Date Assigned
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Date Due
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Section
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Page
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Problems
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Comments
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| 22 Jan |
29 Jan |
§1.1 |
7 |
# 1, 4, 7, 10, 11, 14 |
-
Problem 2, that I had originally assigned, is not stated correctly. The function that is a solution to the heat equation (in one spatial dimension) is
\( u(t,x)=t^{-1/2} e^{-x^2/4kt} \).
More generally, the solution to the heat equation in \( d \) spatial dimensions,
\( u_t = k(u_{x_1x_1}+u_{x_2x_2}+\dots+u_{x_dx_d}) \)
is
\( u(t,x)=t^{-d/2} e^{-(x_1^2+x_2^2+\dots+x_d^2)/4kt} \).
Solutions
|
| 29 Jan |
5 Feb |
§1.2 |
25 |
# 1, 4, 5, 9, 16, 21 |
When the book asks you to "determine the sum of the series"
it's sufficient to draw a graph of the function on the given
interval - including endpoints.
I am not interested in seeing all of your work to evaluate the
integrals to find the Fourier coefficients. You should show what
integral is evaluated, but feel free to use other resources to
complete the integration.
Solutions
|
| 29 Jan |
5 Feb |
§1.3 |
30 |
# 1, 4 |
These problems involve the same ODE, but different boundary conditions.
Be sure to consider the cases of \( \lambda>0 \), \( \lambda=0 \), and \( \lambda<0 \).
Solutions
|
| 5 Feb |
12 Feb |
§2.1 |
60 |
# 3, 4, 6, 9, 11 |
|
| 5 Feb |
19 Feb |
§2.2 |
68 |
# 6 |
|
| 5 Feb |
19 Feb |
§2.3 |
74 |
# 2 |
|
| 12 Feb |
19 Feb |
§3.1 |
105 |
# 7 |
|
| 24 Feb |
Exam 1 |
Chapters 1 -- 3
|
|
| 26 Mar |
1 Apr |
§3.2.4 |
130 |
# 7 |
|
| 26 Mar |
1 Apr |
§3.2.5 |
136 |
# 3 |
|
| 26 Mar |
1 Apr |
§3.3.1 |
139 |
# 1, 3 |
|
| 30 Mar |
13 Apr |
§4.2 |
157 |
# 1, 6 |
|
| 30 Mar |
13 Apr |
§4.3 |
163 |
# 10 |
|
| 30 Mar |
13 Apr |
§4.5.2 |
193 |
# 2 |
|
| 16 Apr (take-home) |
Exam 2 |
Chapters 3 -- 4
|
Due by 5pm on Monday, April 27, 2015.
If you have any questions, please ask me - and only me.
You are not to consult with any other animate object.
If you use any inanimate sources, please document this.
To encourage you to work on this well before this deadline, the last opportunity
to ask any questions about the questions on this test is the end of class on the last class:
2:30pm on Thursday, April 24.
Exam 2
Exam 2 Solution Key
|
| 14 Apr |
|
§5.1 |
193 |
# 2 |
Maple worksheet for computing and plotting Fourier integral representations of functions (from class on 14 Apr 2015).
Here's a link to the Wikipedia page about Buffon's needle. This is a simple experiment that has a surprising answer, involving \(\pi\).
|
| 30 Apr (noon-2:30pm) |
Final Exam |
Chapter 5 |
My plan is to give some problems based on §§ 5.2, 5.5, and 5.6 in the text.
We went over §5.2, about the heat equation on the real line, in detail in class.
The other two sections consider the wave equation (§5.5) and Laplace's equation
(§5.6) on the real line. All involve Fourier integral representations, and are very
similar to what we did earlier on a bounded domain. I am not expecting you to memorize
any facts about these equations. What I am interested in is whether you can apply the
basic ideas (separation of variables, identification of eigenvalues and eigenfunctions,
matching coefficients to satisfy initial conditions, ...). Sample problems would include
#7 (p. 243), #7-15 (pp. 250-2). I'm not going to expect you to be able to perform random
tricks using complex analysis or fancy integrations; focus on the basic ideas that we've
used throughout the semester.
The exam starts at 12:30pm and concludes 150 minutes later - at 3:00pm.
Grades will be posted as soon as they are completed.
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