Multivariable Calculus Questions

Multivariable Calulus

Shool of Mathematis

University of the Witwatersrand

Private Bag 3

P. O. WITS 2050

Johannesburg

South Afria

2002

Shool of Mathematis University of the Witwatersrand

2 Multivariable Calulus MATH204

Contents

1 Dierentiation 5

1.1 Derivatives and Dierentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Vetor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Diretional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Tangents and Normals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Integration 19

2.1 Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Salar Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Vetor Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Double Integrals and Fubini's Theorem . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 Classial Integration Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 26

RECOMMENDED TEXTS

Marsden and Tromba, Vetor Calulus, Freeman Publishers.

Spiegel, Advaned Calulus, Shaum Series, MGraw Hill.

Spiegel, Advaned Mathematis, Shaum Series, MGraw Hill.

Dineen, Multivariable Calulus, Springer Verlag.

3

Chapter 1

Dierentiation

1.1 Derivatives and Dierentials

Denition 1.1.1 We dene the partial derivative of f : R

n

! R at x with respet to x

j

by

f(x)

x

j

= lim

h!0

f(x

1

; :::; x

j1

; x

j

+ h; x

j+1

; :::; x

n

) f(x

1

; :::; x

n

)

h

:

Denition 1.1.2 We dene the derivative of F : R

n

! R

m

to be the matrix

F

0

=

2

6

4

F

1

x

1

: : :

F

1

x

n

.

.

.

.

.

.

F

m

x

1

: : :

F

m

x

n

3

7

5

:

Denition 1.1.3 We dene the dierential of F : R

n

! R

m

at a to be the linear map

from R

n

to R

m

given by dF(a;h) = F

0

(a)h for all h 2 R

n

:

Theorem 1.1.4 Let F;G : R

n

! R

m

and g : R

n

! R then:

1. (F G)

0

= F

T

G

0

+G

T

F

0

;

2. (F+ G)

0

= F

0

+ G

0

for all ; 2 R;

3. (gF)

0

= gF

0

+ Fg

0

.

Theorem 1.1.5 Let p;q : R ! R

3

then

(a) (p q)

0

= p q

0

+ p

0

q

(b) (p q)

0

= p q

0

+ p

0

q.

5


TUTORIAL QUESTIONS

1. Let f; g : R

n

! R and let '(t) = f(t; x

2

; :::; x

n

) and (t) = g(t; x

2

; :::; x

n

) where x

2

; :::; x

n

are onstant.

(a) Give expressions for the derivatives '

0

and

0

in terms of f and g.

(b) From the single variable alulus result that for a; b 2 R we have (a' + b )

0

=

a'

0

+ b

0

, prove that

(af + bg)

x

1

= a

f

x

1

+ b

g

x

1

:

() From the single alulus result that (' )

0

= '

0

+ '

0

, prove that

(fg)

x

1

= g

f

x

1

+ f

g

x

1

:

2. In eah of the following ases: parametrize the urves; and determine whether the urves

are orthogonal where they interset.

(a) The urves y = x

2

and x = y

2

.

(b) The urves y = x

2

and y =

1

2

x

2

.

3. Parametrize the irle of radius 5 having entre (8;3) and hene nd a tangent vetor

and the tangent line to this irle at the point (5; 1).

4. For f(x

1

; x

2

) = x

2

os x

1

nd f

0

(x

1

; x

2

); f

0

(2; 7) and df [(2; 7);h.

5. For f(x; y; z) = (x 2z)e

y

nd f

0

(x; y; z) and df [(x; y; z);h.

6. Find F

0

and dF[a;h where:

(a) F(x

1

; x

2

) =

2

4

lnx

1

x

2

e

2x

1

x

2

2

os x

1

3

5

and hene show that if v is tangent to the urve parametrized

by r(t) =

1

t

then dF[r(t);v is tangent to the urve parametrized by F(r(t)) at

F(r(t)), by diret alulation;

(b) F(x; y; z) =

x 1

xy

1+z

2

and hene show that if v is tangent to the urve parametrized

by r(t) =

2

4

2

t

2

1

3

5

then dF[r(t);v is tangent to the urve parametrized by F(r(t))

at F(r(t)), by diret alulation.

7. Prove Theorem 1.1.4 parts (b) and (). In eah begin by omputing the term in brakets

on the left of the expression. Then ompute the derivative of this expression and expand

the result by using what you know about partial derivatives. Now ompute the right

hand side and ompare your expressions.


University of the Witwatersrand Shool of Mathematis

8. Let '(x

1

; x

2

; x

3

) = x

3

x

2

2

2x

1

and F(x

1

; x

2

; x

3

) =

e

2x

1

+ 5x

2

x

5

3

x

2

2x

1

+ sinx

3

.

(a) Calulate '

0

and F

0

.

(b) HENCE nd ('F)

0

.

9. Let F : R

n

! R

m

. Prove that for eah a the mapping h 7! dF[a;h is linear on R

n

.

10. Let F : R

n

! R

m

and g : R

n

! R. Prove that

d(gF)[a;h = g(a)dF[a;h + F(a)dg[a;h:

11. (a) Let u : R ! R

n

and v : R ! R

n

. By writing u v as a summation, prove that

(u v)

0

= u

0

v + u v

0

:

(b) Let p : R ! R

n

and v 2 R

n

: Assume that v and p

0

(t) are orthogonal for all t 2 R

and that p(0) is orthogonal to v: Prove that v and p(t) are orthogonal for all t 2 R:

() Show that if u(t) is a unit vetor for all t, then u(t) and u

0

(t) are orthogonal for

all t.

(d) Let '(t) = te

t

2

and u(t) =

os 2t

sin 2t

. Find '

0

and u

0

, and hene nd ('u)

0

,

leave your expression for this derivative as the sum of two orthogonal vetors.

12. Let F(r; ; z) =

2

4

rz os

rz sin

z

3

5

.

(a) Find F

0

.

(b) Find det(F

0

).

13. If the motion of a partile in spae is given by r(t) =

2

4

2t

2

t

2

4t

3t 5

3

5

where t represents

time.

(a) Find the veloity and a

eleration of the partile at time t.

