I’m sad to say that this week my progress on KA was pretty average if not underwhelming. I got through seven videos and one exercise which is probably slightly better than average for me, but I didn’t take any notes on the last four videos which I definitely should have done. The exercise I worked through was pretty difficult and I’m happy that I managed to get through it, but I don’t have much of a grasp on why the math worked from that exercise. (Which seems to be a common theme in my last few posts…) I do have a decent understanding of what the questions were talking about, generally speaking, and got pretty good at memorizing how to solve them, which is a bit of a silver lining. As you’ll see below, I had to find the partial-derivative cross-product of a given function in the questions and although I know how to do it, I have no clue why it works. All in all, I’d say my progress was passable but feel like it was a pretty mediocre week, overall. 

The first section I worked through this week was titled Surface Integrals Preliminaries. The section contained a series of three videos, all of which worked through the same question which was about finding the surface area of a torus, a.k.a. a donut. 🤤 Here are a bunch of screen shots from those videos with an explanation under each one of what’s going on:

In this first screen shot you can see the torus in the top left corner. The “cylinder” of the torus, so to speak, encircles the z-axis and is centered on the (x, y) plane where z = 0. The radius from the z-axis to the center of the cylinder is length b and the angle of the cylinder around the z-axis is denoted with t. (The bottom two diagrams illustrate this.) The radius from the center of the cylinder to the surface of the torus is length a and the angle from the center of the cylinder around the cylinder is angle s. (The top right diagram shows this.) The right side of this screen shot states that both angles go from 0 to 2π.

The purple ring on the outside of the diagram shows the outermost circumference of the torus. On this line, s = 0 and therefore the radius of the inside of the cylinder, a, is “flat” on the (x, y) plane going away from the z-axis. Therefore, the purple ring indicates b + a as is travels around the outside circumference of the torus. You can see that Sal made a chart on the left that describes the purple line where s = 0 and t, the angle of the torus around the z-axis, ranges from 0 to 2π. The yellow ring is the exact same thing except that s = π, therefore the radius a again sits “flat” on the (x, y) plane but in the negative direction, i.e. it points towards the z-axis. You can see this denoted on the chart on the left with the yellow vertical line which indicates that s = π and t, once again, ranges from 0 to 2π.

This screen shot shows the top and bottom of the torus drawn around the z-axis and also shows the lines denoting these rings on the chart to the left with the blue and yellow vertical lines. Sal then drew in an orange ring on the torus showing the circumference of the cylinder where t = 0. He also denoted this ring on the chart on the left with an orange line at the bottom of the chart where t = 0 and s ranges from 0 to 2π.

Here Sal drew in three more rings of the cylinder at the angles t = π/2, t = π, and t = 3π/2. He also drew the horizontal lines on the chart on the left which correspond with and denote those rings. 

Here Sal shaded in the bottom left corner of chart. This shaded area represents a 1/16^th side of the torus which he attempted to shade in on the diagram, although it gets a bit hard to make sense of visually. The shaded area on the torus is where both s and t range from 0 to π/2. The part here that blew my mind is that if you shade in a square on the 4×4 chart, it corresponds with shading in a certain 1/16^th side of the torus. 🤯

I wrote out notes for all these screen shots but instead of retyping them out for the next two screen shows I’m just going add a picture of my notes below them:

Reading through these notes now, I still find it all very confusing. I have a general understanding of how/why it all works but don’t know it well enough to give any kind of coherent explanation. 🤷🏻‍♂️

I finished those videos on Wednesday and throughout the rest of the week ended up watching the first four videos in the following section Surface Integrals. I need to rewatch all four videos and make notes on all of them which I didn’t do this week, so I’m going to wait until next week to do that.

I started the one exercise I made it through on Friday but didn’t pass it until Sunday morning. As I mentioned in my intro, I had to find the cross product of partial derivatives in each question which I got pretty good at doing but don’t know why the math works the way it does. I had to watch a few videos to review what exactly the cross product does and found out/remembered that it will 1) give you a “normal” vector (i.e. a perpendicular vector) to the two vectors being used in the cross product calculation, and 2) the magnitude if the cross product tells you the scale (read: size) of the parallelogram created when taking the two vectors in question and lining them up “head to toe”, if that makes any sense… In any case, all the questions I worked through asked me to find vectors that were normal to either a plane, a cylinder, or a sphere and so I had to use the cross product in each question. (Also, I’m not 100% confident that a) I’m saying this correctly or b) that I’m correct in any way in what I’m talking about. I could be completely wrong about what I was doing in the questions.) Here are five examples question I worked through:

Question 1

Question 2

In this question you can see that I didn’t bother deriving the formula for the cross product of a sphere and just grabbed the formula from the question before and inputted the given values. You can also see at the end that I realized why I got the question wrong. I didn’t properly think through the angle of z which would be pointing in the negative direction since the angle of Φ was 3π/4.

Question 3

My notes here clearly don’t make sense but I got this question correct and wanted to add it to this post.

Question 4

Here you can see that I had memorized how to answer these questions and at this point started skipping a ton of steps to get to the answer.

Question 5

So, like I said at the very beginning, you can see that I actually got some decent work done this week and learned some things, although, as I said, I don’t have a crystal-clear picture of what’s going on with all of it. As frustrating as it is, I can tell that I’m on the right track and am making progress, nonetheless. This coming week it’d be great to finish off this section but it might not happen considering I still have nine videos to watch, plus I want to go back and make notes on the first four videos. Hopefully I’ll get through it all and will have a decent grasp of what’s going on by the end of it but if I don’t, the next section is titled Surface Integrals (Articles) so, at the very least, once I get through that section I should (hopefully 🙏🏼) understand surface integrals pretty well.

I’m now 75% of the way through Integrating Multivariable Functions (1,200/1,600 M.P.) so the light at the end of the tunnel is becoming brighter. I’m hoping to get through this unit asap but am not too concerned about how much longer it takes, it either way. After I get through it, I only have ONE MORE UNIT in Multivariable Calculus, then the course challenge, and then I’ll be DONE calculus… 😳 And it will have only taken me ~4.5 years.