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
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 2periodic extensions,
and for each x [0, 1] determine the value to which its Fourier s
section 1.3
Continuity
required reading

Continuous functions extend topological properties from one space to another.
When a subset of Rn is mapped to a subset of some other space Rk then one always needs to know which
topological properties remain inta
MAT 237, Quiz 1
Name
Tutorial section
Your Quiz 1 takes place on Tuesday May 22 during the tutorial. It will be about 15 min. long and will
cover sections 1.1 and 1.2. A quiz is a small model of an evaluation and tries to highlight elements of learning
a
MAT 237, Quiz 2
Name
Tutorial section
Your Quiz 2 takes place on Tuesday May 29 during the tutorial. It will be about 15 min. long and will
cover sections 1.2  1.4. A quiz is a small model of an evaluation and tries to highlight elements of learning
and
section 1.2
generalizing open intervals
required reading

The ordered set R
Recall the set of real numbers is an ordered set, and this ordering was consistent with the algebraic structure
of the reals. Indeed we learned in MAT137 that R was an ordered el
Suplementary reading on preimage of a set
This supplementary reading directly concerns sections 1.3, 1.6 and 1.7, and indirectly relates to
whenever topological arguments are used, such as in section 1.4 and 1.8. This reading introduces a
notation that ma
MAT 237, PS3

Due, Thursday June 7
beginning of tutorial
FAMILY NAME: FIRST NAME:
STUDENT ID:
Please note:
1. Your problem set must be submitted on this form. Please provide your final, polished solutions in the
spaces provided. You may use back of the
from R to Rn
section 1.1
required reading

Why Rn :
This course deals with functions of several variables; this means functions that take more than one variables
as their argument (or their input) denoted by
f (x, y), f (x, y, z)
or
f (x1 , x2 , . . . xn
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
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
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 ktensor on a vector space V .
(e) a kdimensional ma
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 2tensor on Rn .
Show that T is dierentiable at (0, 0) and compute
df (0
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 ktensor on Rn . Prove that T is C as a map Rnk R.
(3) Let M be a union of x
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 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 2tensor on Rn .
Show that T is dierentiable at (0, 0) and compute
df (0,
(1) Let M Rn be a kdimensional manifold. Let be an lform 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 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
Math 257/316 Assignment 2
Due Friday Jan 23 in class
Problem 1. Find the rst six nonzero terms in the power series y = an (x x0 )n of
n=0
the general solution of the following secondorder, linear, homogeneous ODEs, centred at
the indicated point x0 :
a)
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 3
Due Friday Jan 30 in class
1. The steadystate 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 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
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,