Math 3335 Exam 1.
Sanders Spring 2013
This exam has ve problems, and all ve will be graded. Use my supplied paper only.
Return your solution sheets with the problems in order. Put your name, last name rst,
and student id number on each solution sheet you
Math 3335 Exam 1.
Sanders Fall 2014
This exam has ve problems, and all ve will be graded. Use my supplied paper only.
Return your solution sheets with the problems in order. Put your name, last name rst,
and student id number on each solution sheet you tu
MATH 3335 Final Exam. Sanders Fall 2012
This exam has 10 problems, and all 10 problems will be graded. Use my supplied paper
only. Return your solution sheets with the problems in order. Put your name, last name
rst, and student id number on each solution
MATH 3335 Final Exam. Sanders Spring 2013
This exam has 10 problems, and all 10 problems will be graded. Use my supplied paper
only. Return your solution sheets with the problems in order. Put your name, last name
rst, and student id number on each soluti
Cylindrical and Spherical Coodinates
In cylindrical coordinates, we have the change of variables
x = r cos
y = r sin
z = z.
C.1. Compute the normalized basis vectors er , e and ez .
Answers:
0
sin
cos
sin , e = cos , ez = 0 .
er =
1
0
0
C.2. From yo
Math 3335 Exam 2.
Sanders Fall 2012
This exam has ve problems, and all ve will be graded. Use my supplied paper only.
Return your solution sheets with the problems in order. Put your name, last name rst,
and student id number on each solution sheet you tu
Math 3335 Exam 1.
Sanders Fall 2012
This exam has ve problems, and all ve will be graded. Use my supplied paper only.
Return your solution sheets with the problems in order. Put your name, last name rst,
and student id number on each solution sheet you tu
Math 3335 Supplemental Notes and Homework 4
Here well consider the question of dierentiability of functions F : Rn where is an
open subset of Rm . F is dierentiable at x0 if there exists an n m matrix denoted
by Dx F(x0 ) such that
|F(x) (F(x0 ) + Dx F(x0
Math 3335: Orthogonal Change of Variables
In these exercises our goal will be to determine how the d -dimensional gradient transforms in generalized orthogonal coordinates. Consider the transformation x = x(u) . For
example, 2-dimensional cylindrical coor
Math 3335 Homework 1: Vectors, Vector Spaces and Inner Products
At a minimum to qualify as a vector space, vector addition and scalar multiplication must
adhere to certain requirements. Below, let x , y and z be arbitrary vectors in V , and let
and be ar
An Introduction to Complex Functions
A complex number z has standard form z = x + iy where x and y are real numbers and i
has the property that i2 = 1. The real part, resp. imaginary part, of a complex number
z is denoted and given by
(z ) = x R
resp.
(z
Here I derive several identities found in Table 3.2 on page 147 of your text book. Theyre
not in order, but all of the more dicult ones are here. Ones near the top are those I
consider the easiest. As you go down they become more dicult.
3.40: ( ) = 0
(
Homework 2
The Kronecker delta symbol, i,j , and the Levi-Civita epsilon symbol, i,j,k , are dened
as follows:
+1 if (i, j, k) is an even permutation of (1, 2, 3)
1 if i = j
i,j =
i,j,k = 1 if (i, j, k) is an odd permutation of (1, 2, 3)
0 otherwise,
0 ot
Work through all examples from Section 1.7 in your text. Also do exercises 7, 8 and 9 on
page 23.
These next three line integral exercises come from page 190 of your text.
2.1. Compute F dx where F(x) = x2 ex + ey + yz ez and is given by x(t) =
t ex + 2t2
The Chain Rule
Let G : Rl Rm and F : Rm Rn , and suppose G(x) is dierentiable at x = x0
and suppose F(G) is dierentiable at G = G(x0 ) . Then, the composition function
F(G) : Rl Rn is dierentiable at x0 , and its derivative is given there by the formula
D
Homework 6: A Bit of Complex Variables
1. Each of the following can be written as f (x + iy ) = u(x, y ) + iv (x, y ) where u(x, y ) R
and v (x, y ) R . Determine the functions u(x, y ) and v (x, y ) .
1
(a) f (z ) = z 2
(b) f (z ) = ez
(c) f (z ) = cos z
Math 3335, Vector Analysis. Sanders, Fall 2014
Text:
Introduction to Vector Analysis, 7th Ed., by Davis and Snider,
Hawkes Publishing, 2000. ISBN 0-697-16099-8.
Exams:
Two midterm exams ( 2 30% ) and a cumulative nal exam ( 40% )
Reading: Chapter 1: Vecto
Math 3335: Orthogonal Change of Variables
In these exercises we will explore what are called orthogonal change of coordinates. In
general a d dimensional change of coordinates is a mapping
x1 (u1 , . . . , ud )
.
.
x : Rd Rd , x(u) =
.
.
xd (u1 , . . . ,