0

Let's say that I want to perform the functional derivative of a scalar functional $F$, for example

$F_{d}=\int d\mathbf{r}\left(\frac{\alpha(\rho)}{2}|\mathbf{p}|^{2}+\frac{\beta}{4}|\mathbf{p}|^{4}+\frac{K}{2}(\partial_{\alpha}p_{\beta})(\partial_{\beta}p_{\alpha})-v_{1}\frac{\rho-\rho_{0}}{\rho_{0}}\nabla\cdot\mathbf{p}+\frac{\lambda}{2}|\mathbf{p}|^{2}\nabla\cdot\mathbf{p}\right)$

but the derivative is taken in respect to a vectorial function $\bf p$:

$$\mathbf{h}=-\frac{\delta F_{d}}{\delta\mathbf{p}}=-\frac{\partial f}{\partial \bf p}+\nabla \cdot\frac{\partial f}{\partial\nabla \bf p}$$

$$h_i=-\frac{\delta F}{\delta p_{i}}=-\frac{\partial f}{\partial p_{i}}+\partial_{j}\frac{\partial f}{\partial\partial_{j}p_{i}}$$

where I just applied the definition [here]

So, my problem is, if in $f$ I have a term $f_1=\nabla\cdot\bf p$, what is the expression: $$\frac{\partial \nabla\cdot\bf p}{\partial\nabla \bf p}$$

and what happens for terms as $f_2=|\nabla\times\bf p|^2$?

Willie Wong
  • 73,139
usumdelphini
  • 568
  • I think you're just putting together things at random... –  Oct 25 '13 at 15:17
  • Why should I do that, mate? – usumdelphini Oct 25 '13 at 15:18
  • The intended interpretation is, most likely, exactly like you've written it down in the other thread. I.e. each component of $\bf h$ is to be associated with the functional derivative w.r.t. the same component of $\bf p$ (which is just one function in each case, e.g. $p_2(r_1,r_2,r_3)$). – Nikolaj-K Oct 25 '13 at 15:29
  • @Nick Could you help me trying to explicit the calculation in components of with the index notation? Because I cannot see it, sorry. – usumdelphini Oct 25 '13 at 15:33
  • 1
    $\nabla \cdot \mathbf{p} = \delta_{kl} \partial_k p_l$ so $\displaystyle \frac{\partial \nabla \cdot \mathbf{p}}{\partial \nabla \mathbf{p}} = \frac{\partial \delta_{kl} \partial_k p_l}{\partial \partial_i p_j} = \delta_{ij}$. – Willie Wong Oct 25 '13 at 15:36
  • @usumdelphini: write the whole $f$ down in index notation but introduce $A_{ab}:=\partial_a p_b$ for each instance of $\partial_a p_b$ in $f$ as well as $\bf h$. – Nikolaj-K Oct 25 '13 at 15:38

0 Answers0