I feel like I’ve been saying this a lot lately, but I had a pretty weird/interesting week working through KA. It felt very slow, like I wasn’t getting anything done, except I also felt like I had some pretty solid breakthroughs. I only made it through one article, one exercise and two video which I didn’t make any notes on. Nonetheless, I had some pretty significant lightbulb-moments where I got a lot of clarity and insights into some key concepts. One negative aspect of this week, however, was that I had no grasp on what was going on in the exercise I worked on or why the math worked… But by the end of the week I got pretty good at memorizing how to solve the questions (so I have that going for me which is nice). As I’ve said many times before, it’s always disappointing getting through an exercise simply through memorizing the formulas/equations, but a lot of times (maybe the majority of the time?) that seems to be the first step to actually understanding why the math works the way it does. Overall, I’m happy with how this week went but it also could have been better. 🤷🏻‍♂️

It took me a few days to get through the Triple Integrals article. The gist of what triple integrals are and how they work isn’t too difficult for me to understand. In essence, they’re used to calculate the volume of objects given specific bounds using integration across the (x, y, z) axes. The article explains that the difficult part of triple integrals is figuring out how to set the bounds for each component. Like I said, I have a decent, low-resolution understanding of what triple integrals are but in terms of setting up the bounds, I have no idea what’s going on. Here’s one of the example questions from the article:

So… Ya… It took me a day or two just to work through this single example, partly because I couldn’t come up with the equation for the bounds of the circle being projected onto the (x, y) plane. I realized I needed to use the completing-the-square method but couldn’t remember how to do it. I Googled something like “Completing the square visualization” and found this video:

After literally YEARS of hearing about the completing-the-square method, this video helped me to FINALLY wrap my head around what’s going on with this technique. If I thought about it I would have assumed that the technique used a literal square, but I had never taken the time to think through it. I’m not going to reiterate what the video says, but here’s the page from my notes where I wrote out the gist of what’s going on:

I find that visually thinking through this technique is SUPER helpful. I’m guessing that when I need to use this technique going forward it will be much easier for me to work through. BOOM. Breakthrough.

Finally, here are two screen shots from the end of the Triple Integrals article which summarize what the article talked about:

The next section was titled Change of Variables and weirdly only had two exercises in it. I got through the first exercise last week and it was a breeze. The second article I worked on this week was the complete opposite in that it was incredibly hard. It took me three days to get through and although I was able to figure out how to solve the questions (the math was actually pretty straightforward), I had (and still have) no clue what was going on in the questions. To preface the example questions below, the method to solve each question starts by replacing the variables in the integrand with the given vectors, then you find the Jacobian of the function in the integrand with the substituted variables, then lastly you input that value from the Jacobian back into the integrand and multiply it with the function with the substituted variables. (Of all the sentences I’ve written in the past 221 weeks, I’m pretty sure the sentence I just wrote was the most confusing one ever.)

So with that incredibly confusing explanation out of the way, here are five of the questions I worked through:

Question 1

Question2

Question 3

Question 4

(As a side note, this question was definitely one of the most frustrating ones. At this point I had figured out how to solve the questions but had no clue what was going on with changing the variable and, although I solved the question ‘properly’, I got it wrong because for some reason the variable u was supposed to be negative. I don’t understand why, but this is because at the beginning of the question it states that “u < 0” which means that it has to be negative for some reason. Like I said, I have NO clue what’s going on… 😔) 

Question 5

Throughout the week I also watched the first two videos from the next section, Surface Integrals Preliminaries, and had some pretty solid insights come from those two videos. I didn’t make any notes on either video, however, so I’m going to wait until next week to try to explain what that section covers. Having gone through those videos and having a decent grasp on what seemed like a pretty interesting concept, I’m definitely looking forward to getting through that section and the following one titled Surface Integrals!

(Talk about a cliff hanger. 😏)

I’m slowly but surely getting closer to finishing off Integrating Multivariable Functions (960/1,600 M.P.). That said, I still have 17 videos, nine articles and four exercises left to get through… So I’m feeling good that I’m close-ish to FINALLY getting through this unit, but I still definitely have a ways to go. Thirteen of the 17 remaining videos are coming up in the Surface Integrals section so if it’s as interesting as the two preliminary videos I watched made them out to be, hopefully that section and those 13 videos won’t take me too long to get through. I’ve got one more month left in 2023, so I’m hoping I can finish strong and maybe even finish of this unit before the start of the New Year! Fingers crossed. 🤞🏼