Math 21a: Multivariable calculus
Homework 6: Greens theorem
1 [Compare Stewart 13.4: 4] Evaluate the line integral using Greens
Theorem.
F dr
C
where F (x, y) = x, y and C consists of the line segments from
(0,1) to (0,0) and from (0,0) to (1,0) and the p
Math 21a: Multivariable calculus
Homework 5: Theorem of line integrals
1 [Compare Stewart 13.3: 12] a) Find a function f such that F =
f if F (x, y) = x2 , y 2 .
b) Use a) to evaluate C F dr along the parabola y = 2x2 from
(1, 2) to (2, 8).
Solution:
(a)
Math 21a: Multivariable calculus
Homework 8: Flux integral
T
1 [Compare Stewart 13.6: 22] Evaluate the ux integral
for the vector eld
S
F dS
F (x, y, z) = 3z, 3y, 3x ,
where S is the helicoid
r(u, v) = u cos v, u sin v, v , 0 u 1, 0 v
which has an upward
Math 21a: Multivariable calculus
Homework 4: Line integrals
1 [Compare Stewart 13.2: 20, 22] a) Evaluate the line integral
dr, where C is given by the vector function r(t).
C
F (x, y, z) = x + y, y z, z 2
r(t) = t2, t3, t2 , 0 t 1.
b) Evaluate the line in
Math 21a: Multivariable calculus
Homework 7: Curl and Div
This homework is due Friday, 11/15 rsp Tuesday 11/19.
1 [Compare Stewart 13.5 Problem 4] Find a) the curl and b) the
divergence of the vector eld
F (x, y, z) = sin(yz), sin(zx), sin(xy) .
Solution:
Math 21a: Multivariable calculus
Homework 13: Partial dierential equations
1 [Compare Stewart 11.3: 70] Determine whether each of the following functions is a solution of Laplaces equation uxx +uyy = 0.
a) u = 5x2 + 5y 2 b) u = 7x2 7y 2 c) u = x3 + 3xy 2
Math 21a: Multivariable calculus
Homework 9: Stokes Theorem
1 [Compare Stewart 13.7: 4] Evaluate S curl(F)dS, where F (x, y, z) =
x2 y 3 z, sin(xyz), xyz , where S is the part of the cone y 2 = x2 +z 2
that lies between the planes y = 0 and y = 3, oriente
FINAL EXAM
Math 21aF
Name:
MWF 9 Oliver Knill
Start by printing your name in the above box
and check your section in the box to the
left.
MWF 9 Chao Li
MWF 10 Gijs Heuts
Do not detach pages from this exam packet
or unstaple the packet.
MWF 10 Adrian Zah
Math 21a: Multivariable calculus
Homework 10: Divergence Theorem
1 [Compare Stewart 13.8 2] Find the ux of the eld F (x, y, z) =
x2, xy, z through the outwards oriented boundary of the solid
bound by the paraboloid z = 4 x2 y 2 and the xy-plane.
SECTION 1
Math 21a: Multivariable calculus
Homework 14: Cylindrical/Spherical integration
1 [Stewart 12.8: 2,4] Evaluate the following integrals.
a)
b)
2 9r2
r dz dr d
0 0 0
2
2 2
sin d d d
0
/2 1
Solution:
a) The region of integration is given in cylindrical co
Math 21a: Multivariable calculus
Homework 3: Vector elds
1 [Compare Stewart 13.1: 15-18] Match the vector elds F with the
plots labeled I-IV. Give reasons for your choices.
a) F (x, y, z) = 1, 2, 3 , b) F (x, y, z) = 1, 2, z
c) F (x, y, z) = x, y, 3 , d)