(b) Find the omponent of the a

eleration in the diretion of travel (i.e. in the dire-

tion of the veloity) at time t.

14. Consider a plane urve with polar equation = (t) and = t. This urve an be

written parametrially by setting r(t) = (t)

os t

sin t

: Using the Theorem on vetor

dierentiation to show that:

(a) r

0

(t) =

0

(t)

os t

sin t

+ (t)

sin t

os t

;

(b) r

00

(t) = (

00

(t) (t))

os t

sin t

+ 2

0

(t)

sin t

os t

;

Multivariable Calulus MATH204 7


() kr

0

k =

p

2

+ (

0

)

2

.

15. If r(t) lies on the irle of radius R with entre 0 for all t 2 R, prove that kr

0

k =

p

r

00

r.

16. Let r(t) denote the position of a partile at time t. The angular veloity of the partile

with respet to the origin is dened to be ! = r r

0

. Prove that !

0

= r r

00

.

17. Let f : R

2

! R and let h(t) = f(x

1

+ t; x

2

).

(a) Show that h

0

(t) =

f

x

1

(x

1

+ t; x

2

).

(b) If

f

x

1

(x) = 0 for all x 2 R

2

, prove that f(x

1

; x

2

) = f(0; x

2

) for all x

1

; x

2

2 R.

HINT: Try integrating h

0

(t) from 0 to b for a suitable b.

18. Let F : R

n

! R

n

be suh that F(x) is a unit vetor for all x 2 R

n

. Prove that

det(F

0

(x)) = 0 for all x 2 R

n

. HINT: Consider (F F)

0

and proeed by a ontradition

argument.

1.2 Vetor Analysis

Denition 1.2.1 Let f : R

n

! R, we dene the gradient of f , denoted gradf or rf , by

gradf = rf =

2

6

4

f

x

1

.

.

.

f

x

n

3

7

5

:

Denition 1.2.2 Let F : R

n

! R

n

we dene the divergene of F, denote divF or r F, by

div F = r F =

n

X

j=1

F

j

x

j

:


n

! R, we dene the Laplaian of f , denoted r

2

f , by

r

2

f = r rf =

n

X

j=1

2

f

x

2

j

:

If r

2

f 0 we say that f is harmoni.

Denition 1.2.4 Let F : R

3

! R

3

, we dene the url of F, denoted urlF or r F, by

url F = r F =

2

6

4

F

3

x

2

F

2

x

3

F

1

x

3

F

3

x

1

F

2

x

1

F

1

x

2

3

7

5

:

Theorem 1.2.5 For a; b 2 R, f; g : R

n

! R and F;G : R

n

! R

n

we have:



(a) r(af + bg) = arf + brg;

(b) r(fg) = grf + frg;

() r (aF + bG) = ar F+ br G;

(d) r (gF) = (rg) F+ gr F

and if n = 3 we in addition have:

(e) r (aF+ bG) = ar F + brG;

(f) r (gF) = (rg) F + gr F.

(g) r (FG) = (r F) G (rG) F;

Theorem 1.2.6 Let f : R

n

! R, then

2

f

x

i

x

j

=

2

f

x

j

x

i

for all i; j = 1; :::; n.

Theorem 1.2.7 For f : R

3

! R and F : R

3

! R

3

we have:

1. rrf 0;

2. r (r F) 0.

TUTORIAL QUESTIONS

1. For eah of the following funtions f , alulate rf :

(a) f(x

1

; x

2

) = ln(x

2

1

+ x

2

2

);

(b) f(x

1

; x

2

; x

3

) = x

2

3

e

x

1

x

2

+ sin(x

1

x

2

).

2. Let F(x

1

; x

2

; x

3

) =

2

4

x

1

e

x

2

x

2

sinx

3

x

1

x

2

x

3

3

5

(a) Calulate G = url F and div F.

(b) What an you say about r G.

3. Let F(x

1

; x

2

; x

3

) =

0

x

1

x

3

e

x

2

x

1

+ x

2

1

A

and nd url F; div F; r (rF); F (rF) and

r (r F):

4. Determine whih of the funtions below are harmoni:

(i) u(x

1

; x

2

; x

3

; x

4

) = 1 + x

1

x

2

3x

4

+ e

x

3

sinx

2

;

(ii) u(x

1

; x

2

; x

3

) = 3x

2

1

x

3

+ 2x

1

x

3

2x

2

x

3

3

;

(iii) u(x

1

; x

2

) = x

2

1

+ x

2

2

:



5. Prove Theorem 1.2.5 parts (a), (b), (), (d), (f) and (g) by omputing the left and right

hand sides of the identities and expanding the results using what you know about partial

derivatives.

6. Prove Theorem 1.2.6 for the ase of n = 2 by arrying out the following steps.

Let : R

2

! R and (x; y) 2 R

2

be xed. Let h; k 2 R and set

f

s

(t) = (x+ s; t) (x; t)

g

s

(t) = (t; y + s) (t; y)

(a) Show that f

h

(y + k) f

h

(y) = g

k

(x+ h) g

k

(x).

(b) Use the 1st mean value theorem to show that kf

h

0

(y+K

1

) = f

h

(y+ k) f

h

(y) for

some K

1

between 0 and k, and that hg

k

0

(x+H

1

) = g

k

(x+ h) g

k

(x) for some H

1

between 0 and h.

() Hene onlude that kf

h

0

(y + K

1

) = hg

k

0

(x + H

1

) for some K

1

between 0 and k,

and some H

1

between 0 and h.

(d) Thus show that

k

y

(x+h;y+K

1

)

(x;y+K

1

)

= h

x

(x+H

1

;y+k)

(x+H

1

;y)

:

(e) Put p(t) =

y

(x + t; y +K

1

). The 1st mean value theorem gives that there is H

2

between 0 and h suh that hp

0

(H

2

) = p(h) p(0). From this dedue that

y

(x+h;y+K

1

)

(x;y+K

1

)

= h

x

y

(x +H

2

; y +K

1

)

and in a similar manner dedue that there is K

2

between 0 and k suh that

x

(x+H

1

;y+k)

(x+H

1

;y)

= k

y

x

(x +H

1

; y +K

2

):

(f) By ombining (d) and (e) and by letting h and k tend to zero (noting what happens

to H

1

; H

2

; K

1

; K

2

and the sandwih theorem), omplete the proof of the theorem

for n = 2.

