This week was like the inverse of the past few weeks. I actually made it through a decent number of videos, articles and an exercise but I didn’t take many notes or feel like I learned too much. 🤔 I quickly made it through the final two articles from the first section of the Green’s, Stokes’, and the Divergence Theorems unit and managed to finish off all but once exercise from the following section, titled Green’s Theorem, AND made it through the first article in the third section, Green’s Theorem (Articles). The reason why I didn’t take many notes this week was because I didn’t really understand how or why Green’s Theorem works. I think I was able to get a general understanding of what G.T. is but I have a low-resolution understanding of it, at best. Overall, I’m happy with my effort this week but probably could and should have done more. 😒
On Tuesday I finished the last two articles in the first section of the unit, Formal Definitions of Div. and Curl (Optional Reading). I definitely don’t have my mind completely wrapped around the equation for the formal definition of curl, but I have a somewhat decent grasp on it which I’m happy enough about. Here’s an example question I worked through from that article:
One thing that finally clicked for me this week is the notation used for a vector-valued functions. I’m sure I learned this a LONG time ago, but it clicked that a VVF is denoted with a bolded, capital letter, ex. F. Knowing this makes working through this type of question WAY easier to visualize (although I can tell I still have a long way to go before I can completely and clearly understand the formula in its entirety). Sadly, I also still don’t really understand how to parameterize these types of functions. As I was working through this question, I couldn’t figure out that I needed to parametrize F using r(t). I did know that I needed to rewrite the vector using some other functions in place of –y and x, but I thought it would end up being –sin(θ) and cos(θ). I think I could have used θ in place of t, but I’m not really sure and since KA’s answer didn’t use θ, part of me thinks it wouldn’t have been the right way to do it. But I don’t know…
The next section, Green’s Theorem, contained four videos and two exercises. I watched all the videos (two of them I actually watched twice) and finished the first exercise which was quite easy. I skipped to the following section, Green’s Theorem (Articles), without even attempting the second exercise because I only had a vague idea of what G.T. is and thought the article would help make it more clear. Here are some screen shots from that article that summarize what G.T. is all about and then my notes summarizing what I think G.T.’s about:
I’m not going to reiterate my note, but I think it sums up what G.T. is all about, but I’m not 100% positive.
Here are two questions from the exercise I worked through:
Question 1
Question 2
As you can see, this exercise was simply about learning the different names for the different types of functions and “regions” (which is a word that I don’t think I’ve seen used on KA before but that seems to refer to the area of a specific region of a multivariable function).
And that’s all I got through this week… I’m going to go back this coming week and make notes on the four videos from Green’s Theorem, and hopefully will be able to get through the exercise AND the second article in the following section, Green’ Theorem (Articles). Even if I do, I still have a TON of videos left to get through in Green’s, Stokes’, and the Divergence Theorems (80/600 M.P.), but I’ll feel good about my pace of progress through this unit, overall. I’m really starting to see the light at the end of the KA tunnel, at least for the Math section anyways. Hard to believe it will be close to five years of studying by the time I get there! Then on to Physics… 🤓