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|
Problem 2.
(i) Use the GaussGreen formula to deduce the d
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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 Euler-Lagrange equation
_Z: a (6F(.r,u,7u) +3F(m,u,7u)
6:
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, 4, and 7
Homework 4. Problems 2, 3, and 5
1. Prove t
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, 4
Homework 8. Problems 1, 3, 4
1. Use separation of v
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. Problems 5, 7
Homework 4. Problems 2, 4, 5, 6
Homewor
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, Q ] is the commutator dened by [Q, Q ] = QQ Q Q.
E([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 , . . . , xd ) to denote x = x1 e + + xd e . We call x1 , . .
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 given functional L you suppose that u
is a minimizer o
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 = cfw_x R2 : x1 > 0, x2 = 0 .
We rewrite this as
(x2 , x1
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, x) (0, ) ,
x
x .
u(t, x) dx
d
dt
u(t, x) n(x) dS = 0
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 = 0 with v(0, x) = f (x) then u(t, x) = eat v(t, x) solve
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 particular, convolution by 0 is the identity
operator. A
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 to earn partial credit for statements that are true,
pre
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 7 f is injective. Also check that f as defined
really i
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 to earn partial credit for statements that are true, pre
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 policy? How will
your final grade be computed.
(b) Sign u
Math 126 Midterm Information
The midterm will take place on Friday, October 9, during class.
The exam will cover everything up to and including the lecture on Friday Sept 25.
The exam is closed book. No textbooks, notes, or calculators are allowed.
Th
Math 126 Homework 4 (Due Friday Sept 25)
1. Maximum principle: Consider the heat equation
(
ut uxx = 0,
< x < , t > 0
(H)
u(x, 0) = (x), < x < .
As it turns out, there are infinitely many solutions u of the above heat equation. All
but one solution are n
Math 126 Homework 3 (Due Monday Sept 21)
1. Show that the wave equation does not, in general, satisfy a maximum principle.
2. For a solution u(x, t) of the wave equation
utt uxx = 0,
the energy density is defined as e = (u2t + u2x )/2 and the momentum den
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 ) = (0, 0) and
r = 1, prove the general case. As far a
Homework 5 Solutions
Albert Ai
UC Berkeley, Summer 2015
Revised: August 3, 2015
Please call me out on errors as usual.
Exercise 0.1. Let St : S(Rd ) S(Rd ) be the scaling-by-t operator, St f (x) = f (tx). Then show
that
1
d
S
t f = d S1/t f .
t
Extend bot
Math 126 HW3 (Partial) Solutions
2.4.8) Observe that for fixed t,
max S(x, t) = S(, t) = (4kt)1/2 e
2
/4kt
,
|x|<
since S(x, t) decreases as a function of |x|. We would like to take the limit of this expression as t 0.
Intuitively, it should be 0 since ex
Math 126 Final Exam Solutions
1. (a) Give an example of a linear homogeneous PDE, a linear inhomogeneous PDE, and
a nonlinear PDE. [3 points]
Solution. Poissons equation u = f is linear homogeneous when f 0 and
linear inhomogeneous when f 6= 0. The PDE u2
Math 126 HW13 (Partial) Solutions
12.4:1) Taking the Fourier transform in x, we find that
u
(k, 0) = (k).
u
t = k 2
u + (ik)
u
This is a first-order, linear, constant-coefficient ODE in t, which has solution
h
i
2
u
(k, t) = (k)
ek t eikt .
If we can wri
Math 126 Discussion Solutions for 8/4/16
1) Solve the nonlinear equation ut + uux = 0 with auxiliary condition u(x, 0) = x. Sketch some of the
characteristic lines.
Solution: Recall that the characteristics are straight lines with slope dx/dt equal to the
Math 126 HW1 (Partial) Solutions
1.1.4) Let u1 and u2 be solutions to the inhomogeneous linear equation Lu = g. Then,
L(u1 u2 ) = L(u1 ) + L(u2 )
= L(u1 ) L(u2 )
=gg
= 0.
Therefore, u1 u2 is a solution to the homogeneous linear equation Lu = 0.
1.1.12) Ob