7. Prove Theorem 1.2.7 part (b) by diret omputation of the left hand side of the equation

and using Theorem 1.2.6.

8. Prove that if and are harmoni, then r

2

( ) = 2r r :

9. Prove that if f : R

n

! R is harmoni, then r

2

f

2

= 2krfk

2

.

10. Show that for funtions f and g from R

3

to R we have r (frg) = (rf) (rg):

11. Prove that r (grf) r (frg) if and only if rf(x) and rg(x) are parallel for all

x.

12. Let f(x; y) = ax + by

2

+ 2x

2

+ os x for all (x; y) 2 R

2

. Find all possible onstants

a; b; 2 R for whih f is harmoni.

13. Prove that f : R

n

! R is harmoni if and only if r (frf) = krfk

2

: TAKE CARE

HERE!



1.3 Chain Rule

Reall that the 1st mean value theorem said that:

For f : R ! R and for eah t; h 2 R there is k between 0 and h for whih

f(t+ h) f(t) = hf

0

(t + k):

The 1st year single variable Chain Rule:

Theorem 1.3.1 R ! R Chain Rule

Let f : R ! R and g : R ! R, then

d(f g)

dt

(t) =

df

dt

g(t)

dg

dt

(t)

[i.e. (f g)

0

(t) = f

0

(g(t))g

0

(t).

The "mini hain rule", whih forms the mathematial basis for the General Chain Rule:

Theorem 1.3.2 R

n

! R Chain Rule

Let f : R

n

! R and G : R ! R

n

, then

(f G)

0

(t) = f

0

(G(t))G

0

(t)

= (rf)(G(t)) G

0

(t)

=

n

X

i=1

f

x

i

G(t)

dG

i

dt

t

:

Finally the general Chain Rule:

Theorem 1.3.3 General Chain Rule

Let F : R

q

! R

n

and G : R

p

! R

q

, then

(F G)

0

(x) = F

0

(G(x))G

0

(x):

TUTORIAL QUESTIONS

1. In the following proof of the simplest form of the hain rule, give the expressions whih

should replae the apital letters.

Let f; g : R ! R be ontinuous and have ontinuous derivatives.

(a) From the rst mean value theorem

g(t+ h) g(t) = A

where lies between 0 and h.



(b) And similarly from the rst mean value theorem

f(a+ s) f(a) = B

where lies between 0 and s.

() By setting a = g(t) and s = A we see that

f(g(t+ h)) f(g(t)) = C

where lies between 0 and s.

(d) Observing that as h! 0 we have s! 0, hene we may onlude that

lim

h!0

f(g(t+ h)) f(g(t))

h

= D :

2. (a) Let f : R

n

! R and G : R ! R

n

: Using the hain rule give an expression for

[f(G(t))

0

:

(b) Let f(x

1

; x

2

; x

3

) = x

2

1

+x

2

2

x

3

and let G : R ! R

3

be an unknown funtion. Using

the hain rule give an expression for [f(G(t))

0

in terms of the omponents of G

and their derivatives.

() Let f : R

2

! R and G : R ! R

2

: Let A = fxjf(x) = kg where k is a onstant.

If G(t) 2 A for all t 2 R, prove that at eah point G(t) we have that rf(G(t)) is

orthogonal to urve given by G(t):

3. Let f(x; y) = e

x

y

2

.

(a) Find rf .

(b) Given that r(t) =

x(t)

y(t)

, and that f(r(t)) = 0 for all t, use the hain rule to

give an expression for y

0

in terms of x; y; x

0

.

() Using the fat that

dy

dx

=

dy=dt

dx=dt

use what you found in (b) to give an expression for dy=dx.

4. (a) State the hain rule for f G where f : R

n

! R and G : R ! R

n

.

(b) If x and y are funtions of t, using the mini hain rule, give an expression for

d

dt

f(x(t); y(t); t)

in terms of x; y; t;

dx

dt

;

dy

dt

and the partial derivatives of f .

() If f(x(t); y(t); t) is onstant for x(t) = e

t

and y(t) = t

2

, then give an equation

whih the partial derivatives of f must obey.

5. Let f(x; y) = os(x

2

3y).

(a) Find rf .



(b) Given that r(t) =

x(t)

y(t)

, and that f(r(t)) = 0 for all t, use the hain rule to

give an expression for y

0

in terms of x; y; x

0

.

() Using the fat that

dy

dx

=

dy=dt

dx=dt

use what you found in (ii) to give an expression

for dy=dx.

6. Proof of the general hain rule. Let G : R

n

! R

p

and F : R

p

! R

m

.

(a) Using the Theorem 1.3.1, give an expression for

F

i

G

x

j

(x) as a summation.

(b) Rewrite the summation found in (a) as a matrix produt of a row and a olumn

matrix.

() From the denition of the derivative, give (F G)

0

as a single matrix.

(d) Using the expressions found in (a) and (b), write (F G)

0

as a produt of two

matries, hene proving the general hain rule.

7. Let F(r; ) =

r os

r sin

and '(x; y) = x

2

+ y

2

6xy. Using the hain rule nd (' F)

0

.

8. (a) Using the hain rule, show that

d(F G)[a;h = dF[G(a); dG[a;h:

(b) Show that dF[r(t); r

0

(t) = (F(r(t)))

0

and thus dedue that the mapping h !

dF[r(t);h maps tangent vetors at r(t) to the urve parametrized by r(t) to tangent

vetors at F(r(t)) to the urve parameterized by F(r(t)). (I.e. F maps points while

dF maps tangent vetors.)

9. Let F : R

2

! R

2

, and let P be the parallelogram with sides given by the vetors h;k.

Let Q be the parallelogram with sides given by the vetors dF[a;h and dF[a;k. Show

that Area(Q)=det(F

0

(a)) Area(P). Also prove the analogous result for parallelpipeds in

R

3

.

