I had a bad week on KA. 😔 It was close, but I don’t think I reached my weekly goal of studying for five hours. Two weeks ago, in my Week 230 post, I mentioned that I’ve been working on “other exciting projects” and I didn’t want them to interfere with my effort towards KA. Well, that hasn’t been going according to plan. I spent a LOT of time this week thinking about and mapping out this new project I’ve been working on which I’m VERY excited about, but it’s very distracting. I can’t help but feel disappointed in myself that I didn’t put in a solid effort on KA over the past few weeks. I told myself ~4.5 years ago that if I could teach myself calculus, I could do anything I wanted. The new project I’m working on has potential (I think) to have a fairly large, positive impact on the world (which I realize is an audacious thing to say), and I think it will be equally difficult to bring into reality. I want to finish calculus for a number of reasons, but three key reasons are 1) because I hate not finishing things that I start, 2) I’m SO close to finishing calculus as it is so it would be a huge shame if I didn’t, but also 3) the superstitious part of me is saying that if I want to accomplish this new project, I NEED to finish calculus to prove that I can do anything. That makes sense, right? 🤔 🤷🏻‍

I only got through three articles this week but got a much stronger grasp on the vector operator curl. (I also learned that it’s called a vector operator, so I also have that going for me which is nice.) Just below this are a bunch of notes I wrote at the beginning of the week about curl which really helped me wrap my head around what’s happening with the different partial derivatives in the operator. The key is to understand that, for example, ∂y means a tiny, tiny step either ‘up’ (+) or ‘down’ (–) the y-axis and (∂v₁)(î) would be a tiny, tiny step to either the ‘right’ (+) or ‘left’ (–) on the x-axis. Understanding that concept will make understanding these notes a bit easier:

Hopefully my notes do a good job of explaining the gist of curl because I’m not going to reiterate the majority of them here, but I think it’s worth talking about the very last note which gets to the crux of what I found confusing about curl. As it says, the key (for me, anyways) to think through which way a vector is curling is to remember that you FIRST think about a little arrow going ‘up’ or ‘down’ the y-axis (∂y) or ‘left’ or ‘right’ across the x-axis (∂x) and SECOND think about a little arrow going ‘up’, ‘down’, ‘left’ or ‘right’ across either the x- or y-axis depending on whether it’s (∂v₁)(î) or (∂v₁)(ĵ). As far as I understand it, the same thing applies to 3D curl when using the z-axis, but the difference is that you’re calculating the length of the normal vector to the yx-, yz-, and xz-planes. (I don’t actually know if that’s true, but I’m pretty sure it’s true.)

Here are two questions I worked through, one from an article on 2D curl and the second from an article on 3D curl:

Question 1

Question 2

As I mentioned at the very end of my note here, I don’t understand why if the curl vector at (0, 1, 2) is pointing essentially straight along the y-axis in the direction (–1, 17, 0) how, “the rotation of the fluid near this point is almost entirely parallel to the xz-plane.” It seems to me that the fluid is nearly perpendicular to the xz-plane. I’m pretty sure I’m missing something and/or not conceptualizing it properly.

Here’s a final screen shot I took at the end of the article on 3D curl which summarizes it:

The two articles on curl I worked through were “suggested review” from the first article of the first section in Green’s, Stokes’, and the Divergence Theorems. After getting through both of those articles, I finally returned to first section of the unit and got through one of the five articles in that section. It was just a few paragraphs long and explained that the following four articles will go into greater detail of the theory and formal definitions of divergence and curl which should make them easier to understand in the long run, so getting through it definitely wasn’t much to be proud of. (I have a feeling the next four articles are going to be rough…)

So, all in all, not a great week, but I’m still happy that I was able to get a better grasp on curl. I have a long way to go before getting through Green’s, Stokes’, and the Divergence Theorems (0/600 M.P.) so I definitely need to improve my time management going forward if I want to get through it anytime soon. As always, fingers crossed that I can make some decent progress this week! 🤞🏼 🙏🏼 🤞🏼 🙏🏼 🤞🏼