- #1

spaghetti3451

- 1,344

- 34

## Homework Statement

Given that ##Q' = \int d^{3}x \bigg[ \rho \big( \frac{vx}{c^{2}},x,y,z \big) - \frac{v}{c^{2}} j^{x} \big( \frac{vx}{c^{2}},x,y,z \big) \bigg]##, show the following:

##\frac{dQ'}{dv} = \frac{1}{c^{2}} \int d^{3}x \bigg[ x \big( \frac{\partial \rho}{\partial t} - \frac{v}{c^{2}} \frac{\partial j^{x}}{\partial t} \big) - j^{x} \bigg] \bigg|_{t = \frac{vx}{c^{2}}, x, y, z}##

## Homework Equations

3. The Attempt at a Solution [/B]

I understand that the second term can be easily obtained by differentiating ##- \frac{v}{c^{2}} j^{x}## with respect to ##v##. However, I am having trouble differentiating the first term with respect to ##v##.

In essence, I need to show that ##\frac{\partial \rho}{\partial v} = \frac{1}{c^{2}} x \bigg( \frac{\partial \rho}{\partial t} - \frac{v}{c^{2}} \frac{\partial j^{x}}{\partial t} \bigg)##.

Any hints?