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Could I have solved the maximization/minimization using Lagrange multipliers? In particular, define $g_1(x,y) = y-x$, $g_2(x,y) = y+x$, and $g_3(x,y) = -(x^2+y^2)+1$. Then, a solution $(x^*,y^*)$ necessarily satisfies $$\nabla{u} + \lambda_1\nabla{g_1} + \lambda_2\nabla{g_2} + \lambda_3\nabla{g_3} = 0$$ and $$\lambda_1{g_1} = 0, \lambda_2{g_2}=0, \lambda_3{g_3}=0$$

for some $\lambda_{i} \geq 0$.

Then, after finding the points $(x^*, y^*)$, I need to verify that $$\nabla^2{u} + \lambda_1\nabla^2{g_1} + \lambda_2\nabla^2{g_2} + \lambda_3\nabla^2{g_3} $$ is positive definite on the tangent space $T_{x^*,y^*}D$.

Would this approach also work?

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Test 2 / TT2 Problem1
« Last post by Kexin Wang on March 31, 2022, 09:43:26 PM »
Hi Professor,
I did my Term Test 2 today and for problem 1, I don't know if my understanding is correct. I think this question is approachable by both (Inverse) Fourier Transformation and Separation Variable methods. I'm not sure if I have the correct insights, I think both methods work from the information provided.
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Test 2 / Misprints are possible
« Last post by Victor Ivrii on March 30, 2022, 07:17:41 PM »
Misprints are possible
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Test 2 / Re: Sturm Liouville eigenfunctions
« Last post by Victor Ivrii on March 30, 2022, 07:15:38 PM »
they defined up to a constant
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Test 2 / Question about the Steps in Test 2 PS Friday Sitting
« Last post by Jinqiu Liang on March 30, 2022, 01:12:28 PM »
I am confused about the steps in Test 2 problem set Friday sitting question 4.
 
When dealing with the Fourier transform of the even continuation, in the second step, why x of the first part of the integral can be integrated out, but the second not. In other words, can someone explain to me how the second step's equation form based on the first step?

Thank you guys so much! I have attached the problem's picture below.
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Test 2 / Sturm Liouville eigenfunctions
« Last post by Weihan Luo on March 28, 2022, 11:06:57 PM »
In the practice term test 2 Variant A Problem 1, I had to solve the the following Sturm Liouville problem:

$$X''+\lambda x = 0$$ with boundary conditions $$X'(0) = X'(4\pi) = 0$$ In the answer key, the eigenfunction corresponding to the eigenvalue $\lambda_{0} = 0$ is $X_0 = \frac{1}{2}$. However, if we substituted $\lambda_0 = 0$ into the ODE, we get:

$$X'' = 0$$ which the solution is simply $$X(x) = \alpha + \beta x$$ Plugging into the boundary conditions and we get that $$\beta = 0$$ so The solution is $$X(x) = \alpha$$ where $\alpha \in \mathbb{R}$. How do we get that $ X_0 = \frac{1}{2}$?
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Test 2 / Term Test 2 Variant E problem 2
« Last post by Kexin Wang on March 28, 2022, 03:25:10 PM »
Hi, I just want to confirm with someone about the solution for Term test 2 variant E problem2. I think for$\lambda_n$ the n should start from 0, so n should be n = 0, 1, 2 and so on. I don't know if the solution makes a typo or am I missing some insights here.
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Test 2 / Term Test 2 Variant C Problem4
« Last post by Kexin Wang on March 27, 2022, 08:36:36 PM »
Hi Professor,
I was reviewing practice test variant C problem4. I tried to follow your solution but I think where I circled should be the same as the other red circle? I don't know if this is a typo or am I missing some insights here. I've attached a picture.
Thank you
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Test 2 / Term Test 2 Variant E
« Last post by Kexin Wang on March 27, 2022, 03:31:36 PM »
For problem 1 in Practice Test Variant E, I'm wondering if this is a typo because the interval given is from $-\pi$ to $\pi$? (I circled it in red)
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Quiz 5 / Re: Quiz 5
« Last post by Victor Ivrii 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
Quote
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 $[\alpha,\beta]$.
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