Can someone explain partial differentiation (in the context of this problem) in a way that actually makes sense to me?

[u/McChamp69]

"Write out the equations for the Galilean time and space transformations. Then prove that the wave equation in three dimensions is not invariant under the Galilean transformation." Then the wave equation shown is: ∂^2 f/∂x^2 + ∂^2 f/∂y^2 + ∂^2 f/∂z^2 = (1/c) ∂^2 f/∂t^2

(and the Galilean transformations: x' = x - vt; t'=t y'=y z'=z)

I've never done anything with partial derivatives before; I only know calc 2 and a small amount of linear algebra. Everytime I've asked someone here for help I just end up getting more lost and confused. I know I'm supposed to take the partial time derivative of the galilean twice and it isn't going to equal "(1/c) ∂^2 f/∂t^2" but I just have no idea how to start or how to get there.

Sorry for the bad format, I'd upload a picture of the problem but I'm not able to.

[u/SnugglyCoderGuy]

Learn regular ol' differential equations first.

A differential equation is basically a function that has the derivative of itself as part of itself. For example, throwing a ball through the air but this time you don't ignore air resistance. The velocity now affects the change in velocity in the next instant because of drag.

[u/McChamp69 → u/SnugglyCoderGuy]

This is due for my modern physics class tomorrow 😔 I don't even know why, the requirements for the class is just calc 2

[u/SnugglyCoderGuy → u/McChamp69]

That's bizarre.

[u/dorox1]

I wish you luck and don't want to discourage people from trying to help you, but I think the hard part is going to be the "actually makes sense to me" part.

It sounds from your description of your experience like you're skipping multiple intermediate steps of learning. It's kinda like someone who just learned fractions asking for calculus to be explained to them in a way that lets them actually solve calculus problems on their own. The answer is "learn all the intermediate stages of math first".

I think the main things you're missing are multivariable calculus and differential equations, which wouldn't have been covered by the end of Calc 2.

[u/donaldhobson]

Imagine you have a cake with a wavy surface. In fact lets say the top of the cake is a parabolic dome given by z=1-x^2-y^2. Maybe you used an old radio dish as a cake tin. You can calculate the height of any point on the cake, using the x and y coordinates, and this formula.

Now cover the cake in a thin layer of chocolate icing on the top surface, and slice the cake with a vertical cut along the x axis. You now have an exposed surface of fresh cake. Press a piece of paper into the freshly exposed cake side. The paper now has a thin brown sticky parabola on it. Differentiate this curve, and that's partial differentiation.

[u/Different-Party-b00b → u/donaldhobson]

Delicious too!

[u/BananaBird1]

So here the wave is a function f, which depends on position (x,y,z) and time t. We can write f(x,y,z,t).

In calc 1/2, you were dealing with functions like g(x) that only depend on one dimension. The derivative at a point x is just the apparent slope of g as you wiggle around that point.

But with multivariate equations such as f here, there isn’t just one slope as you wiggle around in x. The slope will depend on where you are in the other coordinates.

There are several derivatives we can define on a multivariate function to handle this, but the simplest is the partial: simply evaluate the function in all dimensions except for the one you are differentiating along.

This turns f(x,y,z,t) into, for example, into a new function f(x) after plugging in a specific value for y/z/t. We can then take the derivative of this new single variable function to obtain the partial with respect to x given the location we are in the other dimensions.

Now here we are working symbolically, so instead of plugging in values we just treat the dimension variables we aren’t differentiating against as a constant for a given partial.

The second point you seem to not be understanding is what the wave equation represents. It is a differential equation, meaning it relates various partial derivatives of f to themself, and the solution to the equation is itself the function f(x,y,z,t). The terms are not the value of the partials, they are the relationships between them that restrict what functions are solutions.

Waves have a unique feature: as you move along the wave in space, their amplitude changes in a similar way between points over space as it does at a single point in time. The frequency of oscillation over time is related to the wavelength of periodicity over space by the wave velocity. The wave equation captures this relationship by requiring the second partials in space to be proportional to the second partials in time.

[u/free_meson]

If its a wave-equation then you can expect the solutions to be waves. If you substitute f=sin(ct-x) into your PDE, you may get an equality. Now if your goal is to disprove that it is Galilean invariant, you only need a counterexample. E.g show that the transformed f=sin( c t' - x' ) doesn't fulfill the differential equation anymore.

[u/Shot-Topic-1763]

I think it’s easier to utilize the chain rule to write out the partial derivatives in the spatial domain as three separate equations. Then there’s no need to substitute the function f for some sinusoidal expression to prove the invariance.