I got less work done this week than I have in a long time on KA, but I’m still happy-ish with how the week went, overall. I only made it through four videos and a single exercise which feels quite disappointing, BUT, as you’ll see below, most of my week was spent working on what’s called “higher order partial derivatives” which gave me a lot of good review/practice doing derivatives, in general. There were times this week where I was working on these questions and came up against a derivative I hadn’t seen in a long time and managed to get it correct on my first attempt. (Question 4 below was one in particular.) So even though I didn’t make much progress through this unit, I still feel pretty good overall with where I’m at in terms of my general understanding of calculus and feel like I’m in a good position moving forward.

(Also, the reason why I got nothing done this week was because it was literally my busiest week of the year at work. To be honest, I’m surprised I managed to get through as much as I did so, although I’m disappointed, I’m not beating myself up too badly about it. Next week will be better.)

The first video I watched this week was titled “Symmetry of Second Partial Derivatives”. What it demonstrated was that if you have a multivariable function with two inputs, say x and y, and you take the function’s derivative with respect to x and then with respect to y, it’s the same thing as doing it in reverse order and taking the function’s derivative first with respect to y and then with respect to x:

Grant briefly went through the different types of notations for this type of thing which look like this:

After getting through this video, I spent the next few days working on the last exercise of the first section which was titled “Higher Order Partial Derivatives”. A higher order partial derivative is when you take more than one derivative of a specific variable while treating the other variable like a constant. This is the same thing I was doing last week except that in this exercise I took the derivative of certain functions multiple times. Here are six examples of questions I worked through:

Question 1

Question 2

Question 3

Question 4

(I made a note after working through this question that I was pumped that not only was I able to work through the power rule with the exponent being a fraction, but I did so in about 45 seconds and didn’t to think twice about it. This is something that would have taken me quite a bit of time to work through a few months ago and probably wouldn’t have been able to solve a year ago.)

Question 5

Question 6

I got started on the next section, Gradient and Directional Derivatives, on Friday. Although I got through three videos in this section, I’m not going to go into much detail now about what I learned this week because I still don’t have a great understanding of it. My general, laymen’s-terms understanding is that you can do some fancy, partial derivatives to find the slope at any point in an (x, y, z) field which is known as the gradient. (This makes sense to me since when I would hear the word gradient I always thought of it as the slope of a given surface.) Here’s a note I took that explains the how of what I learned:

As you can see, to find the gradient of a multivariable function you simply find the partial derivative of each input and then put them into a vector. The best part about finding the gradient is the name for the notation used with it which is the upside down triangle you see in my notes, ∇, which is called “Nabla” and which you put in front of the function to indicate that you’re finding the function’s gradient (i.e. ∇f(x, y)). Like I said, I’ll go into greater detail next week, but here are two examples of questions I worked through in the first exercise I attempted but didn’t pass:

Question 7

Question 8

Like I said at the beginning, I think I’m in a pretty good position right now to start making some serious progress though M.C. and this unit, Derivatives of Multivariable Functions (340/2,100 M.P.), in particular. Even though I didn’t get too far this week, I enjoyed working through and practicing the higher order partial derivatives. The questions I worked on felt like little puzzles that I had to solve which over the past 3.5 years has always been my favourite part of KA but something I haven’t had much a chance to do in the last little while. I’m hoping that what I work through over the next few weeks will feel similar to this because, as far as studying math goes, it’s pretty fun! 🤓