I had another pitiful week on KA. I got through 2.5 articles from the first section of Green’s, Stokes’, and the Divergence Theorems and reviewed another article that was suggested on 3D Flux, so in total I worked through 3.5 articles this week. The silver lining is that what was being taught in the articles I made it through was pretty difficult so I have some sympathy for myself for not getting more done. Regardless, I’m not sure if I ended up studying for 5 hours this week and so I’m again disappointed in myself. As I’ve mentioned in my last few posts, I have a number of other exciting things on the go these days, but I also feel in a superstitious kind of was that getting through calculus is key to me succeeding in the other projects/goals I’m working towards, even though they’re unrelated. 🤔

This is going to be a pretty short post, partly because I didn’t get much done this week but also because I literally have 30 minutes to write it. The two articles I worked through from the first section of Green’s, Stokes’, and the Divergence Theorems this week were titled Formal Definition of Divergence in Two Dimensions and Formal Definition of Divergence in Three Dimensions. Here are screen shots from the example question given to me in the first article and my notes working through the question below:

As you can see, this question (and the entire article for that matter) is all about understanding the theory behind the formal definition of divergence, and to be honest I don’t really get it. I can kind of follow along with the question and understand what is going on with most of the steps and how to do them, but when I look at the formula, I don’t see the bigger picture of why each component fits together or why the math works for some of the steps. The good thing is that even though the formula for the formal definition of divergence is a bit intense with some crazy notation, I don’t feel at all intimidated by it, at least not in the same way that I would have a year or two ago. I’ve seen and learned about all the different variables/factors/types of notation in the formula before now and even though I can’t stitch together the bigger picture of how they’re connected in the formula and how they’re related to divergence (not YET, anyways), I’m not concerned about it since I’ve learned about each individual component of the notation in the past.  

Here’s a screen shot of the summary from the first article and a question I ask CGPT based off the first bullet point:

Reading through both of those screen shots, I still find it confusing. I think divergence is likely a bit of an abstract concept kind of like limits. In the same way that when you think of limits, you’re saying to yourself something like, “take the limit as this number goes to 0 but never actually gets to 0 – but we still have to think of it as 0, but it’s not!”, I think that to mathematically prove divergence you probably have to use your imagination in the same type of way. I don’t actually know what I’m talking about though and, as usual, could be completely wrong about all of this.

Here are two screen shots from the second article I worked through:

As I mentioned, I’m able to generally follow along with and understand this breakdown of the different variables/different parts of the notation in the formula, but it’s still definitively a bit confusing and overwhelming, especially everything that’s inside the integrand. I’m not going to reiterate what everything in the formula means here, partly because I don’t understand it all but also because I don’t have enough time right now even if I did. 😅

The last thing I’ll mention is that I was having a hard time remembering/understanding why there were two integrals in the formula. I asked CGPT and got the following answer. I made a note of what I think the reason is for the double integral is which follows the screen shot:

This probably doesn’t make sense but, again, I’m not going to bother reiterating what I wrote out right now. Hopefully next week I’ll 1) understand it better and can give a clearer explanation of what’s happening and 2) have more time to do so.

This upcoming week needs to be a good one. I NEED to get through the last two articles from the first section of Green’s, Stokes’, and the Divergence Theorems (0/600 M.P.) and can then hopefully get through the following section, Green’s Theorem, which has four videos and two exercises in it. Time to turn things around! 😤 😤 😤 😤