10. Let F : R

n

! R

m

and G : R

m

! R

p

:

(a) Using the hain rule, give an expression for [F G

0

:

(b) Suppose that F : R

n

! R

n

; G : R

n

! R

n

and that F(G(x)) = x for all x 2 R

n

:

Denote y = G(x): Prove that

G(x)

x

=

h

F(y)

y

i

1

:

() Let F : R

3

! R

2

and G : R

2

! R

3

where G(x) =

0

x

1

x

2

x

2

1

x

2

2

x

1

+ x

2

1

A

: Give an

expression for [F G(x)

0

involving only the following: D

j

F

i

and x

j

for j = 1; 2; 3

and i = 1; 2:

(d) Let F : R

3

! R

2

and G : R

4

! R

3

where F(y) =

y

1

y

2

2y

1

+ y

2

3

: Given that

G

1

(x)G

2

(x) = x

1

and that 2G

1

(x) + (G

3

(x))

2

= x

3

; prove that G

2

(x)D

1

G

1

(x) +

G

1

(x)D

1

G

2

(x) = 1 for all x 2 R

4

:



1.4 Diretional Derivatives


n

! R and let u be a unit vetor in R

n

. We dene the diretional

derivative of f at x in the diretion u by

D

u

f(x) = lim

t!0

f(x+ tu) f(x)

t

:


n


n

. Then

D

u

f(x) = u rf(x):


n

! R.

1. The diretion of maximum rate of inrease of f at x is rf(x) and the rate of inrease

of f in this diretion is krf(x)k.

2. The diretion of minimum rate of inrease of f at x is rf(x) and the rate of inrease

of f in this diretion is krf(x)k.

TUTORIAL QUESTIONS

1. Let ' : R

4

! R be given by '(x) = x

1

e

x

2

+ x

4

os(x

3

): Find the diretional derivatives

of ' at

0

B

B

2

3

4

5

1

C

C

A

in the diretion of the vetor

0

B

B

5

6

1

2

1

6

1

6

1

C

C

A

:

2. Let ' : R

3

! R be given by '(x) = (x

2

1

x

2

2

)e

3x

3

: Find the diretional derivatives of '

at

0

3

2

1

1

A

in the diretion of the vetor

0

2

1

1

1

A

:

3. Let (x; y; z) = xe

y

2z.

(a) Find r.

(b) Hene nd the rate of inrease of ' at (1; 0; 3) in the diretion of v =

0

2

3

1

1

A

.

() Is the diretion of v the diretion of fastest inrease of ' at (1; 0; 3), if not, what

is the diretion of fastest inrease of at (1; 0; 3), and how fast does ' inrease in

this diretion.

4. (a) Let f : R

n

! R, prove that the diretion of fastest inrease of f at x is rf(x).



(b) Let

f(x

1

; x

2

; x

3

) = (x

2

1

x

3

)e

x

2

:

Find the diretion of fastest inrease of f at x

0

= (2; ln 3;1).

Give the diretional derivative of f at x

0

in the diretion of the vetor v =

0

1

2

5

1

A

.

5. Let f : R

n


n

and x

0

some xed point in R

n

. Show

the following:

(a) If f is onstant along the line parametrized by r(t) = tu+x

0

; t 2 R; thenD

u

f(x

0

) =

0.

(b) If f has a maximum or minimum at x

0

then D

u

f(x

0

) = 0.

() If rf(x) = 0 for all x 2 R

n

, then f is onstant on R

n

.

1.5 Tangents and Normals

Denition 1.5.1 Let ' : R

n

! R and 2 R. A hypersurfae in R

n

is a set of the form

S = fx 2 R

n

j'(x) = g:

A point x 2 S is said to be a regular point of S if r'(x) exists and is non zero, otherwise the

point x is said to be singular.

Denition 1.5.2 A vetor n is asid to be a normal to a hypersurfae S at x if for eah

: R ! S with (t

0

) = x for some t

0

2 R we have n

0

(t

0

) = 0.

Theorem 1.5.3 Let x

0

be a regular point of S, where S is the hypersurfae given by S =

fxj'(x) = g. Then r'(x

0

) is a normal to S at x

0

).

Denition 1.5.4 We dene the set of tangent vetors to the hypersurfae S at x

0

to be the

set

T

x

0

= f

0

(t

0

)j : R ! S; (t

0

) = x

0

g:

Theorem 1.5.5 If x

0

is a regular point of S = fxj'(x) = g, then

T

x

0

= fvjv r'(x

0

) = 0g:

Denition 1.5.6 The tangent hyperplane to S at the regular point x

0

of S is

x

0

+ T

x

0

= fv + x

0

jv 2 T

x

0

g:

TUTORIAL QUESTIONS

1. (a) Give a tangent vetor at G(t) to the urve, C, given by G(t); t 2 R.



(b) If for eah t 2 R all tangent vetors at G(t) to the urve C are orthogonal to

rf(G(t)) prove that f is onstant on the urve C.

2. Let S = fx 2 R

3

: '(x) = 3g where '(x) = x

2

1

x

2

6x

3

3

and let p =

0

3

5

2

1

A

and

q =

0

6

3

3

1

A

:

(a) Show that p lies on the surfae S:

(b) Find a normal vetor to S at p.

() Find T

p

; the set of tangent vetors to S at p.

(d) Find the tangent plane to S at p.

(e) Does q lie in the tangent plane to S at p ? Justify your answer.

(f) Is q tangent to S at p ? Justify your answer.

3. Let '(x; y; z) = z + 2x

2

y

2

and S be the surfae S = f(x; y; z)j'(x; y; z) = 1g.

(a) Show that (1; 2; 1) 2 S.

(b) Find a normal to S at (1; 2; 1).

() Hene nd the set of tangent vetors to S at (1; 2; 1).

(d) Give an expression for the tangent plane to S based at (1; 2; 1) and denote this

tangent plane by P .

(e) Show that the line r(t) =

0

t+ 2

3 + 2t

4t+ 1

1

A

; t 2 R, lies in P .

