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Homework Help: Use of the conservation equation for Noether currents

  1. Jun 23, 2015 #1
    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. jcsd
  3. Jun 28, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
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