I probably didn’t put in enough effort this week on KA, but I’m still happy with the progress I ended up making. I got through the final exercise of the 2D Divergence Theorem section but was still pretty confused with what was going on even by the end of it. After that, I then made it through nine of the 11 videos from the following section, Stokes’ Theorem, and finished off one of the two exercises. To be fair, the exercise was pretty simple and only took me two or three minutes to finish and I did on my first try BUT, even still, I was happy to get through it. Like just about everything else I’ve learned in this unit up to this point, I don’t really understand what’s going on with Stokes’ Theorem, but I can tell that, overall, I’m making progress towards understanding vectors, linear algebra, curl, divergence, etc. which is a relief. 😮‍💨

The exercise from the 2D Divergence Theorem section wasn’t too difficult. Two of the four questions didn’t require any math as the answers for each were simply, “Green’s theorem is not necessarily applicable.” One of the questions I had answered before and remembered the solution, so that was also a freebie. So, there was only one question I really had to think about which was this one:

Question 1

As I said in my notes, given that there are three terms, I don’t really understand how you’re supposed to know which term(s) in the OG double integral’s integrand is/are supposed to go with P and which one(s) is/are supposed to go with Q. It could just be a trial-and-error type of thing, but I really have no clue. I think I kind of have an idea of what’s going on with P, Q, the partial derivatives, and can somewhat understand the theory of what’s going on and visualize it all but, generally speaking, I’m still having a pretty hard time understanding all of this. 😒

Getting into the next section, Stokes’ Theorem, didn’t help much with my clarity of everything that’s going on either. The gist of Stokes’ Theorem is that it takes the same concept of G.T. but applies it to three dimensions instead of just two. The first five videos in the section went through the intuition of the theorem, explained the relationship between S.T. and G.T. and explained the orientation and conditions needed for S.T. to work. The following four videos went through this example question:

As you can see, there are a lot of ‘moving parts’ to these questions. Between parameterizing the variables, the vector notation, understanding the linear algebra of the gradient and determinant and how it relates to curl, understanding the unit vectors, AND integrating the double integrals, I’m honestly surprised and happy that I have even a small grasp on what’s going on with this question. As I said above, I’m slowly starting to wrap my head around the big picture of what’s going on, but because there are a million steps and different formulas to use and so many branches of math all in one questions, I find it super difficult to put all the pieces together in my brain and have it come together in a clear picture. It’s starting to make more sense, but I’m pretty sure I still have a long way to go before it all becomes straightforward for me.

As I mentioned at the start, the first and only exercise that I worked through from Stokes’ Theorem was quite easy. An analogy Sal gave that helped me a lot with two questions from this exercise (only one of which I added below) was that when you spin a bottle cap counter clockwise it goes ‘up’ off the bottle which is the same way you can think about the orientation of curl and the direction of the normal vector. (The first question below is the one where this analogy helped.) The other two questions I had to reason through just by looking at the shapes and thinking through what the coordinates would be if one of the variables was set to 0. I made notes showing how I did that on the last question, although it wasn’t the way KA solved it:

Question 2

Question 3

Question 4

I’m now officially past the halfway point of Green’s, Stokes’, and the Divergence Theorems (320/600 M.P.), but I still have a TON of videos and articles left to get through in this unit. My birthday is coming up at the start of May which I always find motivating and use as a reflection point, and I’m REALLY hoping I can make it through this unit by then. I highly doubt I’ll also be able to get through the Course Challenge by then, but if I’m able to at least get through this unit I’ll be pumped. Regardless of whenever I get through it, I only have one more calculus course left to finish off which is titled Differential Equations. I just looked through it and it’s quite small compared to the past units I’ve gone through. It’s mostly videos and I’ve already watched a bunch of them, AND already done a bunch of the exercises. Considering all of that, I think it’s possible I could finally be done with calculus by the fall, but I also don’t want to get ahead of myself or get my hopes up. BUT, if I do, it will only have taken me five years to learn calculus… 🙃