4. Let S be a surfae in R

3

dened by '(x; y; z) = 36 where '(x; y; z) = 4x

2

+ 9y

2

+ z

2

,

and let r(u; v) =

2

4

3 os u os v

2 sinu os v

6 sin v

3

5

where u 2 [0; 2 and v 2 [=2; =2.

(a) Show that the point x

0

= (3=2;

p

3; 0) lies on S.

(b) Find a normal to S at x

0

.

() Find T

x

0

, the set of tangent vetors to S at x

0

.

(d) Show that '(r(u; v)) = 36 and that r(=3; 0) = x

0

.

(e) Show that

r

u

(=3; 0) and

r

v

(=3; 0) are tangent vetors to S at x

0

and that

r

u

(=3; 0)

r

v

(=3; 0) is a normal to S at x

0

.



5. Let S be a surfae in R

3

dened by '(x; y; z) = 0 and given parametrially by

r(u; v) =

2

4

x(u; v)

y(u; v)

z(u; v)

3

5

:

(a) Prove that

r(u;v)

u

and

r(u;v)

v

are tangent vetors to S at r(u; v).

(b) Thus (or otherwise) prove that

r(u;v)

u

r(u;v)

v

is a normal to S at r(u; v).

6. Let x

0

be a regular point of the hypersurfae S = fx 2 R

n

j'(x) = g. Prove that T

x

0

,

the set of all tangent vetors to S at x

0

, is an n 1 dimensional vetor spae (subspae

of R

n

).

7. Let S = fx j '(x) = 0g be a hypersurfae in R

n

and

T

x

(S) = f

0

(0) j : R ! S; (0) = xg

be the set of tangent vetors to S at x. Let x

0

be a regular point of S and let F : R

n

! R

n

be one-to-one and have detF

0

(x

0

) 6= 0. Denote F(S) = fF(x) j x 2 Sg.

(a) Prove that F(S) = fy j (y) = 0g where (y) = '(F

1

(y)):

(b) Prove that F(x

0

) is a regular point of F(S).

() Prove that F

0

(x

0

)T

x

0

(S) T

F(x

0

)

(F(S)) where

F

0

(x

0

)T

x

0

(S) = fF

0

(x

0

)v j v 2 T

x

0

(S)g:

(d) Now onlude that F

0

(x

0

)T

x

0

(S) = T

F(x

0

)

(F(S)) and thus that

T

F(x

0

)

(F(S)) = fdF[x

0

;hjh 2 T

x

0

(S)g:

1.6 Maxima and Minima


n

! R, we say that f has:

1. a loal maximum at x

0

if there exists > 0 suh that kx x

0

k < =) f(x) f(x

0

);

2. a strit loal maximum at x

0

if there exists > 0 suh that 0 < kxx

0

k < =) f(x) 0 suh that kx x

0

k < =) f(x) f(x

0

);

4. a strit loal minimum at x

0

if there exists > 0 suh that 0 < kxx

0

k < =) f(x) >

f(x

0

);

5. a ritial point at x

0

if rf(x

0

) = 0;



6. a saddle point at x

0

if x

0

is a ritial point whih is neither a loal maximum nor a loal

minimum.

Theorem 1.6.2 If f : R

n

! R has a loal maximum or a loal minimum at x

0

, then x

0

is a

ritial point of f .

Theorem 1.6.3 Several variable Taylor Theorem

Let f : R

n

! R, then

f(x+ h) = f(x) + df [x;h +R(x;h)

where R(x;h)=khk ! 0 as khk ! 0.


n

! R and rf(x

0

) = 0. Dene the disriminant of f at x

0

by

=

2

f

x

2

1

(x

0

)

2

f

x

2

2

(x

0

)

2

f

x

1

x

2

(x

0

)

2

:

1. If > 0 and

2

f

x

2

1

(x

0

) > 0, then f has a strit loal minimum at x

0

.

2. If > 0 and

2

f

x

2

2

(x

0

) > 0, then f has a strit loal minimum at x

0

.

3. If > 0 and

2

f

x

2

1

(x

0

) < 0, then f has a strit loal maximum at x

0

.

4. If > 0 and

2

f

x

2

2

(x

0

) < 0, then f has a strit loal maximum at x

0

.

5. If < 0 then f has a saddle point at x

0

.

TUTORIAL QUESTIONS

1. For eah of the following funtions f nd the ritial points and determine whih are

strit loal maxima, strit loal minima and saddle points.

(a) f(x

1

; x

2

) = x

2

1

2x

2

sin(x

1

):

(b) f(x; y) = x

3

2x

2

y + xy

2

+ y

3

3y.

() f(x; y) = 3xy x

3

+

3

2

y

2

.

(d) f(x; y) = 4xy x

4

2y

2

+ 2.

(e) f(x; y) = (1 + x

2

)(2 + os(x+ 2y)).

(f) f(x; y) = 3x

3

+ 3xy x

2

+ 3y

2

.


Chapter 2

Integration

2.1 Parametrization

Denition 2.1.1 Let r : [a; b! R

n

and where a < b, then a set of the form

= fr(t) : t 2 [a; bg

is alled a urve and the funtion r(t); t 2 [a; b; is alled a parametrization of .

If a give a diretion or orientation (usually indiated by an arrow along the urve), then

is alled a path or oriented urve.

Let is a path parametrized by r(t); t 2 [a; b; if the diretion r(t) moves as t inreases is the

same as the diretion assoiated with , then r(t); t 2 [a; b; is said to be an orientation preserv-

ing parametrization of . Otherwise r(t); t 2 [a; b; is an orientation reversing parametrization

of .

If is an oriented urve, we denote by

, the urve but with reversed orientation.

A urve is said to be pieewise smooth, if the a parametrization r(t); t 2 [a; b; of where r

0

(t)

exists and is non-zero exept at at most nitely many points of . The values of r(t) where

either r

0

(t) does not exists or is zero are alled singular points of .

Note 2.1.2 The parametrization of a urve is not unique.

