This was one of those weeks where I felt like I was quite productive and am happy with the effort I put in but didn’t make very much progress in terms of the number videos and exercises I got through in KA. The majority of my week was spent working on an exercise titled Multivariable Chain Rule. By the end of the week, I started to get a pretty solid grasp on how to work through the questions from that exercise. I still haven’t figured out why these questions work the way they do but I can tell I’m getting closer. I got a LOT of practice this week working through derivatives and which I was quite rusty on which was one thing I was definitely happy about. This is the type of math I enjoy the most, the kind where it feels like I’m solving little puzzles, so overall I enjoyed this week even though I didn’t make much:

The three videos I watched in the section Multivariable Chain Rule more-or-less went through how to solve MVCR questions but, more specifically, tried to explain the intuition as to WHY it works. Unfortunately, I wasn’t able to really figure it out… 😔 Regardless, here’s a note I took where grant was going through how to properly pronounce the notation for MVCR:

I used this formula the entire week trying to work through MVCR questions. The gist of how to use it is you 1) find the partial derivatives of x, y and z, then 2) input a given vector’s coordinates in place of x, y and z, then 3) find the derivative of the given vectors and finally 4) find the dot product of the gradient with the derivative of the given vector.

I’m guessing that explanation was difficult to understand. Here are six example questions I work on this week:

Question 1

Question 2

This question took me a long time to get through because at first I wasn’t able to understand how to find the partial derivatives of x and y. I didn’t realize/couldn’t remember that I needed to use the chain rule for (1/(y – x)). Once I realized that, the calculus too difficult to work through but it still took me a bit of time to figure it all out.

Question 3

Question 4

When I got close to end of this question, after having taken the dot-product of the gradient of f and g’(t) I wasn’t sure if there was a trig-identity that I could use to simplify 2cos(t)/sin(t) + sin(t). I spent a bit of time looking at it trying to remember different potential identities I could use and eventually remembered that cos(t)/sin(t) = cot(t) and decided to go with that. I submitted my answer and got it correct but when looking at KA’s answer they left it as 2cos(t)/sin(t) + sin(t). I just thought it was kind of cool that my understanding of trig functions is good enough now that I can make substitutions like cos(t)/sin(t) = cot(t) and actually know what I’m talking about. 😤

Question 5

Question 6

At this point in the week, I just started skipping steps. I spent the entire week working through this one exercise and by Saturday I knew the formula through-and-through so I didn’t bother writing out the notation, I just jumped straight to finding the gradient of f and derivative of g(t). I was happy and felt like the fact that I was able to skip steps and fill in the blanks in my head was an indication that I had a solid grasp on how to answer these types of questions.

On Saturday I started the next section, Curvature, but only made it through the first two videos. Even though I think I have a decent understanding of what the videos were talking about, I’m not going to go through them now since I’ve just started to scratch the surface as to what curvature is now and will have a way better understanding of it next week. One interesting thing is that this is a unique section in that it has nine videos but no exercises. 🤔

I’m now 39% of the way through this unit, Derivatives of Multivariable Functions (820/2,100 M.P.), but still have 37 videos left to watch (😳), 11 exercises to work through and eight articles to read. If I get through this unit by the end of July, it would be a miracle. Considering that I’m only 17% finished, as I write this I’m just now realizing now that getting through the entire Multivariable Calculus course by the start of September is off the table. This is disappointing but my timeline for learning calculus has been pushed back so many times that at this point it doesn’t even register as a problem. 🤷🏻‍♂️