Math 321
Vector and Complex Calculus
for the Physical Sciences
Fabian Waleffe
Department of Mathematics &
Department of Engineering Physics
University of Wisconsin, Madison
2016/2/27
2
Any fool can know. The point is to understand.
Albert Einstein
Prefac
Part of Hints for to Hw 5
Math 321
Exercises in 1.10
1. ~v = (etan t 1, 12 t t2 + 1 + 21 ln(t + t2 + 1), 2(t2t+1) + 12 arctan t).
~ is
2. This problem is harder than I thought. First of all show that ~vk (parallel to B)
always zero and thus the particle i
Math 321 Quiz 1 (G)
301/302 Feb. 7, 2012
Your Name:
Instructions: This is the quiz for 301/302. (Yes, at last, Ill normalize the quiz scores such
that the sections on Tuesday and the sections on Thursday have the same average quiz
score.) You have 15 minu
Part of Hints for Hw 13
Math 321
By Lei Li
1.5
1. Use the ratio test. Center is 1/2. Radius is 1. Then the domain of convergence is a disk
with radius one centered at 1/2
P
P
2. #4: The series is
(z 2 )n =
(1)n z 2n . Radius of convergence is 1.
n=0
#5:
1
Part of Hints for Hw 9
Math 321
By Lei Li
Some easy problems
1. a). (0 2, 0 2, 0 z 4)
b). dV = dddz
c). 256
3
2. a).
b). 3
25
3
1.5
R
R 2
R
#2. For area, use dSr Area = S R2 sin dd = 0 d 0 R2 sin d = 4R2
R
R R R 2
R
Volume: V = ball r2 sin drdd = 0 dr 0 d
Math 321 Quiz 5W November 2, 2011
Name: EVQVQSxOS D \mqu
Note: Show all your work. No calculators allowed.
1. (3 points) A curve is parametrized in cartesian coordinates as 7" =
a: ; + y 57, + h(:r, y) 63;, where h is a differentiable function. Compute
vi
Y7 o b)o\
Name . 7 . Math 321 Quiz 1, 2010/1/28
1.(5pts) Given a square ABCD, use vectors to ShOW that the two diagonals are perpen-
dicular.
B 2\= H
A i A? B3: (k+>=<-E+i>
2 _\0b+b.,~v_9i
\
:07
.33: fig ML: 0; in
A N N _\
b; L, N
-\~ Rik
I -,'Z
Name . Math 321 Quiz 8, 2010/4/15
1.(10pts) What is the Taylor series of (1:2)2 about 2 = 0? What is its radius of
convergence?
Q01 2( z)- - 2H); at rm < 1
/ 1+2: V: 0 v" :0 I\ < I
Take clerfva+ives
00 n Y!
(+%)2 M120
(>0 W m-
*3 20!) V1 1?
2
E Nl
2 24!)
Math 321 Quiz 4M October 24, 2011
Name: Evakvvsggbg D \mok
Note: Show all your work. No calculators allowed.
1. Let F: m), 27
dF/dt = mi ds = m, df/ds = m and a = d/dt.
(a) (1 point) Show that v = g.
2
(b) (3 points) Prove that if = dii: )
+ (112/071.
. 3030
Name Math 321 Quiz 4, 2010/2/25
1. (i) (3pts) What does (1 - 2: = 0 tell you about (1"?
do
(ii) (3pts) What does ("i X (1: = 0 tell you about ('1?
2 (16 a d5 (1 a 2 2
(111) (4pts) Suppose a - E = 0 and b- :1; = 0. What does 8201 x b) = 0 tell you a
Math 321 Quiz 8M November 28, 2011
Name: EVCMASQLQS D \mau
Note: Show all your work. No calculators allowed.
1. (3 points) Let p(f) = p0 pg(2 . F),'where p0, p and g are constants.
Compute fspdS, Where S is a closed surface and is the outside
unit normal
P2 0 A
Name . (j . Math 321 Quiz 2, 2010/2/4
1. Let fi be a unit vector and let 6 be any vector.
1) (3pm) Find (in.
iii) (2pts) Find (in and (ii if 73 is any nonzero vector (i.e. not necessary a unit vector).
3. = 3;? "A a fox x?
/ if? QJ. - (a )
51
. 3030
1. Let Q be an orthogonal transformation matrix.
i) (3pts) Find the (i,j)-entry of QQT.
T .= M '. T .
(QQ )U' 9&0 )9
Name Math 321 Quiz 3, 2010/2/11
ii) (3pts) Find the (i,j)-entry of QTQ.
T ' T x
(Q Q)U _ (Q )g'k Q13
1" QM Q 5'
iii) (2pts) What ar
Math 321 Quiz 7 November 2123, 2011
Name: EV ouncgzLOS \W\ow
Note: Show all your work. No calculators allowed.
