Math 241: Multivariable calculus, Lecture 1nirobles/files241/lecture01.pdf · Calculus of 1...

Math 241: Multivariable calculus, Lecture 1Introduction, Rn, vectors.

Sections 12.1, 12.2

go.illinois.edu/math241fa17

Monday, August 28th, 2017

go.illinois.edu/math241fa17.

Components of the course

• Lectures will be Monday, Wednesday, Friday in 314 AltgeldHall.

• Discussion sections Tuesday and Thursday.

• Worksheets.• Quiz

• Information about the course will be on the Wiki:go.illinois.edu/math241fa17.

• Webassign homework,https://www.webassign.net/uiuc/login.html

• Piazza, https://piazza.com

• Tutoring room

• Moodle for the grades

• Exams and Final

• Discussion sections Tuesday and Thursday.

• Worksheets.• Quiz

• Tutoring room

• Exams and Final

• Discussion sections Tuesday and Thursday.• Worksheets.

• Quiz

• Tutoring room

• Exams and Final

• Discussion sections Tuesday and Thursday.• Worksheets.• Quiz

• Tutoring room

• Exams and Final

• Tutoring room

• Exams and Final

• Tutoring room

• Exams and Final

• Tutoring room

• Exams and Final

• Tutoring room

• Exams and Final

• Tutoring room

• Exams and Final

• Tutoring room

• Exams and Final

Calculus of 1 variable

In Calculus I and II you study real valued functions

y = f (x)

of a single real variable, x .

Examples:

• f (x) = x2, r(x) = 2x2+xx3−5x+20

, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...

• T (t) = temperature in Champaign-Urbana, t hours aftermidnight on August 27.

• ρ(d) = density of a piece of wire at distance d from one end.

y = f (x)

Examples:

• f (x) = x2, r(x) = 2x2+xx3−5x+20

, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...

y = f (x)

Examples:

• f (x) = x2, r(x) = 2x2+xx3−5x+20

, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...

y = f (x)

Examples:

• f (x) = x2,

r(x) = 2x2+xx3−5x+20

, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...

y = f (x)

Examples:

• f (x) = x2,

r(x) = 2x2+xx3−5x+20

, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...

y = f (x)

Examples:

• f (x) = x2, r(x) = 2x2+xx3−5x+20

h(θ) = sin(θ) + cos(2θ),g(u) = eu,...

y = f (x)

Examples:

• f (x) = x2, r(x) = 2x2+xx3−5x+20

, h(θ) = sin(θ) + cos(2θ),

g(u) = eu,...

y = f (x)

Examples:

• f (x) = x2, r(x) = 2x2+xx3−5x+20

, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...

y = f (x)

Examples:

• f (x) = x2, r(x) = 2x2+xx3−5x+20

, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...

y = f (x)

Examples:

• f (x) = x2, r(x) = 2x2+xx3−5x+20

, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...

Three key concepts from Calculus I, II.

f (x), a function of one variable.

1 The derivative: f ′(x) = dfdx = d

dx f (x) = dydx .

• Rate of change.• Slope of the tangent line to the graph.

2 The integral:∫ ba f (x) dx .

• Signed area under graph.

• Average value 1b−a

af (x) dx .

3 Fundamental Theorem of Calculus: Relates the two.

• f (b)− f (a) =∫ b

af ′(x) dx .

dx f (x) = dydx .

af (x) dx .

• f (b)− f (a) =∫ b

af ′(x) dx .

dx f (x) = dydx .

af (x) dx .

• f (b)− f (a) =∫ b

af ′(x) dx .

dx f (x) = dydx .

• Rate of change.

• Slope of the tangent line to the graph.

af (x) dx .

• f (b)− f (a) =∫ b

af ′(x) dx .

dx f (x) = dydx .

af (x) dx .

• f (b)− f (a) =∫ b

af ′(x) dx .

dx f (x) = dydx .

af (x) dx .

• f (b)− f (a) =∫ b

af ′(x) dx .

dx f (x) = dydx .

af (x) dx .

• f (b)− f (a) =∫ b

af ′(x) dx .

dx f (x) = dydx .

af (x) dx .

• f (b)− f (a) =∫ b

af ′(x) dx .

dx f (x) = dydx .

af (x) dx .

• f (b)− f (a) =∫ b

af ′(x) dx .

dx f (x) = dydx .

af (x) dx .

• f (b)− f (a) =∫ b

af ′(x) dx .

1 variable is too limiting

Functions of a single variable are insufficient for modeling morecomplicated situations.

Examples:

• The temperature depends on location as well as time. Needto specify location, e.g. by latitude x and longitude y , andtime, e.g. t hours after midnight:

T (x , y , t) = temperature at time t in location (x , y).

• Density of a flat sheet of metal can depends on the point inthe sheet, specified by x and y coordinates

δ(x , y) = density of point at (x , y) in a sheet of metal

Examples:

• The temperature depends on location as well as time.

Needto specify location, e.g. by latitude x and longitude y , andtime, e.g. t hours after midnight:

Examples:

• The temperature depends on location as well as time.

