Question

In the wave equation, which is mathematically represented by partial derivatives, how would it make a different if we represented it with ds, instead? That is: d^2 E/d x^2 + ... = ..., instead of partial^2 E/partial x^2 + ... = ...? In other words, what is the significance of the partial derivative? Thanks in advance!

Answer 1

The difference between the partial and full derivative is very tricky. Here is how I understand it. For a function, f(x, y), of variables x and y, one writes the differential expression \[ df = \frac {\partial f} {\partial x} dx + \frac {\partial f} {\partial y} d y \]
The full derivative is hence
\[ \frac {df} {dx} = \frac {\partial f} {\partial x} + \frac {\partial f} {\partial y} \frac {dy} {dx} \]
and is in general different from the partial derivative unless \[ \frac {dy} {dx} = 0 \]
In other words, y must be independent of x. If you have interdependencies among the variables, then the full and partial derivatives will not be the same.

Answer 2

It feels great when a bugging question gets answered in such a manner as your response, Rasmusp. Thanks a lot! :)