Lorentz boost, electric field along x-axis, maths confusion?

[JesseC]

Homework Statement

Given that (φ/c,A) is a 4-vector, show that the electric field component Ex for a
Lorentz boost along the x-axis transforms according to Ex' = Ex.

Homework Equations

$$E_x = -\frac{\partial \phi}{\partial x} - \frac{\partial A_x}{\partial t}$$

$$A_x$$ being the x component of the vector potential

The Attempt at a Solution

So I don't have a problem getting φ'/c or Ax'

Obviously:
$$E_x^{\prime} = -\frac{\partial \phi^{\prime}}{\partial x^{\prime}} - \frac{\partial A_x^{\prime}}{\partial t^{\prime}}$$

But I don't understand how to get the partial derivative w.r.t. x' in terms of x and t. Likewise for the partial derivative w.r.t. t'.

In the solutions it seems:

$$\frac{\partial}{\partial x^{\prime}} = - \frac{\gamma \beta}{c}\frac{\partial}{\partial t} - \gamma \frac{\partial}{\partial x}$$

?? Does it come from the lorentz co-ord transformation?

 

[tiny-tim]

Hi JesseC!

But I don't understand how to get the partial derivative w.r.t. x' in terms of x and t. Likewise for the partial derivative w.r.t. t'.

Chain rule: ∂/∂x' = ∂x/∂x' ∂/∂x + ∂t/∂x' ∂/∂t