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For example, given such energy function:

$E(u) = \int \mathcal{L}(u, \nabla u) dx = min_u \frac{1}{2}\int |u-f|^2 dx + \int |\nabla u| dx$

We can compute the Euler-Lagrange derivative of this equation:

$\frac{\partial \mathcal{L}}{\partial u} - \frac{d}{dx}\frac{\partial \mathcal{L}}{\nabla u} = u - f + \frac{d}{dx} \frac{\nabla u}{|\nabla u|} = u - f - div\big(\frac{\nabla u}{|\nabla u|}\big)$

So my question is, why does $\frac{d}{dx}\frac{\nabla u}{|\nabla u|} = div \big(\frac{\nabla u}{|\nabla u|}\big)$ and not a Jacobian?

Where div() is the divergence function and u is an image, $u:\Omega \rightarrow \mathbb{R} $

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    That makes no sense to me. Could you provide an example / quote? – 2017-02-25
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    @user251257, I have edited my question. I hope it helps to understand. :) – 2017-02-25

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