I was recently reviewing Calculus III. I had forgotten it since high school, and, to be honest, never really understood it at the time; though I blame the curriculum. This is a guide to learning the key ideas in a couple of hours. Please excuse my strong language, I wrote the following rant in an impassioned moment.

The way that Calculus III is taught today is stupid. The notation for the class is all from electrostatics, and the theory of the class is a single theorem from Riemannian Geometry (Summarized, Explained, Additional Lecture), but this is buried in jargon. Rather than mine for the truth, some bozo slapped a stupid river analogy on top of the junk heap, and all the teachers nodded along and decided that this should become the accepted wisdom. This is a good way to make everyone so unbelievably bewildered that they can’t solve a single problem. Even 3B1B can barely make this analogy coherent. If you must have 'flow' imagine a Snap! freeze-frame of the river where the currents are somehow magically preserved and something still moves around in it. But that’s stupid, things don't move through ice! Any toddler can tell you that frozen water is ice and that ice does not have a 'flow' (a bright youngster might mention glaciers, but at that scale everything's different). You know what these mathematics do describe? How about the thing they were literally designed to describe: magnetism! You know what happens if you freeze molten iron in a magnetic current generated by an electric coil? It hardens into a magnet. That’s how magnets are made! Guess what: magnets pull iron filings despite being STATIC. They are the magic ‘frozen river’ these bozos keep trying to describe. Don’t try and invent some stupid analogy for something that is best described by nature. Look at nature; look to experiment, and you can actually begin to have an idea about how all this shit works. If you’re taking the class today and confused out of your mind as much as I was, I give you this advice: watch some demonstrations of magnetism on YouTube or play (carefully) with some toy magnets. You’ll get infinitely more Bayesian updating playing with magnet toys first and refining with the mathematics of Maxwell's Equations and Stokes's Theorem afterwards.