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Thread: Doubt on functional derivative

  1. #1
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    Doubt on functional derivative

    Hi. I want to find the derivative of a functional with respect to a function, in particular, the $\displaystyle L_2$ norm of a function. I haven't done this type of calculations for a while.

    So, let's say my functional is a mismatch between two functions:

    $\displaystyle R=||g(\mathbf{r})-f(\mathbf{r})||^2=\int d\mathbf{r}[g(\mathbf{r})-f(\mathbf{r})]^2$,

    and I want to find $\displaystyle \frac{\partial R}{\partial g(\mathbf{r})}$.

    So, I thought of using the chain rule, such that: [tex]\frac{\partial R}{\partial g(\mathbf{r})}= $\displaystyle \frac{\partial R}{\partial \mathbf{r}}\times \frac{\partial \mathbf{r}}{\partial g(\mathbf{r})}=\frac{\partial R}{\partial \mathbf{r}}\times \left ( \frac{\partial g(\mathbf{r})}{\partial \mathbf{r}}\right)^{-1}$.

    That would give: $\displaystyle \frac{\partial R}{\partial g(\mathbf{r})}=2[g(\mathbf{r})-f(\mathbf{r})]\hat{r}\cdot \left[ \nabla g(\mathbf{r})\right]^{-1}=\sum_i 2[g(\mathbf{r})-f(\mathbf{r})] \left[ \frac{\partial g(\mathbf{r})}{x_i}\right]^{-1}$.

    with $\displaystyle x_i=x,y,z$ for i=1,2,3.

    Is this correct?

    Thanks in advance.
    Last edited by Ulysses; Jul 15th 2019 at 07:07 AM.
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