For the first time in what feels like forever, I’m actually happy with how this week went. It’s a pretty ironic that I’m pleased with how this week went considering I didn’t actually make it through that much content on KA, however I DID study for at least 5 hours this week (if not 6 or 7 hours) which I’m pretty sure I haven’t don’t for a long time. I feel like I finally got myself back on track this week which is a pretty big relief. 😮‍💨 The sad part is I only made it through one article and three videos and didn’t even take any notes on the three videos I watched. The three vids were from the section 2D Divergence Theorem and I think I have a decent grasp on what the three videos were talking about (key word: think), which I’m happy about considering how lost I’ve felt over the past few weeks. I also feel like I made a bit of progress understanding Green’s Theorem, so all in all it was a good week even though I didn’t technically get through too much content. 🤷🏻‍♂️

This is going to be a short post since 1) I didn’t take many notes this week, and 2) I only have about an hour to write this post. I took screen-shots and made notes on two example questions from the article. Here’s the first example:

The thing that I initially found most difficult about these questions was understanding which expression in the line integral’s integrand was P(x, y) and which one was Q(x, y). To be honest, I’m still not 100% sure if I understand it properly, but I believe whichever expression has dx in it is P and whichever one has dy in it is Q. I believe this is the case as P is the function that tells you the degree to which vectors are moving in the x-direction and Q is the function that tells you the degree to which vectors are moving in the y-direction.

I did also struggle with the algebra when factoring out (x² – 4) from the antiderivative of the inside integral. I got the question wrong initially because I tried to expand the expressions. Once I looked at KA’s answer and saw that they factored out (x² – 4), I was able to solve it without actually looking at the algebra step-by-step, so I’ve got that going for me, which is nice.

I’m not going to lie, I don’t really have any clue what’s going on with this question. First of all, I don’t understand why ∂Q/∂x – ∂P/∂y has to equal 1. I think it’ has something to do with ‘s because setting G.T. to equal 1 means you essentially factor it out of the equation and then by solving in the integral, you’re just solving the area of the region. But I literally have no clue if what I just wrote made even a tiny bit of sense, so… ya.

I also got smoked on the calculus at the start of trying to switch x*dx – y*dy to (x(t)dx – y(t)dy)dt. I don’t understand why you add in the dt/dt which is a bit concerning as I feel like that’s probably a pretty ‘basic’ and fundamental concept to parameterizing these types of questions and I don’t really know what’s happening… 😒

I’m not going to go into any more detail about the article (since I don’t really understand it anyways…), but below is a screen shot of the summary of that article. I should also mention that I’m hopeful that the further into this unit I go, G.T. will start to make more sense.

As I mentioned in my into, I finally got started on the following section this week, 2D Divergence Theorem. There are only three videos and one exercise in this section and, as I mentioned, I watched all three videos but didn’t get started on the exercise. I’m going to wait until next week to make actual notes on the videos but I think I have a vague understanding of the big picture of what’s going on in the videos:

My understanding is that divergence is the perpendicular vector valued function to curl. I believe that a curl function tells you how much a vector coming off the border of a particular region is running tangent to the border at any given point. Divergence, however, tells you how much of that vector is coming directly ‘out’ of the border at any given point, i.e. how much of the vector is perpendicular to the border. (It’s hard to explain what I mean by ‘how much’ of the vector is coming ‘out’ of the border. If the vector is coming out at an angle, if you constructed a triangle where the vector is the hypotenuse of a right triangle, then the opposite side of the triangle normal to the border is a how much the vector is coming ‘out’ of the border.) In the same way that the line perpendicular to y = 3x is y = (–1/3)x (i.e. the slopes are the negative reciprocals of each other) the vector valued function for divergence is the negative reciprocal of curl:

• Curl:
  • ∂Q/∂x – ∂P/∂y
    
• Divergence:
  • ∂P/∂x + ∂Q/∂y

For the millionth time, I don’t know if I’m correct about any of this, but I’m pretty sure that’s what’s going on.

This upcoming week I’d like to quickly get through making notes on those three videos and hopefully get through the single exercise in that section early in the week. Then I’ll be starting on the following section, Stokes’ Theorem, which is an absolute beast with 11 videos and two exercises in it. It’d be great to get through half of the videos or more by the end of the week. I’ll still be pretty far away from finishing off the entire unit, Green’s, Stokes’, and the Divergence Theorems (160/600 M.P.), but I’d be happy with that amount of progress, nonetheless. Fingers crossed I can keep this momentum going! 🤞🏼