Some standard parametrizations follow:

1. The line segment from a to b:

r(t) = a + t(b a); t 2 [0; 1:

19


2. The irle entre (a

0

; b

0

) with radius :

r(t) =

a

0

b

0

+

os t

sin t

; t 2 [0; 2:

3. The expliit urve y = f(x) where x goes from a to b and a < b:

r(t) =

t

f(t)

; t 2 [a; b:

4. The expliit urve x = f(y) where y goes from a to b and a < b:

r(t) =

f(t)

t

; t 2 [a; b:

2.2 Salar Path Integrals

Denition 2.2.1 Let the urve be parametrized by (t); t 2 [a; b: Let f be a real valued

funtion dened on . We dene the salar path integral of f over by:

Z

f ds :=

Z

b

a

f((t)) j

0

(t)j dt:

Note 2.2.2 We may thus formally onsider

ds = j

0

(t)j dt:

Theorem 2.2.3

Z

f ds =

Z

f ds:

Note 2.2.4 The length of the urve is

R

1 ds:

TUTORIAL QUESTIONS

1. (a) Let = fr(t) 2 R

n

: t 2 [a; bg and f : R

n

! R: Dene what we mean by

R

f ds:

(b) Let be that part of the ellipse

x

2

1

4

+x

2

2

= 1 from

0

1

to

2

0

taken antilok-

wise. Parametrize and evaluate

R

f ds where f(x) =

p

x

2

1

+ 16x

2

2

:

2. (a) Let = fr(t) 2 R

n

: t 2 [a; bg and f : R

n

! R: Dene what we mean by

R

f ds:

(b) Let be the the urve x

3

= y

2

; y 2 [1; 1: Parametrize and nd the length of :

3. Let be the portion of the ellipse

x

2

4

+

y

2

9

= 1 taken antilokwise from (

p

3;

3

2

) to (2; 0).

Parameterize and evaluate the integral I =

R

f ds where f(x; y) =

x

2

p

13x

2

y

2

.



4. Let = fr(t)jt 2 [1; 1g where r(t) =

2t

3

3t

2

:

(a) Find the length of the path .

(b) Let f(x

1

; x

2

) = jx

1

j

p

x

2

+ 3. Evaluate

R

f ds.

() The average of f over is dened to be the integral of f over divided by the

length of .

5. Let a piee of wire be desribed by the loop in the path r(t) =

t

2

t(t+ 2)

; and have

mass per unit length f(x) =

p

x

1

+ x

2

+ 1 at x. Find the length of the wire.

6. Let denote the path along the urve x = y

2

from (1;1) to (4; 2). Sketh the urve

, parametrize and hene evaluate

R

x

p

1+3x+y

2

ds.

2.3 Vetor Path Integrals

Denition 2.3.1 Let the urve be parametrized by (t); t 2 [a; b: Let F : R

n

! R

n

. We

dene the (vetor) path integral of F over by:

Z

F d :=

Z

b

a

F((t))

0

(t) dt:

Note 2.3.2 We may thus formally onsider d =

0

(t) dt and onsequently, formally we have

d = u(t) ds

where u(t) is the unit tangent vetor to at (t) in the diretion of the orientation of , and

Z

F d =

Z

F u ds:

Theorem 2.3.3

Z

F d =

Z

F d:

Note 2.3.4 If F represents fore, then

R

F d is the work done on a partile by the fore F

in traversing .

Note 2.3.5 Let be parametrized by r(t) =

x(t)

y(t)

; t 2 [a; b, and let F(x; y) =

P (x; y)

Q(x; y)

,

then one may write dr =

dx

dy

and thus

Z

F dr =

Z

P dx+Q dy:



TUTORIAL QUESTIONS

1. (a) Let = fr(t) 2 R

n

: t 2 [a; bg and F : R

n

! R

n

: Dene what we mean by

R

F dr:

(b) Let be that part of the parabola x

2

= x

2

1

+3 from

1

4

to

3

12

: Parametrize

and evaluate

R

F dr where F(x) =

x

1

sin(x

2

)

:

() Let be that part of the urve x

2

= ln(x

1

) from

e

1

to

e

3

3

: Parametrize

and evaluate

R

F dr where F(x) =

x

2

x

1

e

x

2

:

2. Let be the urve x = y

2

y + 1 from (3;1) to (3; 2). Parameterize and determine

whether your parametrization has the required orientation. Hene evaluate the integral

R

v dr, where v(x; y) =

y

2x

.

3. Let be the irle entre (1; 5), radius 3 taken antilokwise. Sketh the urve in

the (x; y)-plane. Parameterize and determine whether your parametrization has the

required orientation. Evaluate the integral

R

v dr, where v(x; y) =

5x y

x

.

4. Let be the ellipse

x

2

a

2

+

y

2

b

2

= 1 where a; b > 0, taken antilokwise.

(a) Parametrize the oriented urve .

(b) Let v(x

1

; x

2

) =

x

1

x

2

x

1

+ x

2

: Evaluate the integral of v over the urve .

5. Let F(x) =

x

jxj

3

. Let

1

(N) be the line segment from (1; 1) to (N;N); N > 0, and let

2

be the irle entre (0; 0) with radius 2, taken antilokwise.

(a) Parametrize

1

(N).

(b) Parametrize

2

.

() Show that F(x) is orthogonal to

2

at x for eah x 2

2

. Hene evaluate

R

2

F dr:

(d) Calulate

R

1

(N)

F dr and hene nd lim

N!1

R

1

(N)

F dr .

6. Let be the irle radius 3, entre (1; 5). Sketh in the (x; y)-plane. Parametrize

and hene evaluate

R

(5x y) dx + x dy:

7. Let be the spiral given by x = os t, y = sin t and z = t

2

, t 2 [2; 2. Find

R

p

x

2

+ y

2

+ 4z ds and

R

(x zy) dx+ (zx + y) dy + (4z) dz.



2.4 Double Integrals and Fubini's Theorem

Denition 2.4.1 Let D be a region in R

2

with nite area and bounded by a pieewise smooth

ontinuous losed urve. Let f : D ! R be ontinuous. For eah N 2 N divide D into N

subregions D

N

i

; i = 1; :::; N; in suh a manner that

lim

N!1

max

i=1;:::;N

diam(D

N

i

) = 0:

Let x

N

i

2 D

N

i

. We dene the integral of f over the region D by

Z Z

D

f da = lim

N!1

N

X

i=1

f(x

N

i

) Area(D

N

i

):

Note 2.4.2 1.

