In terms of getting through videos and exercises, I made very little progress this week on KA but overall, I put in a pretty solid effort and actually learned quite a bit. 😌 I made it through the final article in the section Line Integrals for Scaler Functions (Articles) which took me until the end of Wednesday to finish. I then watched the first two videos in the following section, Line Integrals in Vector Fields, and wrapped up the week by working through the first exercise in that section. My goal was to get through seven more videos and three more exercises than I actually did (lol) so, in that sense, it was definitely a disappointing week. But as I’ve said many times before, as much as I’d like to get through what I’m working on quickly, I’m more concerned with actually learning the material (“mastering” it, as Sal would say) than I am about quickly getting through everything. That said, I’m nonetheless disappointed and annoyed that my progress seems so slow right now. 😒

At the start of the week, the final article I worked through was titled Line Integrals in a Scaler Field. I took notes on three questions from this article which you can find below, all of which helped to understand the different parts of line integrals and gave me a better grasp on what is happening and how they work. I mentioned this in my last post but a line integral can be thought of as the area of a fence that twists and turns in any direction and has a height that varies as you move along it.

In all three of the example questions below, the first thing that’s shown is the curve of the ‘fence’ from a top-down perspective. The curve is a function made up of separate x– and y-functions. In the questions below, x and y are both functions of t, i.e. they’re x(t) = (some function) and y(t) = (some function). This means that the curve of the function as a whole can be thought of as f(x(t), y(t)) = (the curve of the fence). Since both x and y are functions of t, you have to rewrite the integral as a function of t. The integral is equal to the length of the fence (which is equal to the sum of all the tiny, tiny hypotenuses of the curve, i.e. the integral of all the derivatives at every point along the curve which is equal to ∫ √((x’(t))² + (y’(t))²)) dt) times the height (i.e. the z-value) of the fence at every point (meaning you find the dot product of f(x(t), y(t)) and f(x’(t), y’(t))) and put it into an integral.

(I am quite confident that what I just wrote is wrong. Reviewing the article, it seems like I have it right, but then I don’t understand what the difference is between line integrals in scaler functions and line integrals in vector fields, which is below. 😩)

Here are the three questions I made notes on from that article:

Question 1

Question 2

Question 3

This was the final question from the article and it absolutely kicked my ass. I was able to keep up with what was going on for the most part but got rocked on the algebra halfway through the question. I had to simplify (1/t⁴ + 2t² + t⁸) down to (1/t² + t⁴)² and I spent ~30 mins between using Symbolab and looking up videos on YouTube on Criss Cross Trinomial Factoring and still couldn’t figure out how to do it. I eventually gave up and just moved on. 😔 Although I got smoked on that algebra, my general understanding of line integrals was decent which was the entire point of the article and which was why I didn’t feel too bad about moving forward.

The first video in the following section started integrating the math I’ve been working on with concepts and measurements used in physics. I’m not going to go through the different units used (partly because I don’t fully understand them but primarily because I’m super busy today and simply don’t have the time), but here’s a screen shot from the first vid that gives you an idea of what I’m talking about:

The three questions below are from the exercise I worked on which all remind me of the example questions from above in the article on line integrals in scaler fields. Like I said, I don’t understand the difference between line integrals in a scaler field and line integrals in a vector field… Looking back, it seems like you DON’T take the dot product of the function and it’s derivative when solving line integrals in scaler fields but you DO for line integrals in vector fields… I really don’t understand what’s going on so obviously I can’t explain it here. 😣 But in any case, here are four questions I worked on from the exercise:

Question 4

Question 5

Question 6

Question 7

By the end of the exercise, I had a pretty good grasp on how to solve these questions, i.e. I was able to use the formula relatively easily. I don’t understand what is going on or why these questions work which is pretty disappointing. Hopefully as I make my way through the section this coming week, the difference between line integrals in scaler fields VS vector fields will become clearer. (PLEASE math gods, let that be the case. 🙏🏼 🙏🏼🙏🏼)

This week I’m going to give myself the exact same goal I gave myself last week, to get through this current section. That’d mean I’d need to get through seven more videos and three more exercises. I think that should be doable, but I also thought I was going to get through nine videos and four exercises last week and only got through two and one, respectively, so who knows… I’m now 10% of the way through Integrating Multivariable Functions (160/1,600 M.P.) which isn’t too bad but also isn’t that good either. I’m REALLY hoping I can crush this upcoming week so that I’ll feel a bit better in terms of my progression. As usual: 🤞🏼.