Needto specify location, e.g. by latitude x and longitude y , andtime, e.g. t hours after midnight:

Examples:

The setting: n–dimensional space, Rn.

• R1 = 1− dimensional space = R = real line

• R2 = 2− dimensional space= Cartesian plane= {(x , y) | x , y ∈ R}= ordered pairs of real numbers .

• R3 = 3− dimensional space= {(x , y , z) | x , y , z ∈ R}= ordered triples of real numbers.

The numbers x , y in R2 or x , y , z in R3 are the coordinates of thepoint. If P is a point in R2 it has coordinates (x , y). We write

P(x , y) in this case. same for more dimensions.

The numbers x , y in R2 or x , y , z in R3 are the coordinates of thepoint.

If P is a point in R2 it has coordinates (x , y). We write

For any n = 1, 2, 3, 4, ..., we have

Rn = n − dimensional space= {(x1, x2, . . . , xn) | xi ∈ R}= ordered n–tuples real numbers.

x1, . . . , xn are the coordinates of the point.

We will study functions whose domain (and range) is a subset ofRn.

Dimensions 1,2 and 3 will serve as motivation and provideintuition, though much of the theory works for all n (but not all!)

For any n = 1, 2, 3, 4, ..., we haveRn = n − dimensional space

= {(x1, x2, . . . , xn) | xi ∈ R}= ordered n–tuples real numbers.

Plan for this course

Develop calculus to study functions of several variables.

1 Derivatives: Chapter 14 (and 13)

2 Integrals: Chapter 15 (and 13)

3 “Fundamental Theorems of Calculus” : Chapter 16

What do we need in order to do calculus?

Question: What sets calculus apart from algebra, trigonometry,pre-calculus?

Answer: Limits! In one variable:

limx→a

f (x) = L means

“as x approaches a, f (x) approaches L”

This requires a notion of “proximity” and hence of distance.

limx→a

f (x) = L means

Answer: Limits!

In one variable:

limx→a

f (x) = L means

limx→a

f (x) = L means

limx→a

f (x) = L means

limx→a

f (x) = L means

Distance

Given P(x1, . . . , xn),Q(y1, . . . , yn) ∈ Rn, define the distancebetween these points to be

|PQ| = distance from P to Q

(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2

n = 2 (Pythagorean Theorem)

Distance from (a, b) to (c, d)

(a− c)2 + (b − d)2

Distance from (a, b, c) to (p, q, r)

(a− p)2 + (b − q)2 + (c − r)2

(a,b,c)

(p,q,r)

Distance

(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2

(a− c)2 + (b − d)2

(a− p)2 + (b − q)2 + (c − r)2

(a,b,c)

(p,q,r)

Distance

(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2

(a− c)2 + (b − d)2

(a− p)2 + (b − q)2 + (c − r)2

(a,b,c)

(p,q,r)

Distance

(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2

(a− c)2 + (b − d)2

(a− p)2 + (b − q)2 + (c − r)2

(a,b,c)

(p,q,r)

Distance

(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2

(a− c)2 + (b − d)2

(a− p)2 + (b − q)2 + (c − r)2

(a,b,c)

(p,q,r)

Distance

(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2

(a− c)2 + (b − d)2

(a− p)2 + (b − q)2 + (c − r)2

(a,b,c)

(p,q,r)

Distance

(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2

(a− c)2 + (b − d)2

(a− p)2 + (b − q)2 + (c − r)2

(a,b,c)

(p,q,r)

Distance

(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2

(a− c)2 + (b − d)2

(a− p)2 + (b − q)2 + (c − r)2

(a,b,c)

(p,q,r)

Distance

(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2

(a− c)2 + (b − d)2

(a− p)2 + (b − q)2 + (c − r)2

(a,b,c)

(p,q,r)

Spheres: application of distance formula

For C (a, b, c) ∈ R3 and r > 0, the sphere of radius r withcenter C is the set

{P ∈ R3 | |PC | = r}

Then, if P has coordinates (x , y , z),

|PC | = r ⇔√

(x − a)2 + (y − b)2 + (z − c)2 = r

⇔ (x − a)2 + (y − b)2 + (z − c)2 = r2

This is equation for sphere: set of points (x , y , z) satisfying thisequation is a sphere.

Compare equation of a circle (x − a)2 + (y − b)2 = r2.

{P ∈ R3 | |PC | = r}

|PC | = r ⇔√

(x − a)2 + (y − b)2 + (z − c)2 = r

⇔ (x − a)2 + (y − b)2 + (z − c)2 = r2

{P ∈ R3 | |PC | = r}

|PC | = r ⇔

√(x − a)2 + (y − b)2 + (z − c)2 = r

⇔ (x − a)2 + (y − b)2 + (z − c)2 = r2

{P ∈ R3 | |PC | = r}

|PC | = r ⇔√

(x − a)2 + (y − b)2 + (z − c)2 = r

(x − a)2 + (y − b)2 + (z − c)2 = r2

{P ∈ R3 | |PC | = r}

|PC | = r ⇔√

(x − a)2 + (y − b)2 + (z − c)2 = r

⇔ (x − a)2 + (y − b)2 + (z − c)2 = r2

{P ∈ R3 | |PC | = r}

|PC | = r ⇔√

(x − a)2 + (y − b)2 + (z − c)2 = r

⇔ (x − a)2 + (y − b)2 + (z − c)2 = r2

{P ∈ R3 | |PC | = r}

|PC | = r ⇔√

(x − a)2 + (y − b)2 + (z − c)2 = r

⇔ (x − a)2 + (y − b)2 + (z − c)2 = r2

What else do we need to do calculus?