R R

D

1 da = Area(D):

2. If f(x) g(x) for all x 2 D, then

R R

D

[f g da is the volume of the region over D

bounded above by f(x) and below by g(x);x 2 D.

Theorem 2.4.3 (Fubini's Theorem)

Let D R

2

and f : D ! R.

1. If D = f(x; y)jp(x) y q(x); x 2 [a; bg then

Z Z

D

f da =

Z

b

a

"

Z

q(x)

p(x)

f(x; y) dy

#

dx:

ba

y = p(x)

y = q(x)

x

y

D

2. If D = f(x; y)jg(y) x h(y); y 2 [; dg then

Z Z

D

f da =

Z

d

"

Z

h(y)

g(y)

f(x; y) dx

#

dy:

y = g(x)

y = h(x)

c

d

D

x

y



3. If D = f(x; y)jp(x) y q(x); x 2 [a; bg = D = f(x; y)jg(y) x h(y); y 2 [; dg

then

Z

b

a

"

Z

q(x)

p(x)

f(x; y) dy

#

dx =

Z Z

D

f da =

Z

d

"

Z

h(y)

g(y)

f(x; y) dx

#

dy:

TUTORIAL QUESTIONS

1. Consider the integral

R R

D

f(x; y) dx dy where D is the region in R

2

bounded by the

urves y = 2 x

2

and y = jxj; and the funtion f is given by f(x; y) = x+ y:

(a) Sketh the region of integration, D:

(b) Write

R R

D

f(x; y) dx dy as a repeated integral, stating preisely your limits of

integration.

() Evaluate

R R

D

f(x; y) dx dy:


R

1

1

R

1

jyj

(1 + 2y)e

x

2

dx dy:

(a) Sketh the region of integration.

(b) Write the integral as a repeated integral in whih one rst integrates with respet

to y and then with respet to x; stating preisely your limits of integration.

() Evaluate the given integral by use of (b).

3. LetD be the region bounded above by y = 2x and below by y = x

2

. Give a mathematial

expression for the region D and hene evaluate the integral

R R

D

f dx dy where f(x; y) =

p

2x+ y + 1.

4. Consider the integral I =

R

1

1

R

1

jyj

e

x

2

dx dy.

(a) Sketh the region of integration of the integral I.

(b) Give TWO mathematial expressions for the region D.

() Evaluate the integral I by using Fubini's Theorem to reverse the order of integra-

tion.

5. Let D be the region bounded above by y = 4x

2

and below by y = 0. Sketh the region

D. On your sketh draw the line y = 3x. Hene evaluate the integral

R R

D

j3xyj dy dx.

6. Let D be the region bounded above by y = 1 and below by y = jxj.

(a) Sketh the region D in the (x; y)-plane.

(b) Give a mathematial expression for the region D.

() Hene evaluate the integral

R R

D

f dx dy where f(x; y) = ye

x

.


I =

Z

p

p

Z

p

jyj

sin(x

2

) dx dy:



(a) Sketh the region of integration of the integral I.

(b) Give TWO mathematial expressions for the region D.

() Evaluate the integral I by using Fubini's Theorem to reverse the order of integra-

tion.

2.5 Change of Variables

Theorem 2.5.1 Let T : R

2

! R

2

and let D be a region in R

2

. Suppose that D

is a region

in R

2

suh that T is one-to-one on D

and T(D

) = D. Then

Z Z

D

f dx dy =

Z Z

D

f(T(u; v))

T(u; v)

(u; v)

du dv:

Note 2.5.2 In the above theorem, D is the region of integration with respet to the o-

ordinates (x; y) and D

is the region in terms of the o-ordinates (u; v). In addition, if we

write

x

y

= T

u

v

then the above theorem beomes

Z Z

D

f(x; y) dx dy =

Z Z

D

f(T(u; v))

(x; y)

(u; v)

du dv:

TUTORIAL QUESTIONS

1. Using the hange of variables u = x+ y and v = x y, evaluate the integral

Z Z

D

e

xy

x+y

sin[(x + y)

2

dx dy

where D is the region of R

2

bounded by the lines x = y; y = 0 and x + y =

p

.

2. Using the hange of variables x = r os and y = r sin , evaluate the integral

Z Z

D

x

2

+ 2y

2

+

y

4

x

2

e

xy+

y

3

x

dx dy;

where D is the setor of the annulus 4 x

2

+ y

2

9 in the rst quadrant between the

line y = x and the positive y-axis.

3. Using the hange of variables u = xy; v =

x

y

; x; y; u; v > 0; evaluate the integral

R R

D

x

2

+ y

2

dx dy where D is the region in the rst quadrant bounded by the urves

x = y; x = 2y; xy = 4; xy = 9:

4. Let D be the region in the (x; y)-plane given by 1 x

2

y

2

9 and 4 xy 25, where

x > 0 and y > 0. Let u = x

2

y

2

and v = xy, where u > 0 and v > 0.




(b) Using the denitions of u and v as given above, give an expression for D in terms

of u and v. Denote this region in the (u; v)-plane by D

. Sketh the region D

in the

(u; v)-plane.

() Let T (u; v) =

x

y

. Using the fat that

T (u;v)

(u;v)

= 1

.

T

1

(x;y)

(x;y)

T (u;v)

nd

T (u;v)

(u;v)

.

(d) Let f(x; y) = x

2

+ y

2

. Give an expression for f(T (u; v))

T (u;v)

(u;v)

.

(e) By hanging to the variable u and v, evaluate the integral

R R

D

f(x; y) dx dy.

5. Using the hange of variables x = r(3 os + sin ) and y = r(3 os sin ); r > 0; 2

[0; 2; evaluate the integral

Z Z

D

1

5x

2

8xy + 5y

2

dx dy

where D is the region given by 9 10x

2

16xy + 10y

2

36:

6. Let D be the region in the (x; y)-plane bounded by the urves y = x

2

2x + 2, y = x

2

and x = 0. Let x = v and y = v

2

+ 2u.


