#HELP mixed partial derivatives

15 messages · Page 1 of 1 (latest)

acoustic crest
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hey guys
ive been stuck on something like a lot
how do you visualise the mixed partial derivative d^2f/dydx
because when i asked ai and tried to use matlabs to visualise it, it didnt really show it and there is no video online

subtle swallowBOT
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subtle swallowBOT
keen knot
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Let's suppose that the partial derivatives are continuous in a neighborhood of (a,b). To visualise $\frac{\partial^2f}{\partial x\partial y}(a,b)$ I would first consider a stick in the plane containing $(a,b)$ and parallel to $(xOz)$ tangent to the curve $x\mapsto f(x,b)$. This stick embodies the slope of $f$ in the $x$ direction. Now if we move around $f(a,b)$ in the $y$ direction, $i.e.$ look at how much the stick rotates if we look at the curves $x\mapsto f(x,b+\varepsilon)$ and $x\mapsto f(x,b-\varepsilon)$ for small $\varepsilon>0$, it gives a flavour of $\frac{\partial^2f}{\partial x\partial y}(a,b)$. Same for a stick in the plane $(yOz)$ and moving around $a$.

haughty kilnBOT
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mimshell

572174658661515273.png
acoustic crest
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how does that prove anything visually?

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or how does it prove it at all?

keen knot
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I thought you asked to visualise the mixed partial derivative to get an intuition of it. You did not state you wanted a visual proof of a result involving it.

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At any rate given the hypothesis I suggested one is supposed to visualise the movements of the stick in accordance with Schwarz's theorem or geometric restatements of it.

acoustic crest
acoustic crest
keen knot
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Maybe use finite differences and see them as lines. If the functions and its partial derivatives are smooth enough, it does not matter which coordinate one uses first.

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Or look at the image of the square (a,b), (a+ε,b), (a+ε,b+ε), (a,b+ε). One can go from (f(a),f(b)) to (f(a+ε),f(b+ε)) in two ways, whatever lengths the distorted sides may have.

subtle swallowBOT
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@acoustic crest

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acoustic crest