Math 126 - Fall 2014 - Homework 3 - Solutions
Problem 1. Let a > 0. Write an integral formula for the solution to
ut u + au = 0 for (t, x) (0, ) Rd
u(0, x) = f (x)
for x Rd .
If v(t, x) solves vt v =
Math 126 - Fall 2014 - Homework 1 - Solutions
Problem 1. Let r : Rd R be dened by r(x) = |x|. Compute the following:
x2
xi
x
1
i
(ii) r = |x|
(iii) xi xi r = |x| |x|3
(iv) r = d1
(i) xi r = |x|
|x|
Pr
Math 126 - Fall 2014 - Final Practice Problems
Be sure to review the homeworks, with an emphasis on the following problems:
Homework 1. Problems 2, 5, 7, 8
Homework 2. Problems 3, 4, 5
Homework 3.
Math 126 - Fall 2014 - Practice Problem Set 2
Be sure to review Homeworks 58, in particular the following problems:
Homework 5. Problems 3, 4
Homework 6. Problems 2, 3, 5
Homework 7. Problems 2, 3,
1832832885
11:11 5188422415 UCE MIN LIERRV
MATH 126 FINAL EXAM 30E. L- EVCL of:
Problem #1. Recall from the calculus of variations that minimizers
of the energy
EM 2 / F(s:,u, Va) do:
D
satisfy the Eu
Math 126 - Fall 2014 - Practice Problem Set 1
Be sure to review Homeworks 14, in particular the following problems:
Homework 1. Problems 2, 5
Homework 2. Problems 3, 4, and 5
Homework 3. Problems 1
Math 126 - Fall 2014 - Homework 5 - Solutions
Problem 1. Let Rd be an open, bounded
smooth solution to
ut u = 0
u(0, x) = f (x)
u(t, x) n(x) = 0
Show that
set with smooth boundary . Suppose u is a
(t
Math 126 - Fall 2014 - Homework 8 - Solutions
Problem 1. Use the method of characteristics to solve the following.
(i)
x1 x2 u x2 x1 u = u x ,
u(x) = g(x)
x ,
where = cfw_x R2 : x1 > 0, x2 > 0 and = c
HW 2, Math 126, Spring 2017
3. The characteristic function of a set E is defined as follows:
1 for x E
E (x) =
0 for x
/ E.
Find a formula for the convolution [0,1] [0,1] (x). What is the support of
Samantha Wu
HW 3, Math 126, Spring 2017
1. Show that the only solution u D0 (R) of u0 = 0 is u = c, where c is a constant
function.
R
Want to show c R s.t. u() = R c (x)dx. To do this, first show that
Math 126 - Fall 2014 - Homework 9 - Solutions
Remark. The solution to all of these problems follows essentially the same scheme, which goes as
follows. First consider the unconstrained problems. For a
Math 126 - Fall 2014 - Homework 11
Hard copy due: Monday 12/8/2014
at 1:00pm
Problem 1. Let Q, Q be hermitian operators on L2 and let L2 . Show that
E([Q, Q ]) := , [Q, Q ]
is pure imaginary, where [Q
14.
Quantum mechanics
17
1. Introduction/background
1.1. Notation. The standard basis for Rd is denoted by cfw_e1 , . 1 . , ed , with d 1 = (1, 0, . . . , 0),
.
e
etc. For x Rd we write x = (x1 , . .
HW 2, Math 126, Spring 2017
3. The characteristic function of a set E is defined as follows:
1 for x E
E (x) =
0 for x
/ E.
Find a formula for the convolution [0,1] [0,1] (x). What is the support of
Homework Solutions
Albert Ai
UC Berkeley, Summer 2015
Revised: August 4, 2015
Please call me out on errors as usual.
Exercise 0.1. (Save for Homework 6) Given the heat mean value property for (t0 , x0
Homework 2 Solutions
Albert Ai
UC Berkeley, Summer 2015
Revised: July 13, 2015
Exercise 0.1. Show that for f D0 , and any partial ,
( 0 ) f = f,
where everything is in the sense of distributions. In p
Math 126 Midterm
UC Berkeley, Summer 2015
July 15, 2015
Name:
Justify your work within reason. For instance, its always nice to highlight places where given
information is used. Note its much easier t
Homework 2 Solutions
Albert Ai
UC Berkeley, Summer 2015
Revised: July 6, 2015
Exercise 0.1. Let : C(Rd ) D0 (Rd ) be given by
Z
f () = f (x)(x) dx.
Use the previous proposition to show that the map f
Math 126 Midterm
UC Berkeley, Summer 2015
July 15, 2015
Name:
Justify your work within reason. For instance, its always nice to highlight places where given information is used. Note its much easier t
Math 126 Homework 1 (Due Friday Sept 4)
1. Course logistics.
(a) Read the entire course syllabus: https:/math.berkeley.edu/~jcalder/126F15
When are the midterm and final exams? What is the homework po
Math 126 Homework 2 Solutions
1. For i = 1, 2, let ui be a solution of the Dirichlet problem
uit
kuixx = 0
ui (x, 0) = 'i
9
for 0 < x < l and t > 0,>
=
for 0 x l,
>
;
ui (0, t) = g i (t) and ui (l, t)