I didn’t have the most productive week, but I FINALLY passed the Integrating Multivariable Functions unit test. 😮‍💨 For the most part I knew how to solve all the questions and generally knew what was going on in the big picture with all the questions. But for most of the questions I still didn’t have a firm grasp on why the match worked and I can tell it will be a while before I completely wrap my head it all. I started the following unit, Green’s, Stokes’, and the Divergence Theorems, on Thursday and spent the rest of the week reviewing the divergence and curl operations. (Or functions? I’m not sure.) I already had a decent understanding of divergence so it didn’t take me long to review that part. I found the curl operation still pretty hard to understand but think I made a bit of progress figuring it out this week. I’m going to wait until next week to talk about both operations once I had a better understanding of each. I wish I would have studied more this week but am PUMPED to finally be done Integrating Multivariable Functions. (And it only took me 18 weeks… 🙄)

Here are six questions I worked through this week on the unit test:

Question 1

I find these questions both fun but also infuriating at times. The calculus and algebra here is pretty straightforward and feels like I’m solving a puzzle when I work through it, which is fun. It’s frustrating, however, that there are SO many steps to these questions and if you get one of the millions of steps wrong you get the question wrong. (And have to restart the entire unit test! 🥵) I ‘cheated’ on this question and a few others by checking my answer in Symbolab before submitting. I’m glad I did though as I mistakenly thought that 2⁵ = 32 when it of course equals 64. Even though I made that careless mistake, I’m happy that I can work through these types of questions relatively easily at this point.

Question 2

I redid my notes for this question but it actually took me four pages to work through. I had to spend a bunch of time reviewing log rules which was disappointing that after 4.5 years of studying math those rules still haven’t sunk in yet. To be fair, I haven’t had to use log rules much over the past few months and the quick examples I made for myself to work through to prove the rules were easy for me to think through and accurate. Once I remembered the necessary log rules, this question wasn’t too difficult to work through. Not to gloat, but I’m pretty happy with myself looking at my notes from this question. I would never have been able to do the math I worked through on this question a year ago. So I have that going for me which is nice.

Question 3

Having worked through these types of questions so many times over the past few weeks, I was able to solve this question pretty quickly, but I still felt unsure of what I was doing as I was solving it. (Which speaks to the fact that I know how to solve this type of question but still don’t solidly understand why it works.) That said, I’m feeling more confident with using linear algebra and taking the dot product of vectors which I’m definitely happy about. I’m hoping that as I finish off calculus and once I get started in the Linear Algebra subject it will all become much clearer. 🤞🏼

Question 4

I found this question a bit tough. Although there wasn’t anything about it that was differentially more difficult for me to understand, it was more that I found the algebra and log rules a bit tricky to think through. I was also a bit confused that the inner integral had brackets around it and wasn’t sure if that made it different than if there were no brackets around it. 🤔 So there’s nothing too noteworthy about this question but I just wanted to add it here because I found it a bit hard.

Question 5

I probably should have done at least some of the math on this question but was happy that I actually understood why the third answer was correct. As I mentioned in my note above, it takes me awhile but I can now think through and visualize what’s happening with the 3D coordinates such as this one. I also know what this type of question is asking when it talks about normal vectors and them being pointed inwards and outwards. Moreover, I’m glad that I knew that the cross product of the partial derivatives of the parameters of the cylinder (…😐) at the point in question tells you the normal vector AND that the vector with the opposite sign (i.e. the third answer) would tell you the inward facing vector. Boom. Smart.  

Question 6

This was the final question I answered on the unit test and I was STRESSED working through it. I was ~75% sure I knew how to answer it and what I was doing but I wasn’t positive, so when I got it correct it was a relief to say the least. I assumed I needed to replace the x and y variables with the given expressions which turned out to be correct. I also thought that since I was integrating the new expression I’d need to add in the absolute value of the determinant to scale the expression to the change of variables. Again, I don’t really know why this is the case, but I’m happy that 1) I simply remembered that I needed to do it in the first place and 2) I remembered how to actually solve the determinant. I ALMOST got this question wrong, not because of the calculus of linear algebra but because I almost said that 5/(4*48) = 1/38 but it’s actually just 5/192. I used a calculator to double check and am glad I did as I would have lost my 💩 if I got the final step of the final question wrong because of a stupid fraction.

As I mentioned at the start of this post, I reviewed divergence and curl in the second half of the week by reading through an article on each operation. Again, I’ll make notes next week at least about curl but maybe not divergence since I find it pretty straightforward.

Thank the math gods I’m FINALLY moving forwards into Green’s, Stokes’, and the Divergence Theorems (0/600 M.P.). I realized just now as I’m writing this post that the title of the first unit is Formal Definitions of Div. and Curl (Optional Reading) which makes sense given what I reviewed at the end of this past week. The section has five articles in it so hopefully I can get through it all before the end of this coming week. Although this unit only has 600 M.P., it has a ton of videos in it and a bunch of articles so I expect I won’t get through it for at least few months. If I could get through the unit before my birthday on May 5^th, I think that would be pretty decent. What an incredibly cool birthday present it would be to finish Multivariable Calculus on my birthday. 😎 🎂