Of all the subpar weeks I’ve had over the last few months, this week was up there for one of the worst. I did manage to watch all the remaining eight videos, so technically I’m through all the content in Green’s, Stokes’, and the Divergence Theorems, but I didn’t know what was going on in the last five videos and didn’t put enough effort into making notes on them or trying to understand them. On the plus side, the first three videos were super easy to understand and I got through them all on Tuesday. However, that makes the rest of my week feel worse by comparison since I had four days to watch and make notes on five videos and I didn’t get around to doing it. 😒 To use the same excuse I’ve used for the past 400 blog posts, I still feel like crap these days and don’t have much energy to put towards anything. It’s hard to say but I think it’s possible that I’m starting to feel better, so fingers crossed that it won’t be long for me to start to feel more like myself and have more energy to put into KA. 🤞🏼

As I mentioned, the first section I worked through this week had three videos in it and was titled Types of Regions in Three Dimensions. They were a series of videos that explained the definitions of what Type 1, Type 2 and Type 3 regions are in 3D. Here are some screen shots from those videos and my notes about them below:

I think the first page from my notes does a pretty good job summing up the definition of Type 1, 2 and 3 regions, but another way to think about it is that a Type 1 region runs parallel to the z-axis, a Type 2 regions runs parallel to the x-axis, and a Type 3 region runs parallel to the y-axis. As I said in my notes, the volume of the object MUST be ‘filled’ in from the ends of the surface, i.e. the object cannot be ‘hollow’ (so to speak) at any point from one end to the other. 

The second page from notes that shows the notation used to describe Type 1, 2, and 3 regions looks confusing, but if you read statement and match up the underlined sections with each part of the notation in the equation, I think it makes it easier to understand what the equation is actually saying.

As I said, I got through that section by Tuesday and then started the following section, Divergence Theorem Proof, on Wednesday. The section had a series of five videos in it that, as you’d assume from the title of the section, worked through a proof for the divergence theorem. Here are four screen shots from those five videos that show how Sal worked through the proof:

I watched all five videos at least twice and don’t really have any clue what’s going on with all of this… I believe the gist of what’s being proved is that if there was some surface (which in this example is shown as a cylinder in the second screen shot) where, for example, the ‘wind’ was blowing through it (i.e. the vector field denoted with F and the arrow above it), the amount of wind blowing through the inside of the object would be equal to the amount of wind going through the surface of the object. 

(From this point forward, I have no clue if what I’m writing is correct in any way.) I believe the key to this proof is that Sal splits up the vector field, F ⃗, into three separate components, P, Q, and R, which would be three separate functions denoting the respective value for the x-, y-, and z-components of a vector at any (x, y, z) coordinate. (There’s <1% chance that what I just wrote is true and makes sense. 🙃) Sal works through the third component of the vector field, R, which would be the function for the z-components of the vector field. He says that’s the proof for R would be the exact same as the proof for P and Q.

I’m realizing now that don’t understand what’s going on at all well enough to even try to explain it. (I just tried and it was a complete disaster.) My incredibly low-resolution, laymen’s understanding of it is that Sal figures out the difference between how much ‘wind’ is going through the two surfaces parallel to the (x, y) plane (although the surfaces don’t necessarily have to be flat) and then equates that to finding the integral of the divergence in the z-dimension, which he does in the very last screen shot and is shown as the first integral inside the triple integral.

I’m not going to lie, I’m pretty sad as I type all of this out realizing how little I understand of what’s going on. I know that one day I’m going to understand it all, but I feel like a bit of an idiot right now. 🙁

This coming week I want to rewatch all five videos from the divergence theorem proof and make my own notes on it. Even if I don’t fully understand what’s going on, copying the proof into my notes will likely help me make a bit of progress understanding it. Hopefully I can get through those notes by at least Wednesdays which would give me three days to work on the unit test. I highly doubt I’ll get through the unit test by the end of the week, but if I can have a go at it, it’s possible that I could get through it by the end of next week. (Unlikely! But possible.) It doesn’t feel like it right now, but I’m actually SO close to finishing off Green’s, Stokes’, and the Divergence Theorems (400/600 M.P.) and will then FINALLY be starting the course challenge for Multivariable Calculus. 🙏🏼

At some point, the day will come when I finally make it through all the math on KA. I’m becoming more and more pessimistic by the week about how far away that day is though…