- #1

spaghetti3451

- 1,344

- 34

## Homework Statement

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.

## Homework 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}##

## 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?