Exponential functions and Euler’s formula
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A Cluttered Mind
Math, anecdotes, recipes
Latest posts
Quick Proof of Cauchy Integral Formula
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Homotopy invariance of Contour Integral
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Chain Rule for Maps
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Harvard Way to Define Trace
$\newcommand\F{\mathbb{F}}\newcommand\R{\mathbb{R}}\newcommand\C{\mathbb{C}}\newcommand\Z{\mathbb{Z}}\newcommand\tr{\operatorname{trace}}\newcommand\End{\operatorname{End}}$ The trace of a sqaure matrix $A$ is defined to be the sum of the elements along the diagonal. A basic fact is that for any invertible matrix $M$, \begin{equation}\tag{*} \tr(M^{-1}AM) = \tr(A). \end{equation}
Singular Value Decomposition
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Visualization of Moore-Penrose Pseudoinverse
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Orientation of a manifold
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Definition of a manifold
$\newcommand{\R}{\mathbb{R}}$ I’ve always disliked the standard definition of a manifold $M$ in differential geometry. First, the definition assumes that $M$ is a topological space. I don’t understand why this is needed. I prefer to show that the topology of $M$ is a natural consequence of the definition. Second, the definition always uses two technical terms, second countable and Hausdorff. I…
What’s the intuition behind the coarea formula?
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