An ordinary differential equation that generalizes the notion of "path of steepest descent." For questions on "gradients" of a function, use (multivariable-calculus) instead.

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### What's the equation of the gradient descent path in an elliptical field from a starting point (x,y)?

The following figure is generated from $f(x,y)$ = $x^2\over36$ + $y^2\over 16$. The black trajectory is the gradient descent trajectory from $(5,5)$ to $(0,0)$. Is there an equation for this ...
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### Gradient flow remains on manifold

Consider a gradient flow $x(t) = -\nabla f(x)$ with the property that for some manifold $M$, $x(0) \in M$ and for all $x \in M$, $\nabla f(x) \in T_x(M)$ where $T_x(M)$ denotes the tangent space at $x$...
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I was reading some physics when I read this particular paragraph: enter image description here I understand that along $\hat \theta$, $ds=rd\theta$ and also the preceding argument about $dr$ but I ...
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### Computing $L^2$ Gradient Flow for a “compactness energy” over a weighted graph.

I'm reading the following paper on applying Mean Curvature flow to gerrymandering: https://www.math.ucla.edu/~majaco/papers/gerrymandering.pdf The setting is that we have a weighted graph of vertices,...
I saw the gradient of a function, $f(x,y) = \| x - y \|^2$ as $\nabla f_x = 2x^T(x-y)$ where $x$ and $y$ are column vectors. I really would love to know the Mathematics behind this. Also, I thought ...