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June 25

Recalculating mixed partial derivatives with a rotated basis

Suppose that for some point on a surface, we know $\partial f \over \partial x$ , $\partial f \over \partial y$ , $\partial ^{2}f \over \partial xy$ , $\partial ^{2}f \over \partial x^{2}$ , $\partial ^{2}f \over \partial y^{2}$ , $\partial ^{3}f \over \partial x^{2}y$ , $\partial ^{3}f \over \partial xy^{2}$ , and $\partial ^{4}f \over \partial x^{2}y^{2}$ . Is there a way to apply a rotation to all these values—especially the mixed ones?

I know directional derivatives can be calculated from $\partial f \over \partial x$ and $\partial f \over \partial y$ , and double directional derivatives can be calculated from the Hessian matrix. But is there a similar way to find the corresponding mixed derivatives? --benadhem (talk) 00:56, 25 June 2012 (UTC)[reply]

$T_{ijkl}={\frac {\partial ^{4}f}{\partial x^{i}\partial x^{j}\partial x^{k}\partial x^{l}}}$

transforms as a tensor under orthogonal transforms. Count Iblis (talk) 19:38, 25 June 2012 (UTC)[reply]

Yes. You need to use the chain rule. Imagine you have a differentiable map

(x,y)\mapsto (u(x,y),v(x,y))

. (It doesn't need to be a rotation, any differentiable function will do.) Using subscripts for partial differentiation, we have

f_{x}=u_{x}\cdot f_{u}+v_{x}\cdot f_{v}\,,

f_{y}=u_{y}\cdot f_{u}+v_{y}\cdot f_{v}\,.

We can write this is matrix notation to save space:

\left[{\begin{array}{c}f_{x}\\f_{y}\end{array}}\right]=\left[{\begin{array}{cc}u_{x}&v_{x}\\u_{y}&v_{y}\end{array}}\right]\left[{\begin{array}{c}f_{u}\\f_{v}\end{array}}\right]

You can invert this to give you

f_{u}

and

f_{v}

:

f_{u}={\frac {v_{y}f_{x}-v_{x}f_{y}}{u_{x}v_{y}-u_{y}v_{x}}}\,\ \ \ \ \ \ \ \ f_{v}={\frac {u_{x}f_{y}-u_{y}f_{x}}{u_{x}v_{y}-u_{y}v_{x}}}\,.

Don't worry about the mixture of us, vs, xs and ys; every thing's really a function of x and y. We're working out the current derivatives, and then the derivatives at an image point of the new function in the new coordinates. You may similarly use the chain rule a second time, and your results from this time, to get a system of simultaneous equations to solve to get

f_{uu},\,f_{uv}

and

f_{vv}

. It's not difficult, but it's certainly messy!

— Fly by Night (talk) 02:09, 26 June 2012 (UTC)[reply]

Fourier vs Harmonic

I had always been under the impression that Harmonic analysis and Fourier analysis were two different terms for exactly the same branch of mathematics, but I have just noticed that they are two different articles on Wikipedia. However, their descriptions do seem very similar (and if they are different, they are obviously closely related). Is there a difference between Harmonic and Fourier analysis? Widener (talk) 06:29, 25 June 2012 (UTC)[reply]

No! Bo Jacoby (talk) 08:23, 25 June 2012 (UTC).[reply]

If they were exactly the same thing, why would we need separate articles for them? -- ♬ Jack of Oz ♬ ^{[your turn]} 10:24, 25 June 2012 (UTC)[reply]

I think the Fourier analysis article explains it best: "Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis." - Letsbefiends (talk) 11:05, 25 June 2012 (UTC)[reply]