Derivatives (and integrals) require limit and displacement:

f ′(x) = limh→0

f (x + h)− f (x)

Need displacement from x to x + h (and from f (x) to f (x + h)).This requires vectors....

f ′(x) = limh→0

f (x + h)− f (x)

Need displacement from x to x + h (and from f (x) to f (x + h)).

This requires vectors....

f ′(x) = limh→0

f (x + h)− f (x)

Need displacement from x to x + h (and from f (x) to f (x + h)).This requires vectors....

Vectors in R2.

A vector in R2 is an arrow. It represents a quantity with bothdirection and magnitude.

Two vectors are equal if they have the same direction andmagnitude.

Notation: book v (bold face) or written ~v (arrow over).

Vectors in R2.

Operations and constructions

Addition: Parallelogram rule. This defines the sum of two vectors:v + w.

Scalar multiplication: Scale magnitude, and reverse direction ifnegative. What is cv if c = 1, 0, 2,−1?

Displacement vector from A to B is−→AB, tail at A and tip at B.

Addition: Parallelogram rule.

This defines the sum of two vectors:v + w.

Scalar multiplication: Scale magnitude, and reverse direction ifnegative.

What is cv if c = 1, 0, 2,−1?

Position vectors

Every vector is equal to one with tail at the origin O = (0, 0):

Vectors in R2 ←→ R2

−→OP ←→ P

−→OP = position vector of the point P. (vectors and points aredifferent objects)

Write ~v = 〈v1, v2〉 (or sometimes just (v1, v2)).

The numbers v1, v2 are the components of ~v .

Position vectors

−→OP ←→ P

Position vectors

−→OP ←→ P

Position vectors

−→OP ←→ P

Position vectors

−→OP ←→ P

Calculations in terms of components

For ~u = 〈u1, u2〉, ~v = 〈v1, v2〉, and c ∈ R we have

~u + ~v = 〈u1 + v1, u2 + v2〉

c~u = 〈cu1, cu2〉

‖~u‖ =√

u21 + u22 = magnitude of ~u.

Sometimes write |~u| instead. Also call it the norm or length of ~u.

A(a1, a2) and B(b1, b2), the displacement vector is−→AB = 〈b1 − a1, b2 − a2〉.

~u + ~v = 〈u1 + v1, u2 + v2〉

c~u = 〈cu1, cu2〉

‖~u‖ =√

~u + ~v = 〈u1 + v1, u2 + v2〉

c~u = 〈cu1, cu2〉

‖~u‖ =√

~u + ~v = 〈u1 + v1, u2 + v2〉

c~u = 〈cu1, cu2〉

‖~u‖ =√

~u + ~v = 〈u1 + v1, u2 + v2〉

c~u = 〈cu1, cu2〉

‖~u‖ =√

~u + ~v = 〈u1 + v1, u2 + v2〉

c~u = 〈cu1, cu2〉

‖~u‖ =√

Vectors in Rn

Vectors in Rn ←→ Rn

−→OP ←→ P

−→OP = position vector of the point P.

~v = 〈v1, . . . , vn〉 = position vector of the point (v1, . . . , vn).

v1, . . . , vn are the components.

Displacement vector from point A(a1, . . . , an) to B(b1, . . . , bn):

−→AB = 〈b1 − a1, . . . , bn − an〉.

“Arrow from A to B”

Vectors in Rn

−→OP ←→ P

−→AB = 〈b1 − a1, . . . , bn − an〉.

Vectors in Rn

−→OP ←→ P

−→AB = 〈b1 − a1, . . . , bn − an〉.

Vectors in Rn

−→OP ←→ P

−→AB = 〈b1 − a1, . . . , bn − an〉.

Vectors in Rn

−→OP ←→ P

−→AB = 〈b1 − a1, . . . , bn − an〉.

Vectors in Rn

−→OP ←→ P

−→AB = 〈b1 − a1, . . . , bn − an〉.

Vectors in Rn

−→OP ←→ P

−→AB = 〈b1 − a1, . . . , bn − an〉.

Vectors in Rn

−→OP ←→ P

−→AB = 〈b1 − a1, . . . , bn − an〉.

Physical quantities

We will also use vectors to denote physical quantities:

Example: Forces have a direction and magnitude, so we canrepresent them with vectorsMultiple forces acting on an object, then the net force on object(or resultant force) is the sum of the forces.

Example: Wind has speed and direction, so can represent it with avector.

Physical quantities

Example: Forces have a direction and magnitude, so we canrepresent them with vectors

Multiple forces acting on an object, then the net force on object(or resultant force) is the sum of the forces.

Physical quantities

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