- #1

- 170

- 18

## Homework Statement

Using the chain rule, find a, b, c, and d:

$$\frac{\partial}{\partial x'} = a\frac{\partial}{\partial x} + b\frac{\partial}{\partial t}$$

$$\frac{\partial}{\partial t'} = c\frac{\partial}{\partial x} + d\frac{\partial}{\partial t}$$

## Homework Equations

Chain rule:

$$\frac{\partial f(x,t)}{\partial x'} = \frac{\partial x}{\partial x'}\frac{\partial f}{\partial x} + \frac{\partial t}{\partial x'}\frac{\partial f}{\partial t}$$

The same form can be used for t'.

Gaililean tranformation

$$x' = x - vt$$

$$t' = t$$

## The Attempt at a Solution

For x',

$$a = \frac{\partial x}{\partial x'} = \frac{\partial}{\partial x'} (x' + vt) = 1$$

$$b = \frac{\partial t}{\partial x'} = \frac{\partial}{\partial x'} (\frac{1}{v}(x-x')) = -\frac{1}{v}$$

Similarly, for t',

$$c = \frac{\partial x}{\partial t'} = \frac{\partial}{\partial t'} (x' + vt) = \frac{\partial}{\partial t'} (x' + vt')= v$$

$$d = \frac{\partial t}{\partial t'} = \frac{\partial t'}{\partial t'} = 1$$