ y22 f (x, y)=

9x2 y2

f (x, y)= 4x2+ y2

f (x, y)=9x2 y2

RemarkAll the above notions can be extended to functions of 3 or more variables.In particular, we are also interested in functions of 3 variables.

Outline

1 Functions of Two or More VariablesBasic IdeasGraphs of Functions of Two or More Variables

2 Limits of Functions of Two or More VariablesLimits Along PathsIntuitive Notion of LimitsBasic Limit Theorems

Limits Along PathsLet f (x, y) be a function of two variables.

Let C be a smooth curve on thexy-plane with equation y = g (x) whose points are in the domain of f . Let(x0, y0) be a point on C . The limit of f (x, y) as (x, y) approaches thepoint (x0, y0) along the curve C is:

lim(x,y)(x0,y0)

f (x, y)= limxx0

f (x,g (x))

In the case that C has equation x = g (y), then

lim(x,y)(x0,y0)

f (x, y)= limxy0

f (g (y), y)

Limits Along PathsLet f (x, y) be a function of two variables. Let C be a smooth curve on thexy-plane with equation y = g (x) whose points are in the domain of f .

Let(x0, y0) be a point on C . The limit of f (x, y) as (x, y) approaches thepoint (x0, y0) along the curve C is:

lim(x,y)(x0,y0)

f (x, y)= limxx0

f (x,g (x))

In the case that C has equation x = g (y), then

lim(x,y)(x0,y0)

f (x, y)= limxy0

f (g (y), y)

Limits Along PathsLet f (x, y) be a function of two variables. Let C be a smooth curve on thexy-plane with equation y = g (x) whose points are in the domain of f . Let(x0, y0) be a point on C .

The limit of f (x, y) as (x, y) approaches thepoint (x0, y0) along the curve C is:

lim(x,y)(x0,y0)

f (x, y)= limxx0

f (x,g (x))

In the case that C has equation x = g (y), then

lim(x,y)(x0,y0)

f (x, y)= limxy0

f (g (y), y)

Limits Along PathsLet f (x, y) be a function of two variables. Let C be a smooth curve on thexy-plane with equation y = g (x) whose points are in the domain of f . Let(x0, y0) be a point on C . The limit of f (x, y) as (x, y) approaches thepoint (x0, y0) along the curve C is:

lim(x,y)(x0,y0)

f (x, y)= limxx0

f (x,g (x))

In the case that C has equation x = g (y), then

lim(x,y)(x0,y0)

f (x, y)= limxy0

f (g (y), y)

Limits Along PathsLet f (x, y) be a function of two variables. Let C be a smooth curve on thexy-plane with equation y = g (x) whose points are in the domain of f . Let(x0, y0) be a point on C . The limit of f (x, y) as (x, y) approaches thepoint (x0, y0) along the curve C is:

lim(x,y)(x0,y0)

f (x, y)= limxx0

f (x,g (x))

In the case that C has equation x = g (y), then

lim(x,y)(x0,y0)

f (x, y)= limxy0

f (g (y), y)

Examples

Example

1 Find lim(x,y)(0,0)

x2

x2+ y2 along the path y = 0, x = 0 and y = x.

2 Find lim(x,y)(0,0)

x9y

(x6+ y2)2 along the path y = x and y = x2.

RecallFor functions of one variable:

There two ways of approaching x0: from the left and from the rightConcept of one-sided limits: only two typesSince the limit is the unique value that the y-values approach as thex-values approach x0 from both sides, lim

xa f (x) exists if and only iflimxa+

f (x)= limxa f (x)

RecallFor functions of one variable:

There two ways of approaching x0: from the left and from the right

Concept of one-sided limits: only two typesSince the limit is the unique value that the y-values approach as thex-values approach x0 from both sides, lim

xa f (x) exists if and only iflimxa+

f (x)= limxa f (x)

RecallFor functions of one variable:

