Messages

1

Chapter 10 / Re: S 10.1 Lemma

« on: April 02, 2023, 05:15:13 AM »

This condition is not necessary. However more conditions to $\phi$ lead to a lesser class of admissible $\phi$ in the assumption  and thus to weaker assumption of Lemma 1, and thus to stronger and more general statement.

2

Test 2 / Re: 2022 midterm 2 solution: Possible typos?

« on: March 25, 2023, 06:48:46 AM »

Thanks! Fixed

3

Chapter 6 / Re: Question while calculating the solution to Laplace using separation of vars

« on: March 24, 2023, 08:28:50 PM »

Section 4.1ODE is not enough. You need to take into account also boundary conditions. See Section 4.1

4

Chapter 4 / Re: When is it a good idea to assume solutions are separable?

« on: February 27, 2023, 09:28:36 AM »

If domain is "simple" in the appropriate coordinate system it is a good idea to try to find such solutions. For linear equations we even can construct general solutions as linear combinations of such solutions. Details in the class later.

5

Chapter 3 / Re: Solving Heat equation

« on: February 13, 2023, 08:26:20 PM »

I will post solutions to show that all the integrals can be computed using elementary functions and $\operatorname{erf}(.)$

6

Quiz 1 / Re: Quiz 1 solution

« on: January 29, 2023, 08:22:28 AM »

Yes––as long as you post your variant solution

7

Chapter 1 / Re: Ch.1.1 - 8

« on: January 23, 2023, 06:41:49 AM »

yes, it is
Please write more legibly. Look how I corrected your previous post

8

Chapter 1 / Re: Ch.1.2(16)

« on: January 22, 2023, 03:32:12 PM »

Part $u$ is also linear and homogeneous$. Thus the correct and precise answer is linear homogeneous.

9

Final Exam / Re: Alternative solution to the optimization in Problem 2 on the practice final

« on: April 25, 2022, 07:51:30 AM »

Yes, it can be solved using Lagrange multiplies. However note, if restrictions are $g_1\le 0$, $g_2\le 0$, $g_3\le 0$ you need to consider

• $g_1=0$ (and $g_2\le 0, g_3\le 0)$); there will be only one Lagrange multiplier at $g_1$. Two other cases in the similar way
• $g_1=g_2=0$ (and $ g_3\le 0)$); there will be  two Lagrange multipliers. Two other cases in the similar way

It will be, however, more cumbersome. Note that (1) corresponds to two rays and one arc, (2) to two corners.

No, you need not consider quadratic forms after you found all suspicious points. It would serve no purpose.

10

Test 2 / Misprints are possible

« on: March 30, 2022, 07:17:41 PM »

Misprints are possible

11

Test 2 / Re: Sturm Liouville eigenfunctions

« on: March 30, 2022, 07:15:38 PM »

they defined up to a constant

12

Quiz 5 / Re: Quiz 5

« on: March 27, 2022, 11:44:46 AM »

My question for Quiz5 was to decompose a function into full Fourier Series [0, $\pi$]. I wonder is it equivalent as decompose into full Fourier Series on [$-\pi$, $\pi$]?

No

I understand it is equivalent when the function is even, but I'm wondering what should I do when the function is odd.

What is the problem? There are formulae for interval $[\alpha,\beta]$.

13

Chapter 5 / Re: Theorem 3 in chapter 5 question

« on: March 18, 2022, 08:17:41 AM »

It is the derivative of Fourier transform; otherwise it would be $\widehat{f'}(k)$ and covered by another property

14

Chapter 5 / Re: Chapter 5.3 Problem 1.1

« on: March 13, 2022, 04:19:38 PM »

You are right, it was a misprint

15

Test 1 / Re: Test 1 coverage

« on: March 03, 2022, 12:11:32 PM »

Look at samples posted