Theorems of Vector Calculus

Notes on the physical meanings of the theorems of Green, Stokes and Gauss covered in MA1505 Chapter 10

Introduction
Those who are familiar with the Mathematics module (the one specially designed to traumatise engineering freshmen) would readily identify the following formulas to be the point where it all starts to fall apart (ignoring Fourier series).

At first glance, these formulae all seem to be coincidence, as what can there be to relate integration over a line with integration over a surface, or to relate integration over a surface with that over a volume? The notes give no helpful explanation either... but the underlying idea is quite simple to understand!

Disclaimer
This following is a demonstration, not a proof. The goal is understanding, not certainty. If one is interested, there are pretty good proofs for Green's theorem as well as the theorems of Stokes and Gauss (TAYLOR Peter D., 1992).

Idea
Special case assumption: When you sum up small loops, they become one big loop.
As seen in this tile of four loops, the interior arrows cancel out and leave the outer boundary of the loop (which has nothing to cancel out with).

We apply this principle everywhere else rather thoughtlessly. Does anyone remember Carnot engines from Physics PC1431? How do you reduce a reversible process to a sum of Carnot cycles?
Yes.
General assumption: The interior boundaries cancel out.
The important thing is that the R.H.S of the three theorems deal with summing up quantities for which the internal boundaries cancel out each other. This is why summations (integral) over a plane/volume can equate to summations over the plane/volume's outer boundary.
This general assumption is necessary to bring Gauss' Theorem into the discussion.

Stokes:A Loopy Proposition
Aside: In practise, it is easier to use Green's theorem to prove Stokes' Theorem rather than the other way round, for an idea how this is done, please refer to Thomas' Calculus (12th ed.), Chapter 16. However, for this section, only Stokes' theorem will be demonstrated (Green's theorem follows as its 2-D speical case, a fact which you may have learnt in class).

The basic idea of Stoke's theorem is to add up the curl (small loops) of the vector field to a closed-path integral (the big loop).
(1)It can be seen that in a 3-D space, many different surfaces can share the same boundary.

(2)As follows from the assumption, any closed loop can be split up into smaller loops, and then split up into even smaller loops, and so on ad infinitium. The sum of these loops, as expected, equals the big loop i.e. the boundary of the surface. This ensures that even for different surfaces, the integral can be the same if the surfaces share the same boundary!

(3)If the big loop can be thought of as the sum of infinitesimal loops, we can approximate each infinitesimal loop to the curl at a point (normal to the surface).

What is the meaning of curl?
Curl is what textbooks describe as "circulation density". Basically, it is how much vectors swirl in the immediate neighbourhood of a given point. The curl vector points upwards if the net circulation is counter-clockwise from above, and vice versa.

The big last step: Approximating infinitesimal loop to curl
After obtaining your infinitesimal surface dS, it is time to match the 2 quantities
(1) Line integral over the infinitesimal loop
(2) Curl over the point

Admittedly, I have not found a way to settle this step yet, but here are 2 important observations:

(1) The quantities a)line integral over infinitesimal closed loop and b)curl derive their values from the same 4 vectors in the neighbourhood of small surface area dS.

(2) Only the components of a) and b) aligned with the surface of dS are taken:


Gauss: Analogy using Jigsaw Pieces


The cutest formula in the bunch also looks the most devastating, not least for its prominent feature of a triple integral. However, the fact that features a divergence operator rather than a curl operator makes it infinity times friendlier to the brain.

Basically, what is says is:
The volume flow rate over a closed surface
is equal to
The net divergence in the volume enclosed by the surface.

What is the meaning of divergence?
Divergence at a point is a measure of the net tendency of nearby vectors to point inwards into the point or outwards away from the point.
Basically,

The cube is a small volume dV in the space enclosed in the surface.
In the jigsaw analogy, we simplify each small volume dV into a jigsaw piece.
The tendency for outflow at one edge, contributing to positive divergence, is represented by a protruding knob.
The tendency of inflow at one edge, contributing to negative divergence, is represented by a hole.

The big step: Cancelling out the interior boundaries
The boundaries must cancel out in between volumes. This is because any vector which contributes to positive divergence in one small volume dV also contributes to the same amount of negative divergence in an adjacent dV, and vice versa.


This is the same as saying in the jigsaw analogy that 2 adjacent jigsaw pieces fit each other smugly.

The "divergence" is found by adding up the knobs (1 per knob) and holes (-1 per hole) along the edge. The "divergence" in the result sum can be thought of as the sum of the "divergences" in the individual jigsaw pieces. It can also be thought as the sum of knobs and holes along the edge of the result (i.e. by giving it a count).

A final observation: The key to understand why Gauss' Theorem works is the same key to understand why a completed puzzle always has straight edges (a particular case of zero divergence) even though all the pieces in it are jaggedy!

-- END OF DEMONSTRATION --