There two ways of approaching x0: from the left and from the rightConcept of one-sided limits: only two types

Since the limit is the unique value that the y-values approach as thex-values approach x0 from both sides, lim

xa f (x) exists if and only iflimxa+

f (x)= limxa f (x)

RecallFor functions of one variable:

There two ways of approaching x0: from the left and from the rightConcept of one-sided limits: only two typesSince the limit is the unique value that the y-values approach as thex-values approach x0 from both sides, lim

xa f (x) exists if and only iflimxa+

f (x)= limxa f (x)

For functions of two variables:

Intuitive Notion of LimitsFor functions of two variables:

The point (x, y) may approach (x0, y0) via an infinite number of paths.Intuitively, the limit of f (x, y) as (x, y) approaches (x0, y0) is theunique value that f (x, y) approaches as (x, y) approaches (x0, y0) viaall these possible paths!This makes the computation of limits of functions of two variablesmuch more complex.We restrict our computation of limits to very simple types offunctions: polynomial and rational functions.

Intuitive Notion of LimitsFor functions of two variables:

The point (x, y) may approach (x0, y0) via an infinite number of paths.

Intuitively, the limit of f (x, y) as (x, y) approaches (x0, y0) is theunique value that f (x, y) approaches as (x, y) approaches (x0, y0) viaall these possible paths!This makes the computation of limits of functions of two variablesmuch more complex.We restrict our computation of limits to very simple types offunctions: polynomial and rational functions.

Intuitive Notion of LimitsFor functions of two variables:

The point (x, y) may approach (x0, y0) via an infinite number of paths.Intuitively, the limit of f (x, y) as (x, y) approaches (x0, y0) is theunique value that f (x, y) approaches as (x, y) approaches (x0, y0) viaall these possible paths!

This makes the computation of limits of functions of two variablesmuch more complex.We restrict our computation of limits to very simple types offunctions: polynomial and rational functions.

Intuitive Notion of LimitsFor functions of two variables:

The point (x, y) may approach (x0, y0) via an infinite number of paths.Intuitively, the limit of f (x, y) as (x, y) approaches (x0, y0) is theunique value that f (x, y) approaches as (x, y) approaches (x0, y0) viaall these possible paths!This makes the computation of limits of functions of two variablesmuch more complex.

We restrict our computation of limits to very simple types offunctions: polynomial and rational functions.

Intuitive Notion of LimitsFor functions of two variables:

The point (x, y) may approach (x0, y0) via an infinite number of paths.Intuitively, the limit of f (x, y) as (x, y) approaches (x0, y0) is theunique value that f (x, y) approaches as (x, y) approaches (x0, y0) viaall these possible paths!This makes the computation of limits of functions of two variablesmuch more complex.We restrict our computation of limits to very simple types offunctions: polynomial and rational functions.

Theorem1 lim

(x,y)(x0,y0)c = c

2 If f (x, y) is a polyomial function, then lim(x,y)(x0,y0)

f (x, y)= f (x0, y0)

Examples

1 lim(x,y)(1,2)

2x3y =

2 13 2= 22 lim

(x,y)(0,0)1000= 1000

Examples

1 lim(x,y)(1,2)

2x3y = 2 13 2=

22 lim

(x,y)(0,0)1000= 1000

Examples

1 lim(x,y)(1,2)

2x3y = 2 13 2= 2

2 lim(x,y)(0,0)

1000= 1000

Examples

1 lim(x,y)(1,2)

2x3y = 2 13 2= 22 lim

(x,y)(0,0)1000=

1000

Examples

1 lim(x,y)(1,2)

2x3y = 2 13 2= 22 lim

(x,y)(0,0)1000= 1000

Theorem (Properties of Limits)If lim

(x,y)(x0,y0)f (x, y)= L and lim

(x,y)(x0,y0)g (x, y)=M , then

1 lim(x,y)(x0,y0)

f (x, y) g (x, y)= LM2 If c R, then lim

(x,y)(x0,y0)c f (x, y)= c L.

