I wish I would have gotten more done this week but I’m still happy with the progress I made and what I was able to learn. I finished the section Laplacian and have a relatively decent understanding of what it is and how it works but still need to get a better grasp on why it works. Halfway through the week I got started in the final section of this unit, Jacobian, which I don’t really understand at all. I managed to get through the five videos in that section and one of the two exercises. That said, very little of what I watched/worked through seems clear to me. The how of the Jacobian (i.e. how to solve the questions using the Jacobian formula) was quite simple to memorize and use. They why, on the other hand, is something I don’t understand at all. I also quickly worked through a few questions this week which I would never have been able to get through a few months ago so even though I didn’t get as far as I hoped I would, I can tell that I’m really starting to wrap my head around calculus which feels pretty good. 😌
I started my week working through the single exercise from Laplacian titled Finding the Laplacian. As you’ll see below, the questions I worked through from section were not too difficult in the sense that the algebra and calculus wasn’t hard but the questions were tough because there were multiple steps that needed to be done. I would often get one part wrong in a question and then have to redo the entire exercise.
Before I go through some example questions from that exercise, at the end of the week I was still having a hard time understanding what the Laplacian was, in general. I asked ChatGPT to eli5 it for me and this is what it said:
Even after reading ChatGPT’s explanation, the concept still hasn’t fully sunk in for me yet. It makes sense to me but it still seems confusing… Regardless, here are a few questions I worked through from that exercise:
Question 1
Question 2
Initially I got these questions, ones that asked me to determine if a function’s Laplacian was harmonic, wrong. (Not sure if that’s how to phrase that.) I realized that if a Laplacian is harmonic, it means that all the terms cancel out and the Laplacian equals 0. My understanding is that if a function’s Laplacian is 0, the gradient of the given function is flat. I think of it as a given point in question the surface around that point is like a piece of paper that may or may not be tilted but is completely flat. This could be wrong, however.
Question 3
Question 4
I made a note that both the third and fourth question above only took me about one minute to work through. I’ve been finding a lot of the concepts that I’ve learned over the past few months hard to wrap my head around, so getting through these questions as quickly as I did was a nice reminder that I AM getting better at math, especially algebra and calculus.
As I mentioned, there were five videos in the Jacobian section, all of which I watched. I don’t understand it at all, however, and am going to wait until next week to give an explanation as to what I think it is. I just asked CGPT what it is and this is what it told me:
Here are two questions from the one exercise I made it through which was titled Finding the Jacobian:
Question 5
Question 6
As you can see, the how of how to solve these questions (i.e. using the formula of using the partial derivative matrix) is pretty straight forward. I don’t at all understand why the formula works or even what it’s expressing, but hopefully I’ll be able to explain it next week.
Like I said, I’m not thrilled with how this week ended up going and wish I would have been able to get more done. Nonetheless, I know I still made some progress and, if nothing else, I know that I have a MUCH stronger grasp on math, in general, than I did even a few months ago. Some of these bigger concepts are hard for me to understand right now but I’m confident the fog will lift and I’ll understand them soon enough. I’m SO close to getting though this unit, Derivatives of Multivariable Functions (1,620/2,100 M.P.), with only one exercise left to get through before the unit test. (Although, as I mentioned, I’m still going to rewatch the videos in the Jacobian section which will take some time next week.) I’m really hoping I can get through this unit by the end of this upcoming week but, at a minimum, I’m pretty sure I’ll be through it before the end of Week 207. It’ll be nice to start a new unit when I hit the FOUR. YEAR. MARK. Crazy to think I’ll be going into my fifth year of KA soon… 🤯