Solutions for Math 317 Assignment #7
(1) Consider the following equation
x2 + y + sin(xy) = 0.
(a) Prove that this equation has a unique continuous solution
y = y(x) such that y(0) = 0 in a small neig
Solutions for Math 317 Assignment #5
(1) Find the volume of the set
B = cfw_x2 + y 2 + z 2 4, 3 x + 2y + 2z 3 R3 .
Solution. Let f : R3 R3 be a linear map given by
f1 (u, v, w)
u
f2 (u, v, w) = A v
Math 317 Assignment #9
Due Apr 4, 2011
(1) Let fn (r) be the content of the n-dimension ball of radius r,
i.e.,
fn (r) = (Br (0) = (cfw_x2 + x2 + . + x2 r2 ).
1
2
n
(a) Show that
r
fn+1 (r) =
fn ( r2
Solutions for Math 317 Assignment #1
(1) Let S1 Rm and S2 Rn be two bounded sets. If either
(S1 ) = 0 or (S2 ) = 0, then (S1 S2 ) = 0 in Rm+n .
Proof. WLOG, we assume that (S1 ) = 0. So for all r > 0,
PROOF OF CHANGE OF VARIABLES
We will give a proof of the following theorem.
Theorem 0.1. Let U Rn be an open set, C 1 (U, Rn ) and K U
be a compact set with content. Suppose that is injective and det
Math 317 Winter 2014 Homework 3 Solutions
Due Feb. 5 2p
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answers.
Question 1. Calculate the radius
Math 317 Winter 2014 Homework 1 Solutions
Due Wednesday Jan. 15, 2014 2pm
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answers.
Question 1. Ar
Math 317 Winter 2014 Homework 2 Solutions
Due Jan. 29 2p
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answers.
Question 1.
a) Prove the root t
Math 317 Winter 2014 Homework 4 Solutions
Due Feb. 26 2p
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answers.
1 x>0
Question 1. Calculate th
ID:
Name:
Math 317 Quiz 2 Solutions
Jan. 20, 2014
The quiz has three problems. Total 10 + 1 points. It should be completed
in 20 minutes.
Question 1. (5 pts) Prove the convergence of the series
n=1
(n
ID:
Name:
Math 317 Quiz 3 Solutions
Feb. 3, 2014
The quiz has three problems. Total 10 + 1 points. It should be completed
in 20 minutes.
Question 1. (5 pts) Prove by denition that
n x3 + 5
3 n2 + 7
fn
ID:
Name:
Math 317 Quiz 1 Solutions
Jan. 8, 2014
The quiz has three problems. Total 10 + 1 points. It should be completed
in 20 minutes.
Question 1. (5 pts) Prove by denition or Cauchy criterion that
Math 317 Winter 2014 Homework 5 Solutions
Due Mar. 12 2p
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answers.
Question 1. Let x 2 (0; 1). Rec
Name:
ID:
Math 317 Quiz 4 Solutions
Mar. 3, 2014
The quiz has three problems. Total 10 + 1 points. It should be completed
in 20 minutes.
n
Question 1. (5 pts) Find a power series
n=1 an (x x0) such th
Name:
ID:
Math 317 Quiz 5 Solutions
Mar. 17, 2014
The quiz has three problems. Total 10 + 1 points. It should be completed
in 20 minutes.
Question 1. (5 pts) Let A be the set of all nite rectangles in
Math 317 Winter 2014 Homework 6 Solutions
Due Mar. 26 2p
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answers.
Question 1. In the following a