3 lim(x,y)(x0,y0)

f (x, y) g (x, y)= L M .

4 If M 6= 0, then lim(x,y)(x0,y0)

f (x, y)

g (x, y)= LM

.

Remark

As a consequence of the last two theorems, if h(x, y)= f (x, y)g (x, y)

is a rational

function where f and g are polynomial functions,

lim(x,y)(x0,y0)

h(x, y)= f (x0, y0)g (x0, y0)

provided that g (x0, y0) 6= 0.

Examples

1 lim(x,y)(0,0)

x22y +2x y +1 =

2

1= 2

2 lim(x,y)(0,0)

x2

x2+ y2 =theorem does not apply!!!

Examples

1 lim(x,y)(0,0)

x22y +2x y +1 =

2

1= 2

2 lim(x,y)(0,0)

x2

x2+ y2 =

theorem does not apply!!!

Examples

1 lim(x,y)(0,0)

x22y +2x y +1 =

2

1= 2

2 lim(x,y)(0,0)

x2

x2+ y2 =theorem does not apply!!!

Announcements

Midterm Exam: February 3, Monday, 7-9 AM, MBLH, conflict need tosign up with respective recit teachersScope of Midterm: until todays lesson

No recitation class tomorrowMath, Physics (Materials Science) and Chem Majors are invited to theIMSP Research Colloquium

Tilings, Quasicrystals and PinwheelsSpeaker: Dr. Dirk Frettleoh from Bielefeld University, Germany

To celebrate the International Year of Crystallography

Take home SW 3-1 due on February 5, Wednesday (will be uploadedtomorrow)Get Chapter 1 and 2 SW bluebook from your recitation teachers,Chapter 2 Quiz (if available)

Announcements

Midterm Exam: February 3, Monday, 7-9 AM, MBLH, conflict need tosign up with respective recit teachersScope of Midterm: until todays lessonNo recitation class tomorrow

Math, Physics (Materials Science) and Chem Majors are invited to theIMSP Research Colloquium

Tilings, Quasicrystals and PinwheelsSpeaker: Dr. Dirk Frettleoh from Bielefeld University, Germany

To celebrate the International Year of Crystallography

Take home SW 3-1 due on February 5, Wednesday (will be uploadedtomorrow)Get Chapter 1 and 2 SW bluebook from your recitation teachers,Chapter 2 Quiz (if available)

Announcements

Midterm Exam: February 3, Monday, 7-9 AM, MBLH, conflict need tosign up with respective recit teachersScope of Midterm: until todays lessonNo recitation class tomorrowMath, Physics (Materials Science) and Chem Majors are invited to theIMSP Research Colloquium

Tilings, Quasicrystals and PinwheelsSpeaker: Dr. Dirk Frettleoh from Bielefeld University, Germany

To celebrate the International Year of Crystallography

Take home SW 3-1 due on February 5, Wednesday (will be uploadedtomorrow)Get Chapter 1 and 2 SW bluebook from your recitation teachers,Chapter 2 Quiz (if available)

Announcements

Midterm Exam: February 3, Monday, 7-9 AM, MBLH, conflict need tosign up with respective recit teachersScope of Midterm: until todays lessonNo recitation class tomorrowMath, Physics (Materials Science) and Chem Majors are invited to theIMSP Research Colloquium

Tilings, Quasicrystals and PinwheelsSpeaker: Dr. Dirk Frettleoh from Bielefeld University, Germany

To celebrate the International Year of Crystallography

Take home SW 3-1 due on February 5, Wednesday (will be uploadedtomorrow)Get Chapter 1 and 2 SW bluebook from your recitation teachers,Chapter 2 Quiz (if available)

Functions of Two or More VariablesBasic IdeasGraphs of Functions of Two or More Variables

Limits of Functions of Two or More VariablesLimits Along PathsIntuitive Notion of LimitsBasic Limit Theorems