1._(3 points) Consider the vector eld 17107) = f(7")f, Where 7" = [7"].
Compute f0 F - df, Where C is a curve from a point A to a point B. Is
th
Part of anwsers to Hw 2
Math 321
The problems with are for the ones who like math.
3
P
Section 1.4 #1: ~ej w
~=
wi ji = wj
i=1
#4:
3 P
3
P
vi wj~ai ~aj
i=1 j=1
#5: (i) ai (ii) a1 b1 + a2 b2 + a3 b3 (iii) 3
rn rn
n
P 2 P 2
P
xi
yi
Extra (a). Component form
Part of Hints for to Hw 3
Math 321
Section 1.6
P
2.(a). The final answer is ai . I dont want to write the corresponding
notation here.
However, if you are trying to understand Einsteins convention, you should do this.
(c). To overcome this difficulty, we
Part of Hints for Hw 10
Math 321
By Lei Li
One important concept problem
b). d~r ~ex = (dx~ex + dy~ey ) ~ex = dx. In the cases, dx = x0 (t)dt and dx = x0 (y)dy. Similar
for dy
c). Just use the relationship n
= t ~ez
3.4
1. You would get 2Area(A). Area of
Math 321
Vector and Complex Calculus
for the Physical Sciences
Fabian Waleffe
Department of Mathematics &
Department of Engineering Physics
University of Wisconsin, Madison
December 18, 2016
2
Any fool can know. The point is to understand.
Albert Einstei
M ATH 321 S ECTION 001 S PRING 2017
H OMEWORK 2 - S OLUTIONS ( FOR PERSONAL USE ONLY )
D UE DATE : JANUARY 30
Please show all your work and justify your answers.
Problem 1. An airplane travels at airspeed V with heading (clockwise from north). The wind ha
M ATH 321 S ECTION 001 S PRING 2017
H OMEWORK 4 - S OLUTIONS ( FOR PERSONAL USE ONLY )
D UE DATE : F EBRUARY 13
Please show all your work and justify your answers.
Problem 1. Prove that a + b bisects (divides in half) the angle between arbitrary vectors a
M ATH 321 S ECTION 001 S PRING 2017
H OMEWORK 3 - S OLUTIONS ( FOR PERSONAL USE ONLY )
D UE DATE : F EBRUARY 6
Please show all your work and justify your answers.
Problem 1. Consider v (t ) = a + tbb for t R, where a ,bb E3 are arbitrary fixed vectors.
(a
M ATH 321 S ECTION 001 S PRING 2017
H OMEWORK 5 - S OLUTIONS ( FOR PERSONAL USE ONLY )
D UE DATE : F EBRUARY 27
Please show all your work and justify your answers.
Problem 1. Find a vector equation for (a) the line passing through the two points r 1 , r 2
FW Math 321
Homework 5, F 2016/10/14
Problems due beginning of lecture on Thursday 10/20/2016, 9:30am.
Late homework not accepted. No email or web submissions.
You can discuss the homework with other students currently enrolled in Math 321 (no former
stud
FW Math 321
Homework 6, F 2016/10/21
Problems due beginning of lecture on Thursday 10/27/2016, 9:30am.
Late homework not accepted. No email or web submissions.
You can discuss the homework with other students currently enrolled in Math 321 (no former
stud
FW Math 321
Homework 4, F 2016/9/30
Problems due beginning of lecture on Thursday 10/6/2016, 9:30am.
Late homework not accepted. No email or web submissions.
You can discuss the homework with other students currently enrolled in Math 321 (no former
studen
FW Math 321
Homework 8, F 2016/11/4
Problems due beginning of lecture on Thursday 11/10/2016, 9:30am.
Late homework not accepted. No email or web submissions.
You can discuss the homework with other students currently enrolled in Math 321 (no former
stude
FW Math 321
Homework 3, F 2016/9/23
Problems due beginning of lecture on Thursday 9/29/2016, 9:30am.
Late homework not accepted. No email or web submissions.
You can discuss the homework with other students currently enrolled in Math 321 (no former
studen
FW Math 321
Homework 6, F 2016/10/28
Problems due beginning of lecture on Thursday 11/3/2016, 9:30am.
Late homework not accepted. No email or web submissions.
You can discuss the homework with other students currently enrolled in Math 321 (no former
stude
Part of anwsers to Hw 1
Math 321
Without specific declaration, the scalar set is always R, i.e. the set of all real numbers. The
problems with are for the ones who like math.
Section 1.1 #3:
Idea: Denote the set V . For any f, g V , we need to show that f