Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #11
1. The Minimal Surface Equation. Show that the minimal surface equation,
(1 + u2 )uxx 2ux uy uxy + (1 + u2 )uyy = 0,
y
x
(1)
is an elliptic PDE.
Solution: In this case, a = 1 + u2 , b = 2ux uy and c
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #9
1. Dene N : C 1 (R2 ) C(R2 ) by
N (u)(x, t) =
[u(x, t)] + u(x, t) [u(x, t)],
t
x
for all (x, t) R2 ,
and all u C 1 (R2 ). Show that N is not a linear operator.
Solution: Apply N to cu, where c is a s
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #8
1. Let R denote an open subset of R3 with smooth boundary, R, and f : R R
and : R R denote C 1 functions. Use the result of Problem 3 in Assignment
#1 to derive the following integration by parts for
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #7
Background and Denitions
Dirichlet Variational Problem. Let R denote a bounded region in R2 with smooth
boundary R. Let g denote a real valued function that is continuous in a neighborhood of R. Dene
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #5
1. Dene f : R R by
0,
e1/t ,
f (t) =
for t 0;
for t > 0.
(1)
Show that f is dierentiable at 0 and give a formula for computing f (t) for all
t R.
Solution: We rst show that
f (0 + h) f (0)
exists.
h0
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #3
1. Derive the Transport Theorem:
The Transport Theorem. Let f denote a C 1 scalar eld dened in a region
R in space in which a uid with velocity eld u is owing. Let B be any open
bounded subset of R a
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #6
1. In assignment #5 you showed how to construct a function : Rn R such
that
supp() = B1 (0) = cfw_x Rn | |x| 1,
the closed ball of radius 1 around the origin in Rn . Furthermore, > 0 in
B1 (0).
Given
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #1
1. Let R denote an open subset of R3 and f : R R denote a continuous function.
Suppose that
f dV = 0
(1)
B
for all bounded subsets, B, of R with smooth boundary. Show that f (x, y, z) = 0
for all (x,
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #2
1. Let f and g denote C 1 scalar elds dened in R. Use the denition of the
material derivative to verify that
D
Dg
Df
[f g] = f
+g
.
Dt
Dt
Dt
(1)
Solution: Apply the Product Rule to compute
d
d
[f (x(
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #4
1. Give a formula for computing the heat energy, Q(t), contained in the rod in the
section between x = a and x = b.
Solution: Since the mass in a small section of the rod of length x is, approximatel
Math 182. Rumbos
Spring 2014
1
Exam 1
Due on Friday, March 14, 2014
Name:
This is an opennotes, open text exam; you may consult your own notes, or the class
notes in my courses website at http:/pages.pomona.edu/~ajr04747/, or the text
for the course.
Stud
Math 182. Rumbos
Spring 2014
1
Exam 2
Due on Friday, April 25, 2014
Name:
This is an opennotes, open text exam; you may consult your own notes, or the class
notes in my courses website at http:/pages.pomona.edu/~ajr04747/, or the text
for the course.
Stud
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #14
1. Use the fact that
e
s2
0
ds =
2
(1)
to deduce that
(a) lim Erf(x) = 1; and
x
(b) lim Erf(x) = 1,
x
where Erf : R R is the error function dened by
x
2
Erf(x) =
2
es ds,
for x R.
(2)
0
Solution:
(
Math 182. Rumbos
Spring 2014
1
Solutions to Exam #1
1. Consider the system of linear rst order PDEs
u v = 0;
x y
(1)
u v
+
= 0,
y x
where u and v denote C 2 functions dened in an open region, R, of R2 . The
system of PDEs in (1) is known as the CauchyR
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #12
1. Find a solution to the initial value problem
u + u = 0,
x R, t > 0;
t
x
u(x, 0) = f (x),
x R,
where f (x) = 1 x2 for 1 x 1, f (x) = 0 for |x| > 1. For various values
of t, sketch the solution u
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #15
1. Let R denote the open square cfw_(x, y) R2 | 0 < x < , 0 < y < . Find all
values of for which the following BVP
(uxx + uyy ) = u in R;
u = 0, on R,
(1)
has nontrivial solutions. Those values are
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #16
1. Derive the following integrations identities:
sin(n) cos(m) d = 0,
for all m, n = 1, 2, 3, . . . ;
(1)
cos(n) cos(m) d =
0,
,
if m = n;
if m = n;
(2)
sin(n) sin(m) d =
0,
,
if m = n;
if m = n.
(3
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #13
1. Assume that u solves Laplaces equation in R2 . For xed (x, y) in R2 , dene
= x x;
= y y,
(1)
and set
v(, ) = u(x, y),
where x and y are given in terms of and by inverting the transformation
equ
Math 182. Rumbos
Fall 2014
1
Solutions to Assignment #10
1. Laplacian of Radially Symmetric Functions in R3 . A function u : R3 R
is said to be radially symmetric if there exists a realvalued function of a
single variable, f : [0, ) R, such that u(x, y, z