Intro
Most theorems in mathematics are named after some white dude. For a long time, women weren't given the chance to do higher mathematics.
The theorems we'll cover here obey that rule. And just to make matters worse, two of the guys were named George. Diversity wasn't a thing back then.
Here, we'll learn about Green's theorem, Stokes' theorem and Gauss' theorem.
George Green and Sir George Stokes were both physicists. They made important contributions to fluid dynamics. Carl Friedrich Gauss was a German mathematician, widely regarded as one of the greatest minds of our time.
So on the mathematical wall of fame, there'd mainly be guys. Hopefully that'll change soon.
Koncept
All these vector calc theorems might seem quite disconnected. The first two ones, Green's theorem and Stokes' theorem, are about curl. The third guy, the Divergence theorem, is about divergence. What do they have in common?
Here's the punchline: an integral along a boundary can be transformed to an integral along a line. That's what these theorems do. They give you recipes for switching between dimensions. And that greatly simplifies calculations (which is always a good thing).
So in this section, we'll discover the big three: Green's theorem, Stokes' theorem and the Divergence theorem.
Summering
There are three main theorems of vector calculus. They are listed below, followed by some words of advice.
1) Green's Theorem
2) Stoke's theorem:
3) Gauss' theorem or the Divergence theorem:
As we read the theorems, they might seem scary and abstract. But fear not: the theorems have real physical meanings which are not that complicated, if look at them one part at a time.
So first, focus on the individual components of the theorems, trying to understand them. Then continue by putting the pieces together to one story. This story will also be explained in the lecture notes.
Green's theorem
We've now reached the pinnacle of our journey through Multivariable Land. Learning about partial derivatives was like 'meh'. Optimisation was like 'aha'. And now we'll learn about a few multivariable calc theorems - the really cool stuff.
But you'll need a solid foundation to understand these theorems. So if you feel a bit shaky on line integrals, curl and divergence, do have a look at those lecture notes again.
As we've seen, line integrals can be really nasty. It takes quite a bit of time to compute in itself! And then you've got to plug in into and compute the entire integral. Ewww...
This is where Green's theorem comes into the picture. Green's transforms a line integral into a normal double integral. It says that
Here, is a closed curve, which is piecewise continuously differentiable. The curve encloses the region . In addition, we require that and be continuously differentiable.
The thing on the right hand side is just the 2D curl. And indeed, if you'll take a look at the following pictures, doesn't it seem natural that the line integral increases as the curl over the region increases?
Here are a few of examples, so you can see Green's theorem in action.
Example 1
Calculate
where is the positively oriented boundary of the quarter disk described by
Solution: In this example, we have that
We can use Green's theorem to calculate
We change to polar coordinates
Example 2
Calculate
where is the positively oriented boundary of the ellipse described by
Solution: In this example, we have that
We can use Green's theorem to calculate
The last integral is the area of , which of course is . Thus,
Stokes' theorem
If you've got a curve in 2D, go with Green's theorem. If you've got a curve in 3D, well, Green's theorem breaks down. And how about Green's theorem for curves in higher dimensions? No way, friendo.
But there's a really slick theorem which holds for higher dimensions. Stokes' theorem basically says the same thing as Green's theorem. It's just that we've tagged on a few factors and integral signs, so we can handle higher dimensions.
Basically, the theorem says that
Here, is some curve, and a surface. To apply Stokes' theorem, take care that the unit normal vector is oriented in the right direction. Otherwise your answer will be off by a factor of .
Alright, but why does this hotchpotch of and and all make sense? Have a look at the three cases here. Clearly, the curl on the surface matters for orientation.
But we can actually study whichever surface we like. Why? Well, our closed line integral can be broken down into two separate closed line integrals. On the surface, they'll cancel out. So the boundary is the only thing that matters, really.
Example
Calculate
where is described by
and
Solution: is the part of a sphere with radius centered at the point lying above the -plane.
The condition means that the -component of the normal to should be positive.
The boundary of is where the sphere intersects the -plane. When intersect the -plane, it means that . Thus, is described by
We can now use Stoke's theorem to calculate
But observe now that is also the boundary to the disc described by
Thus, we can use Stoke's theorem again
Since the disc is lying in the -plane, its normal is parallel to the -axis
The dot product means that we only care about the -component of , which is
Thus,
The last integral gives us the area of the disc . Finally, we can conclude that
Divergence theorem
Now we've arrived at our final theorem. Drumroll, please? This theorem is ubiquitous in physics. So it's not just one of those theorems mathematicians came up with to pickle your brain: this one is really important.
So let's suppose that you're a magician standing (floating, hovering?) in the middle of a closed water tank. You've created yourself a bubble in which you can breathe. Have a look at the sketch, and you'll see what I mean.
As you say 'abrakadabra', you magically create 3 gallons of water, which flow outward from the center of the container.
But water is incompressible: the molecules can't be packed more densely. Since water is created within the tank, it must flow out somewhere. What goes in goes out, you know. This means that water will flow out from our tiny outlet at the top right corner.
The Divergence theorem centers around this idea. It says that
Here, is a continuously differentiable surface, encapsulating the volume . Moreover, the vector field should be continuously differentiable.
The Divergence theorem is a bit like Stokes' theorem, but for the divergence rather than the curl.
Now we'll work an example to help you absorb everything.
Example
The sun radius is , and its distance from the earth is .
The energy per square meter from the sun on the earth surface is . Calculate the energy generated per cubic meter from the sun. Assume that the energy generated is evenly spread out.
First, we calculate a surface integral around a sphere of the radius this gives us the total radiated energy. Note that the energy dissipated from the sun has a direction since it travels in space. This direction is the same direction as the unit normal of the sphere
Next according to the divergence theorem we know that
From the assignment we where told that the energy created by a cubic meter of the sun is equal to and constant inside the sphere. Therefore we know that . We know that the sun only creates energy inside of it and therefore we find that, we set
Next we note that
Isolating we find that
Again it is worth repeteting that
