Use of the conservation equation for Noether currents

[spaghetti3451]

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?

 

[mark726]

The concept of total derivative can be generalized to an arbitrary number of dimensions using the chain rule. In this case, the integrand can be written as:

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

Using the chain rule, we can write:

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

Since $t = \frac{vx}{c^{2}}$, we can write $\frac{\partial t}{\partial v} = \frac{x}{c^{2}}$. Substituting this in the above expression, we get:

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

Using the conservation equation $\partial_{\alpha} j^{\alpha} = 0$, we can write:

##\frac{dQ'}{dv} = \frac{1}{c^{2}} \int d^{3