Divergence theorem - multivariable calculus

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

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