CAAM 402/502 Spring 2013
Homework 1
Solutions
1. (a) Prove that there is no real valued function f : V R, dened on a set V Rn , that has
directional derivates at a give c V , satisfying Du f (c) > 0, for all u Rn , u = 1.
Proof. Suppose that there exists

CAAM 402/502 Spring 2013
Homework 2
Solutions
1. Problem XVII.1.1 in Lang. Let E be a vector space and let v1 , ., vn E be a basis for E . Show that
any linear map : E F into the normed vector space F is continuous.
Proof. Let us denote the norm of F by F

CAAM 402/502 Spring 2012
Homework 3
Solutions
1. Consider the map f : [1, ) R dened by
f (x) =
x1
+.
2x
Prove that this map is contractive and conclude that it has unique xed point. What is the xed
point?
Proof. First we need to show that f ([1, ) [1, ).

CAAM 402/502 Spring 2013
Homework 4
Solutions
1. Let f : V Rn be a continuous function dened on the open set V Rn . Suppose that f is injective
on V , has all the rst order partial derivatives, and its Jacobian Jf (x) satises det Jf (x) = 0 for all
x V .

CAAM 402/502 HW5
Spring 2013
Problem 1
Prove that there exist functions u and v dened on R4 , with values in R, that are continuously dierentiable
on some ball B R4 , centered at the point
(x , y , z , w ) = (2, 1, 1, 2)
such that
u(x , y , z , w ) = 4, v

CAAM 402/502 Spring 2013
Homework 1
Solutions
1. (a) Prove that there is no real valued function f : V R, dened on a set V Rn , that has
directional derivates at a give c V , satisfying Du f (c) > 0, for all u Rn , u = 1.
Proof. Suppose that there exists

CAAM 402/502 Spring 2013
Homework 2
Solutions
1. Problem XVII.1.1 in Lang. Let E be a vector space and let v1 , ., vn E be a basis for E . Show that
any linear map : E F into the normed vector space F is continuous.
Proof. Let us denote the norm of F by F

CAAM 402/502 Spring 2012
Homework 3
Solutions
1. Consider the map f : [1, ) R dened by
f (x) =
x1
+.
2x
Prove that this map is contractive and conclude that it has unique xed point. What is the xed
point?
Proof. First we need to show that f ([1, ) [1, ).

CAAM 402/502 Spring 2013
Homework 4
Solutions
1. Let f : V Rn be a continuous function dened on the open set V Rn . Suppose that f is injective
on V , has all the rst order partial derivatives, and its Jacobian Jf (x) satises det Jf (x) = 0 for all
x V .

CAAM 402/502 HW5
Spring 2013
Problem 1
Prove that there exist functions u and v dened on R4 , with values in R, that are continuously dierentiable
on some ball B R4 , centered at the point
(x , y , z , w ) = (2, 1, 1, 2)
such that
u(x , y , z , w ) = 4, v