Partial derivative w.r.t. another partial derivative

[cc94]

Homework Statement

Given
$$L = \left(\nabla\phi + \dot{\textbf{A}}\right)^2 ,$$

how do you calculate $$\frac{\partial}{\partial x}\left(\frac{\partial L}{\partial(\partial\phi / \partial x)}\right)?$$

Homework Equations

By summing over the x, y, and z derivatives, the answer is supposed to show that $\nabla^2 \phi = 0$.

The Attempt at a Solution

My vector calculus isn't too good, so here's my guess ( $\partial\phi / \partial x \equiv \phi_x$):

$$\begin{align}
\frac{\partial L}{\partial(\phi_x)} &= 2(\nabla\phi + \dot{\textbf{A}})\cdot\left[\frac{\partial}{\partial\phi_x} (\nabla\phi + \dot{\textbf{A}})\right] \\
&= 2\left(\langle\phi_x + A_x,\phi_y + A_y,\phi_z + A_z\rangle\right)\cdot\left[\frac{\partial}{\partial(\phi_x)}\langle\phi_x + A_x, \phi_y + A_y, \phi_z + A_z\rangle\right] \\
&= 2\left(\langle\phi_x + A_x,\phi_y + A_y,\phi_z + A_z\rangle\right)\cdot\langle 1,0,0\rangle\\
&= 2(\phi_x + \dot{A}_x)
\end{align}
$$

and so

$$\frac{\partial}{\partial x}\frac{\partial L}{\partial(\phi_x)} = 2(\phi_{xx} + \dot{A}_{xx})$$

and summing gives $\nabla^2\phi + \nabla^2 \dot{A}$.

So did I calculate $\frac{\partial L}{\partial(\phi_x)}$ correctly? If so, why is there still an A term at the end?

 

[tnich]

$$L = \left(\nabla\phi + \dot{\textbf{A}}\right)^2 ,$$

This is a meaningless equation. The power operator is not defined for vectors. $\vec x \vec x =$ ?
If it were written as $L =|\nabla\phi + \dot{\textbf{A}}|^2$, it would be equivalent to $L =\left(\nabla\phi + \dot{\textbf{A}}\right) \cdot \left(\nabla\phi + \dot{\textbf{A}}\right)$. Then your result would be correct except for a factor of 2.

 

[cc94]

This is a meaningless equation. The power operator is not defined for vectors. $\vec x \vec x =$ ?
If it were written as $L =|\nabla\phi + \dot{\textbf{A}}|^2$, it would be equivalent to $L =\left(\nabla\phi + \dot{\textbf{A}}\right) \cdot \left(\nabla\phi + \dot{\textbf{A}}\right)$. Then your result would be correct except for a factor of 2.

Er, yeah the power is supposed to be the dot product like you wrote. Ok, so I just have to figure out why the A term is zero. Thanks!