This was one of the most satisfying and productive weeks I’ve had in a LONG time. I almost reached my goal of getting through the section Double Integrals but fell just short. I watched all six videos and getting through three of the four exercises in that section. Plus, at the start of the week I got through the last article in Line Integrals in Vector Fields (Articles) and FINALLY finished the last exercise in the previous section, Line Integrals in Vector Fields. (And surprisingly, I actually understood both.) The majority of my week was spent on the latter section and found the “what” and “how” of what was being taught (which was double integrals) very easy to understand. (By that I mean I quickly understood what Sal was talking about and how to use the formulas that were shown.) I feel like I’m close to understanding why double integrals work the way they do, which is always the final piece of the puzzle. With all that said, I didn’t spend any time this week actually making any notes about what double integrals are, but I did make a ton of notes on the questions I worked through. So this post will be a bit different than usual in that it will mostly just be examples of the questions I worked through without much explanation of what’s going on.

Before I started Double Integrals on Wednesday, I finished off both Line Integrals in Vector Fields and Line Integrals in Vector Fields (Articles) on Tuesday. The last article I had to read, Constructing a Unit Normal Vector to Curve – an absolutely terrible title, was pretty straightforward. The layman’s explanation of what it talked about was that if you have a curve created by some vector function you can come up with “unit normal vector” which will be another vector function that will tell you the coordinates for a vector with a length of 1 unit that’s 90 degrees to the OG vector function at every point along that function. (I’m guessing that doesn’t make sense.) Here are a handful of screen-shots I took from the article which summarize is:

I found the final vector in the last screen-shot above a bit hard to understand. I couldn’t visualize why you’d need to divide each component in the vector by the root, of the sum, of the square of each component in order to normalize the vector. I made this note to help me understand it:

After getting through that final exercise, I then started and finished the fourth and final exercise from Line Integrals in Vector Fields. Here are two example questions from that exercise:

Question 1

Question 2

I don’t know why these questions work, but the way to solve them is by first understanding that the notation f = ∇F means that in order to find the value of F on its own, you need to find the integral of both sides. Then, when you integrate ∇F, instead of adding a + C to the end of the antiderivative, you add a H(y) and G(x) to the end of the x- and y-components of the vector, respectively. You set each antiderivative to equal the other and if, for example, the x-component side of the equation has an extra function of y in it that the y-component side of the equation doesn’t have, then that’s what the H(y) represents.

(I’m fairly positive that doesn’t make sense… I don’t know how to explain it or why it works, but I know how to do it. 🙃)

As I said at the start of this post, I didn’t make any notes this week on what double integrals are or what they do. Without going into detail, the gist of what double integrals do is they help you find the volume of objects. (I’m guessing they can do more than that too, but that’s all that was taught to me about them this week.) In the same way that finding the antiderivative of an ordinary 2D function will tell you the area underneath that function, if you find the antiderivative of a 3D object’s x-dimension and the antiderivative of the object’s y-dimension you get the area of the base of that object. You can then multiply that area by the object’s gradient of its z-dimension to get the volume. (I think.) It would be like if you had a house with a wonky roof and you found the area of the base of the house and broke it up into a trillion little hypothetical columns starting from the base of the house going up to their respective heights to the roof of the house and then summed the value of all the columns to find the volume of the house. (Again, I’m guessing that example doesn’t make very much sense. 😔)

Below are eight questions I worked on between the three exercises I made it through. Questions 3 – 5 are from the first exercise, Questions 6 – 8 are from the second exercise, and Questions 9 and 10 are from the third exercise:

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

This was the first question I worked from this exercise and had no clue what to do to solve it initially. I assumed that I needed to input x = 0 and x = 2 into the given solutions to see if the range of y at those x-coordinates matched the image. This seemed like a bit of a brute-force/”trial and error”-ish way to solve the question but it worked and I’m pretty sure that’s essentially the way I was supposed to solve this question. I did the same thing for this following question:

Question 10

For this coming week, the next section I’ll be working on is titled Double Integrals (Articles). My plan is to hopefully get through the four articles and one video from that section this coming week and then be able to give a better explanation as to what double integrals and how/why they work once I’ve gotten through those articles. I’m now 43% of the way through Integrating Multivariable Functions (690/1,600 M.P.) so I’m certainly a lot closer to finishing it off now than I was at the end of last week, but I still have quite a long way to go. My goal now is to get through this unit by the end of the year, which gives me ~2 months. I think that’s a reasonable timeline to finish it off, especially if I keep making as much progress in the coming weeks as I did this week! 🤞🏼