This week was the same as my past few weeks working through KA. I felt like I didn’t make much progress and only got through half an article and two exercises, but the silver lining is that the exercises worked out to be worth 10% of the M.P. in the unit. I’m frustrated that I didn’t have much of a grasp of why most of the math I worked through works the way it does. I’d say I have a tenuous, low-resolution grasp on what’s going on with triple integrals, but don’t have any real understanding of why the math works which makes me feel like I have no clue what’s going on… The good thing is that I definitely got better at memorizing how to solve the questions. I’ve memorized what substitutions to make for x, y, and z when switching from Cartesian to Polar Coordinates. I also memorized the different Jacobian scalers that you need to input into the integrand for cylindrical and spherical integrals. But, like I said, I have no idea why I’m supposed to input those scalers into the integrand in the first place, so I feel like I have no idea what’s happening. As I’ve said a billion times before, learning how the math works seems to be more-or-less a prerequisite for me in order to understand why a given concept works, so in that sense I suppose I made a bit of progress this week… 😒

The article and exercises I worked through this week were all from the section Polar, Spherical and Cylindrical Coordinates. The article was titled Triple Integrals in Spherical Coordinates. I’m not going to go through the details, but the gist of what the article talked about was that if you have some function, f(x, y, z), and want to switch it to spherical coordinates, the function would switch to f(r, Φ, θ). In the spherical coordinate function, r is the radius, Φ is the angle along the z-axis (measured where positive z = 0° and negative z = π), and θ is the angle around the z axis on the (x, y) plane. I explained this in my last post, but in order to convert the Cartesian coordinates to Polar coordinates, you need to make the following substitutions for x, y, and z:

• x = r * sin(Φ) * cos(θ)
• y = r * sin(Φ) * cos(θ)
• z = r * cos(Φ)

Lastly, and I mentioned in my intro that I don’t understand why this is the case, you have to add in the Jacobian of the conversion from Cartesian to Polar coordinates which is equal to r²sin(Φ). (I’m 99.99999% sure what I just said about the conversion made zero sense.) In the very last question at the end of this post I work through how to find the Jacobian for this type of problem, but just below is an example question from the second article. You can see in the first step of the solution that the integrand has r²sin(Φ) added to it, but my notes for this particular question don’t show how the Jacobian scaler is derived. Anyways, here’s the question:

Question 1

So ya… these questions are pretty intense. I just about had a nervous breakdown trying to work through this question. I was happy by the end of it once I figured it out but was also pretty demoralized with how difficult it all was and my overall lack of understanding of what was happening. 😕

I finally got through that article on Wednesday and then started the first exercise on Thursday. I got the first question wrong but restarted the exercise and ended up getting all four questions correct in a row and so passed the exercise on that second attempt. Again, I sort of knew what the questions were expressing with the bounds and the given angles, but I didn’t understand why the math worked. In any case, here are two example questions from that exercise (only the first of which has notes of my own):

Question 2

Question 3

By the end, I was able to think through the solutions for these questions in my head having been given the bounds, the angle from the diagrams, and having memorized the conversions for x and y which is why I didn’t make any notes after the first one.

Finally, it took me a few attempts but I made it through the second exercise by Sunday. As you can see in my notes from the one example question below, I didn’t understand these questions well at all, but I was still able to make the substitutions and solve them. This exercise was pretty much the same as the first one except I had to use spherical and cylindrical coordinates to find the volume of a given function as opposed to calculating the area in a specific region given a specific function. The last three pages of my notes for this question work through the proof/solution of the Jacobian for the conversion from Cartesian to Polar Coordinates. (Again, pretty sure that’s not how you’re supposed to say that…) Here’s the question:

Question 4

I’m sure I don’t need to say this, but I find these questions very hard. As you can see, there are many steps to them and are therefore tons of opportunities for me to make a careless mistake. Working through the Jacobian alone, there are a ton of Trig derivatives with two different angles being used, plus the factoring involved afterwards, all of which which lead me to spending about an hour just trying to write out the solution for the Jacobian alone, after having ALREADY looked at the answer… This question was worth working through for the practice but it was also exhausting and pretty demoralizing.

This coming week I’ll be starting on the section Surface Integral Preliminaries which only has three videos in it, two of which I’ve already watched. I imagine I should get through those videos pretty quickly and will then be able to get started on the following section, Surface Integrals, which has THIRTEEN videos and two exercises… 😳 I’m REALLY hoping I can make a bit of a dent in that section so that I’ll be somewhat close to finally (FINALLY) finishing off Integrating Multivariable Functions (1,120/1,600 M.P.). At this point, I’m not that optimistic about it but, as usual, my fingers are crossed. 🤞🏼