It wasn’t the best week for me working through KA. I managed to make a bit of progress understanding Green’s Theorem but only made it through two videos, and no exercises or articles. I made notes on the third and fourth videos from the Green’s Theorem section which helped me understand the ‘how’ of how to use G.T., but I still am far off from having a ‘high-resolution’ understanding of what G.T. does and why it works. It took me until Friday just to get through the two videos and make notes on them and so I didn’t manage to get through the final exercise in the section. 😔 BUT, I had a few attempts at the exercise where I got three of four questions correct before getting the final question wrong. I think that’s a good sign that I’ll be able to get through the exercise quickly at the start of next week. I don’t completely understand what’s going on in all the questions from the exercise, but I have the feeling that when I get further into this unit, that’s when I’ll get a better grasp on Green’s Theorem. That’s the hope anyways! 🤞🏼

Here are some screen shots and my notes below from the first video I worked through this week, which was the third video from the Green’s Theorem section:

The image denoted with R and the equations written in white and pink at the top of this screen shot are just there to state the G.T. formula and the formula for a general vector field, f^->(x, y). Sal did this to remind the viewer of the equations. Everything written in yellow, purple, and green have to do with the particular question being solved in the video.

The region being calculated in this question is represented with the image in the bottom left corner of the screen shot and is denoted with R: { (x, y) ; 0 ≤ x ≤ 1 , 2x² ≤ y ≤ 2x }. This is saying, “the region in question (R) is on the (x, y) plane ((x, y)), where the lower and upper bounds of the region across the x-axis are from 0 to 1 (0 ≤ x ≤1) and the lower and upper bounds of the region across the y-axis are a function of x and are from 2x² to 2x(2x² ≤ y ≤ 2x).

(That was confusing…)

The line integral (or what I visualize as being a ‘curtain’ with different heights across it, i.e. different z-values across it) is denoted with the function f^->(x, y) = (x² – y²)î + (2xy)ĵ. The function to calculate the area of the line integral (i.e. the area of the curtain that’s “wrapping” around the region in question) is ∮_C (x² – y²)dx + (2xy)dy.

I believe the reason why (x² – y²)dx is associated with P(x, y) is because P calculates the vectors going across the x-axis (which is why it’s paired with î in the equation in the top right corner) and so the expression in the line integrals integrand that’s paired with dx is the P(x, y) vector. Similarly, the expression with dy is the Q(x, y) vector for the same reasons.

(I don’t know if anything that I just wrote made sense, is true or accurate. 🙃)

Using G.T., you then switch the line integral to a double integral with the x- and y-bounds stated for the region (which is what Sal did in the bottom center/right corner) and then find the partial derivative of P(x, y) with respect to y and partial derivative of Q(x, y) with respect to x and put those into the integrand of the double integral using the expression ∂Q/∂x –  ∂P/∂y.

After writing out the double integral, you simply solve the double integral which you can see led to the solution 16/15. And boom goes the dynamite. 🧨💥 

Here’s a screen shot and my notes from the second video I worked through:

You can see that my notes from this question go through a lot of the same concepts I talked about in the previous question, but that they also mention how the direction that the line integral is being calculated affects the sign of the solution. In this video, Sal said to switch (∂Q/∂x –  ∂P/∂y) to (∂P/∂y  –  ∂Q/∂x) since the line integral is being calculated in a clockwise direction, however from the exercise I finished the week working on, it seems easier not to switch the partial derivatives and just multiply the solution by –1 at the end. 🤷🏻‍♂️

Speaking of the exercise, here are three questions I worked through from it at the end of the week:

Question 1

This was the first question I attempted and, as I mentioned at the bottom of my notes, I had no clue what I was doing but made the correct substitutions and ended up getting it correct. I started to get a feel for how to solve G.T. from these questions which was definitely a relief. 😮‍💨 Plus I got a bit of review using the derivative product rule in this question which I found helpful.

Question 2

I was thrown off when I first saw this question because I had to use G.T. in reverse. The reason why I got this question wrong was because I mixed up which expression in the integrand was P(x, y) and which one was Q(x, y). You can then see from my second page of notes that I came up with my own little way of finding the integral of Q(x, y) which is a method I used working on other questions from the exercise and it seemed to work. It’s not what KA did to solve the question though, so I’m not sure if my method is going to work for me going forward, but I’m pretty sure it technically is a ‘valid’ way of solving this type of question. 

Question 3

This is the same type of question as the one just above and you can see that my method of finding the integral of each expression to solve for Q and P worked. One thing that’s key to remember is that since G.T. is Q_y – P_x, given that the integrand in this question is presented as ye^x + x²y², it’s helpful to switch the expression to ye^x – (–x²y²) and then integrate both terms with the negative signs in place of the positive sign. 

And that was all I got done this week. Like I said at the start, not too bad but not that great either. I’m starting to once again feel a bit demoralized at how slow my progress working through Green’s, Stokes’, and the Divergence Theorems (130/600 M.P.) is… It’s not the end of the world, but after ~4.5 years of working through KA, I’m so close to reaching my goal of learning calculus and yet the finish line seems like it’s receding. 🤬 I’m sure I’ll get there eventually and, as I’ve said a million times before, I’d rather go slow and have a decent grasp on what I’m learning than go quickly and not retain or understand the concepts I’m working through. But even still… PLEASE math gods, let me have a productive week this coming week!!! 🙏🏼 🙏🏼 🙏🏼 😩