(b) Using the denitions of u and v as given above, give an expression for D in terms

of u and v. Denote this region in the (u; v)-plane by D

. Sketh the region D

in the

(u; v)-plane and give a mathematial expression for D

.

() Let T (u; v) =

x

y

. Find

T (u; v)

(u; v)

:

(d) Let f(x; y) = e

yx

2

+ x. Give an expression for f(T (u; v)).

(e) By hanging to the variable u and v, evaluate the integral

R R

D

f(x; y) dx dy.

2.6 Classial Integration Theorems

Theorem 2.6.1 Green's Theorem

Let D be a region in R

2

with boundary D oriented antilokwise, then for F : R

2

! R

2

we

have

Z Z

D

F

2

x

1

F

1

x

2

dx

1

dx

2

=

Z

D

F dr:

Note 2.6.2 Some other forms of Green's Theorem are

Z Z

D

Q

x

P

y

dx dy =

Z

D

P dx +Q dy

and if we identify F(x) with the vetor in R

3

given by

2

4

F

1

F

2

0

3

5

then Green's Theorem an be

written as

Z Z

D

(r F) e

3

dx

1

dx

2

=

Z

D

F dr:



We only prove Green's Theorem for the ase of a region D in R

2

of the form

D = f(x; y)jp(x) y q(x); x 2 [a; bg = D = f(x; y)jg(y) x h(y); y 2 [; dg:

We prove it by rst proving the following two Lemmata.

Lemma 2.6.3 Let D be a region in R

2

of the form D = f(x; y)jp(x) y q(x); x 2 [a; bg,

then

Z

D

f

0

dr =

Z Z

D

f

y

dy dx:

Lemma 2.6.4 Let D be a region in R

2

of the form D = f(x; y)jg(y) x h(y); y 2 [; dg

then

Z

D

0

g

dr =

Z Z

D

g

x

dx dy:

TUTORIAL QUESTIONS

1. Prove the seond Lemma.

2. Let D be the region in the (x; y)-plane given by x 2 [a; b, g(x) y f(x), with

g(a) = f(a) and g(b) = f(b). Let

1

and

2

be the urves bounding D from below and

above, and oriented so as to make

1

[

2

= D.

(a) Parametrize

1

and

2

and hek whether your parametrization are orientation

preserving or reversing.

(b) Using your parametrizations from (a), give an expression for I =

R

D

P

0

dr:

Simplify your expression for I as far as possible.

() Using the information you have about the region D, Write out and evaluate as far

as possible J =

R R

D

P

y

dy dx:

(d) Hene show that I = J .

3. Using Green's Theorem, evaluate the path integral

Z

e

x+2y

+ xy

2e

x+2y

+ sin y

dr

where is the triangle with verties (1; 1); (2; 1) and (2; 3) taken antilokwise.

4. Using Green's theorem prove that the area of a region D in R

2

is given by

R

D

ax dy +

(a 1)y dx for all a 2 R, where D denotes the boundary of the region D taken

antilokwise.

5. Using the previous question, nd the area enlosed by the loop in the urve r(t) =

t t

3

1 + t

2

:

6. Let D be the polygon with verties (x

i

; y

i

); i = 1; :::; N; listed antilokwise. For onve-

niene denote x

0

= x

N

and y

0

= y

N

.



(a) Parametrize the line-segment,

i

, from (x

i1

; y

i1

) to (x

i

; y

i

).

(b) Using your parametrization, evaluate

Z

i

y

0

dr:

() Noting that D =

1

[ ::: [

N

, give an expression for the area of the polygon D

in terms x

0

; :::; x

N

; y

0

; :::; y

N

.

7. Suppose that at the point x 2 R

3

a uid has veloity v(x). We say that the uid has

angular veloity w(x) = x v(x) at x relative to the origin.

(a) Let x =

2

4

x

1

x

2

0

3

5

and v(x) =

2

4

v

1

(x)

v

2

(x)

0

3

5

. Compute w(x) and show that w(x) =

2

4

0

0

W (x)

3

5

for some funtion W : R

3

! R.

(b) Let r(t) =

2

4

os t

sin t

0

3

5

; t 2 [0; 2. Let C

= fr(t)jt 2 [0; 2g. Show that

Z

C

v dr =

Z

C

W ds:

() State Green's theorem and hene show that

Z

C

W ds =

Z Z

D

(r v) e

3

dx

1

dx

2

where D

is the dis entre (0; 0) with radius in the (x

1

; x

2

)-plane.

(d) By the mean value of W over C

we mean the integral of W over C

divided by

the length of C

. By the mean value of (r v) e

3

over D

we mean the integral

of (r v) e

3

over D

divided by the area of D

. Show how the mean value of W

on C

and the mean value of (r v) e

3

over D

are related.

8. Fundamental Theorem of Vetor Calulus

Let r(t); t 2 [a; b; parametrize a path in R

n

. Let A = r(a) and B = r(b). Using the

hain rule prove that for ' : R

n

! R we have

Z

r' dr = '(B) '(A):

9. The \baby" Gauss Divergene Theorem

Let F : R

2

! R

2

and set G =

F

2

F

1

and let D be region in R

2

with boundary D

oriented antilokwise.

(a) Using Green's theorem and G as given above, show that

Z Z

D

r F =

Z

D

F

1

dy F

2

dx:



(b) Let D be parametrized by r(t); t 2 [a; b. Show that the unit outward normal, n,

to D at r(t) is

n =

1

jr

0

(t)j

dr

2

dt

(t)

dr

1

dt

(t)

:

() Thus show that

Z Z

D

r F =

Z

D

F n ds:

10. Dierentiation of Integrals.

Let H(x; y; z) =

R

y

x

f(z; p) dp and r(t) =

2

4

'

1

(t)

'

2

(t)

t

3

5

.

(a) Compute H

0

(x; y; z).

(b) Using the hain rule, ompute (H r)

0

(t).

() Hene show that

d

dt

Z

'

2

(t)

'

1

(t)

f(t; p) dp =

Z

'

2

(t)

'

1

(t)

f(t; p)

t

dp+ f(t; '

2

(t))'

0

2

(t) f(t; '

1

(t))'

0

1

(t):


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Multivariable Calculus Questions