Practice test
(1) Give the denitions of the following notions.
(a) an open set in Rn ;
(b) a boundary point of a set A Rn ;
(c) a function f : Rn Rm dierentiable at a point p;
(d) a directional derivative of a function f : Rn Rm at a point p.
(2) Find the
Math 257/316 Assignment 6
Due Monday Mar. 2 in class
1. For the function
f (x) =
1 2x 0 x < 1/2
1
1/2 x 1
dened on [0, 1], sketch (several periods of) its even and odd 2-periodic extensions,
and for each x [0, 1] determine the value to which its Fourier s
Math 257/316 Assignment 1
Due Friday Jan 16 in class
Problem 1. Find the solution to the initial value problem for the ODE
dy
1+y
=
dx
1+x
for each of the initial conditions
(a) y(0) = 1,
(b) y(0) = 1,
(c) y(0) = 2
Problem 2. Find the general solutions to
Math 257/316 Assignment 7
Due Monday Mar. 9 in class
1. The concentration u(x, t) of a reactive chemical diusing in one dimension satises
0 < x < 2, t > 0
ut = uxx u,
u(0, t) = 1, u(2, t) = 1
.
u(x, 0) = 0
where the loss term represents a reaction which
MAT 257Y
Solutions to Term Test 1
(1) (15 pts) Let (X, d) be a metric space. Let A X
be a compact subset. Using only the denition of
compactness prove that A is closed.
Solution
Let U = X\A. We need to show that U is open.
Let p U . Let Un = cfw_x X| such
Math 257/316 Assignment 5
Due Monday Feb. 23 in class
1. For the triangle wave function
f (x) =
x
0x1
2x 1x2
dened on [0, 2], compute its
(a) compute its Fourier sine series
(b) computes its Fourier cosine series
(c) by evaluating f (1), use each of your
Math 257/316 Assignment 3
Due Friday Jan 30 in class
1. The steady-state temperature distribution y(x) along a wire 0 x 1, cooled by its
surroundings, and held xed at zero temperature at its left endpoint, solves:
d
p(x)y y = 0,
y(0) = 0,
dx
where p(x) 0
Math 257/316 Assignment 4
Due Friday Feb. 6 in class
1. Consider the heat conduction problem:
u
2u
= 5 2,
t
x
0 < x < 3, t > 0,
with homogeneous boundary conditions
u(0, t) = u(3, t) = 0.
Find the solution for each of the initial conditions (using formula
Math 257/316 Assignment 2
Due Friday Jan 23 in class
Problem 1. Find the rst six non-zero terms in the power series y = an (x x0 )n of
n=0
the general solution of the following second-order, linear, homogeneous ODEs, centred at
the indicated point x0 :
a)
MAT 257Y
Practice Term Test 1
(1) Find the partial derivatives of the following functions
(a) f (x, y, z) = sin(x sin(y sin z)
2
(b) f (x, y, z) = xyz
(2) give an example of a nonempty set A such that the set of limit points of A is the
same as the set of
(1) Let M Rn be a k-dimensional manifold. Let be an l-form on
M . recall that is called smooth if it can be extended to a smooth
form on an open set containing M .
a) Prove that is smooth if and only if its locally smooth. Here a
form on M is locally smoo
MAT 257Y
Practice Final
(1) Let A Rn be a rectangle and let f : A R be
bounded. Let P1 , P2 be two partitions of A . Prove
that L(f, P1 ) U (f, P2 ).
(2) Let T : R2n = Rn Rn R be a 2-tensor on Rn .
Show that T is dierentiable at (0, 0) and compute
df (0,
MAT 257Y
Practice Final 2
1. Let A Rn be a rectangle. Let f : A R be integrable.
Let
f+ (x) =
f (x) if f (x) 0
0 if f (x) < 0
Prove that f+ is also integrable on A.
2. Mark True or False. If true, give a proof. If false,
give a counterexample.
(a) Let S R
MAT 257Y
Practice Term Test 3
(1) Let v1 , . . . , vk Rn where n k. Prove that volk P (v1 , . . . , vk ) = 0 if and only if
v1 , . . . , vk are linearly dependent.
(2) Let T be a k-tensor on Rn . Prove that T is C as a map Rnk R.
(3) Let M be a union of x
MAT 257Y
Practice Final 1
(1) Let A Rn be a rectangle and let f : A R be
bounded. Let P1 , P2 be two partitions of A . Prove
that L(f, P1 ) U (f, P2 ).
(2) Let T : R2n = Rn Rn R be a 2-tensor on Rn .
Show that T is dierentiable at (0, 0) and compute
df (0
Past Final Exam
1. (12 pts) Give the following denitions
(a) an open set in Rn .
(b) a dierentiable function f : Rn R at a point p.
(c) an integrable function f on a rectangle A Rn .
(d) an alternating k-tensor on a vector space V .
(e) a k-dimensional ma
Chapter 3
THE IMPLICIT FUNCTION
THEOREM AND ITS APPLICATIONS
In this chapter we take up the general question of the local solvability of systems
of equations involving nonlinear differentiable functions. The main result is the
implicit function theorem,