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Jacobian of non-square matrices
Let $m>n$ and $L\colon \mathbb{R}^m \to \mathbb{R}^n$ be a linear map(=matrix). The "$n$-dimensional Jacobian" $$ J^n(L) = \sqrt{\det(LL^t)} ...
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Integration by parts theorem proof.
Theorem statement: Let $u(x)$ and $v(x)$ be differentiable functions on some interval $I$ and let $u'(x)v(x)$ have a primitive function on interval $...
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Arithmetic Sequence Formula
I am currently reviewing some pre-calculus and I remember the part of sequences and series from taking it in academic format and a trivial question hit me...
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Can the argument of a logarithm and its base be negative at the same time?
I'm struggling to understand why a $\log_x y$ is only true for $y > 0$. I know that if $y$ was $0$ , $x$ could only be $0$, but if it was $0$, the...
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Find inverse Fourier transform of Lorentzian using complex integration
I am trying to prove the inverse Fourier transform relation of a Lorentzian $$F(\omega) = \dfrac{2b}{(\omega-a)^2+b^2} = \dfrac{1}{b+i(\omega-a)}+\dfrac{1...
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Proof that the limit of $\frac{1}{x}$ as $x$ approaches $0$ does not exist
Hello I was hoping that someone might be able to verify that the following proof that $\lim_{x\to 0} {1\over x}$ does not exist is correct. First assume t...
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What exactly is the difference between weak and strong induction?
I am having trouble seeing the difference between weak and strong induction. There are a few examples in which we can see the difference, such as reaching...
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How to prove $O(\log n)$ is true for a binary search algorithm?
I have already looked at the answer here. I'm trying to understand how the poster got: f(n) = O(1) = O(nlogba) So far I have O(1) = T(n) - T(n/2). Ho...
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Placing the numbers $1$ through $6$ into a triangular array such that the sum of the three numbers on each side is $9$ (or $10$, or $11$)
I was going through this question given to my friend at university, and I can do it with a little calculation. Is there a set algorithm that will work or ...
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Show continuity at $(0,0)$ of $f(x,y)=\frac{xy}{\sqrt{x^2+y^2}}$ for $(x,y)\neq (0,0)$ and $f(0,0)=0$
Show continuity at $(0,0)$ of $f(x,y)=\frac{xy}{\sqrt{x^2+y^2}}$ for $(x,y)\neq (0,0)$ and $f(0,0)=0$. A solution I saw was to write $$\lim_{(x,y)\to 0}x\...
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