# Use of the conservation equation for Noether currents

1. Jun 23, 2015

### spaghetti3451

1. The problem statement, all variables and given/known data

Given that

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

use the conservation equation $\partial_{\alpha} j^{\alpha} = 0$ to show that the integrand is a total derivative.

2. Relevant equations

$\bigg( \frac{\partial}{\partial x} j^{x} \big( \frac{vx}{c^{2}},x,y,z \big) \bigg)_{y,z} = \bigg[ \frac{v}{c^{2}} \frac{\partial j^{x}}{\partial t} + \frac{\partial j^{x}}{\partial x} \bigg] \bigg|_{t = \frac{vx}{c^{2}}, x, y,z}$

3. The attempt at a solution

I understand that there must be a total derivative before the integrand. I understand that $\frac{d}{dt}$ is one example of a total derivative. But, how is the concept of total derivative generalised in an arbitrary number of dimensions?

2. Jun 28, 2015