This week compared to the past few weeks wasn’t too bad. I didn’t study for five hours which is always my goal, so in that sense it was a fail, but I made it through a bunch of videos and three exercises, AND surprised myself by getting the correct solution to all the questions in the final exercise on my first attempt. I didn’t make much progress this week at better understanding what these theorems are all about or why they work, but I do think I might a bit of progress understanding how to use them. The notation is slowly starting to sink in which is helping to make a difference. In most of the subjects I’ve studied on KA, I usually first understand how to solve the questions before figuring out why the math works, so hopefully that means I’m getting closer to wrapping my hear around all of this. 🤞🏼

For the millionth time in a row, I once again don’t have very much time to write this post this week. I’ll go into a bit of detail on some of the questions below but not much. Here are three questions from the first exercise I worked through which was titled Circulation of Green’s Theorem:

Question 1

I’m pretty sure my theory from last week was correct when I said that in these questions the way to solve them is just to look at the terms in the integrand and decide which term would be easier to integrate with respect to which variable. This seems so stupid to me that I have a hard time believing that it’s true. However, 1) that’s what I did and it worked and 2) I’m pretty sure that’s what KA’s answer is saying. It just seems so dumb that it’s so vague and ambiguous and there’s not a more rigorous way of proving/confirming which variable to associate and integrate with each term. Regardless, like I said, that’s what I did for this and the last question below and got them both correct, so I guess that’s the way to do it. 🤷🏻‍♂️

Question 2

Question 3

The question just below is from the second exercise I worked through this week from the Stokes’ Theorem section which was titled Orientation and Boundaries. I think I have a decent understanding of what these questions are talking about and how to think through the solution, but the problem with the questions from this exercise was the phrasing of the questions. Here’s one that I worked through:

Question 4

I’m at a point now where I can relatively easily understand why the cylinder is shaped and oriented the way that it is in diagram based on the functions and bounds stated at the start of the question, so I’ve got that going for me which is nice. This particular question isn’t too hard to interpret so it’s not a great example of what I’m talking about, but there were a few others where the phrasing used to describe the object, it’s surface and the boundary, C, was really confusing. A few of the other questions also talked about the normal vector on the surface either pointing inwards or outwards. Those questions I found particularly difficult but I apparently I didn’t have enough sense to screen shot them and add them in tot his post… 

I made it to the second exercise in the section (also titled Stokes’ Theorem) late in the week and didn’t think I would pass it before the week finished. I managed to get through the exercise on my first attempt which I would have been happy about regardless, but I also found the questions very challenging/confusing so I was extra pumped to get through the exercise on the first attempt. Here are two of the questions I worked through:

Question 5

I looked at this question for about five minutes before I wrote anything down. I had a vague recollection that I had to 1) use the curl operation and 2) input a capital letter into the matrix where I didn’t have the expression/function. (As a side note, I thought the variable should be R since it usually goes P, Q, R for the x, y, z vectors, and since the expression/function that I was solving for was for z it made sense to use R as the variable. KA’s answer used P so now I don’t know if I did it wrong or if KA just didn’t care which variable to use and went with P. 🤔) After setting up the matrix I just found the determinant and made a guess at what I was supposed to do from there and somehow managed to get it correct. Looking at my notes, it looks like my math was more-or-less correct but I could be wrong and maybe just got lucky. At the very least, I don’t have any clue why the math works. I got lucky at the end trying to ‘reverse engineer’ the middle term in the integrand of the double integral underlined in green with the ĵ expression from the curl of F. (There’s no way that made any sense…) Nonetheless, I still got it right which I was PUMPED about. 

Question 6

This question was much easier than the previous one as I just had to set up the curl of F and then solve it. Again, I still don’t know what the hell is going on, but I’m getting better knowing how to solve these questions.

And that’s all I got done this past week and all I have time to write about. For the upcoming week, I have two more videos to rewatch and one more exercise to redo before I can start the unit test again. I think the last time I did the unit test I got about half the questions wrong, and I think I’ll do much better in my next attempt. I also definitely think it’s possible that I’ll be able to get through the test this coming week but at the same time wouldn’t be surprised if it doesn’t happen. I’ve been working on Green’s, Stokes’, and the Divergence Theorems (400/600 M.P.) since Week 231 so I’m ready to be done with it. Finishing it off on a nice round number, Week 250, would also be quite satisfying to me so fingers crossed I can get it done! 🤞🏼