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Examples of Stoke’s Theorem and the Divergence Theorem Using Python
You might not realize that they are important in physics — but you pretty much need both Stoke’s Theorem and the Divergence Theorem for vector stuff (like Maxwell’s Equations). However, they can be a little difficult to comprehend.
I’m not going to derive these two equations, instead I am just going to show you that both sides are equivalent. But wait! Instead of showing the equations with surface and volume integrals, I’m going to use Python and random numbers. Trust me, this will be fun.
Stoke’s Theorem
I’m going to start with Stoke’s Theorem. I think it’s a little easier to use since you only need a path integral and a surface integral. Here’s what it looks like.
In short, Stoke’s Theorem (I’m just going to call it “Stokes” now because we are close friends and give each other nicknames) gives a relationship between a path integral and a surface integral for a vector field (I’m using the vector field F). There’s so much here, so let’s just get to it.
What is a vector field? In physics, we have LOTS of vector fields. Force, electric field, magnetic field, gravitational field are the most common ones. But basically, at every point in space (x,y,z) there is a vector value F(x,y,z). That’s it. Oh, for this example we are going to need a vector field (but any vector field should work). Something like this:
I used two different notation for vector components — but from now on, I’m going to use the angle bracket notation <1, 2, 3>. I hope that’s OK.
Path Integral
Next, let’s look at the path integral. If you think about F as a force, then this would be the work done by that force along a path — oh, and it’s a close path. That means that it starts and ends at the same point. Yes, if it’s a conservative force then this path integral would be zero — but remember that Stokes works for all vector fields. Note: I assume there are some weird fields that violate Stokes, but let’s just worry about that later.
Since we are using Python to find the path integral, the path will be broken into small pieces. During each piece (like a small step) we can assume that the force is constant. Here is a diagram.
