Solutions to Homework 4
1. For each function below, identify the functions domain and range.
Then sketch at least three non-empty level curves of each function.
Finally, determine if the domain is open, closed, or neither.
(a) f (x, y ) = y x
Solution: y
LB 220 Homework 3 (due Monday, 01/28/13)
Directions. Please solve the problems below. Your solutions must begin
with a clear statement (or re-statement in your own words) of the problem.
You solutions should be clear, legible, and demonstrate at minimum p
LB 220 Homework 2 (due Tuesday, 01/22/13)
Directions. Please solve the problems below. Your solutions must begin
with a clear statement (or re-statement in your own words) of the problem.
You solutions should be clear, legible, and demonstrate at minimum
LB 220 Homework 1 (due Monday, 01/14/13)
Directions. Please solve the problems below. Your solutions must begin
with a clear statement (or re-statement in your own words) of the problem.
You solutions should be clear, legible, and demonstrate at minimum p
Solutions to Homework 8
Section 12.4 # 6: Convert the integral to an integral in polar coordinates
and then compute the integral:
4y 2
2
(x2 + y 2 ) dx dy
0
0
Solution: Begin by sketching the region. Since 0 x 4 y 2 and
0 y 2, one determines that the regi
Solutions to Homework 9
Section 12.7 # 12: Let D be the region bounded below by the cone
z = x2 + y 2 and above by the paraboloid z = 2 x2 y 2 . Setup
integrals in cylindrical coordinates which compute the volume of D.
Solution:
The intersection of the pa
Solutions to Homework 10
Section 12.8 # 6: Use the transformation u = x y and v = 2x + y to
evaluate the integral below, where R is the region in the rst quadrant
bounded by the lines y = 2x + 4, y = 2x + 7, y = x 2, and y = x + 1.
(2x2 xy y 2 ) dx dy
R
S
Solutions to Homework 5
1. Let z = f (x, y ) be a twice continously dierentiable function of x and
y . Let x = r cos and y = r sin be the equations which transform
polar coordinates into rectangular coordinates. Show that
2z
2z
2z
1 2 z 1 z
+ 2= 2+ 2 2+
.
LB 220 Homework 6 (due Wednesday, 04/10/13)
Directions. Please solve the problems below. Your solutions must begin
with a clear statement (or re-statement in your own words) of the problem.
You solutions should be clear, legible, and demonstrate at minimu
Solutions to Homework 7
1. Determine the work done by the vector eld
F = x y 2 , y z 2 , z x2
on a particle that moves along the line segment from (0, 0, 1) to
(2, 1, 0).
Solution: This vector eld is not conservative since one can check
that curl F = 0. S
LB 220 Homework 1 Solutions
1. The bond angle of the methane molecule.
The methane molecule consists of four hydrogen atoms and a single
carbon atom. The hydrogen atoms can be regarded as positioned at
the four vertices of a regular tetrahedron. In this c
A MISCELLANY OF MATHEMATICAL FORMULAE
Triple Integrals in Cylindrical & Spherical Coordinates
2 = x2 + y 2 + z 2
r 2 = x2 + y 2
r = sin
z = cos
x = r cos = sin cos
y = r sin = sin sin
dV = dx dy dz = r dz dr d = 2 sin d d d
Change of Variables for Dou
1
Vectors and the Scalar Product
1.1
Vector Algebra
vector in Rn : an n-tuple of real numbers
v = a1 , a2 , . . . , an .
For example, if n = 2 and a1 = 1 and a2 = 1, then w = 1, 1 is vector in R2 . Vectors are
represented by directed line segments. Every
A TOUR OF CALCULUS III VIA PROBLEM SOLVING
Chapter 11 Review, p. 829
1: How do you compute the component form of P Q given coordinates for P
and Q? How are vector addition, scalar multiplication, and magnitude
dened? What is a unit vector?
3: How do you d
Greens Theorem
Hypothesis: Let C be a counter-clockwise oriented, simple, closed,
piecewise smooth curve in R2 . Let F = M , N be a C 1 vector eld on an
open neighborhood of R, the bounded region enclosed by C .
Conclusion:
N
M
M dx + N dy
dA =
x
y
R
C
In