I had another solid week this week and managed to get through 11 videos, one article and two exercises. I finished off the section Divergence, which ended up being fairly straightforward, and almost made it through the following section, Curl, but was one video and two exercises short. I likely would have made it through that section on Saturday but got sidetracked trying to understand why the determinant of a 2×2 matrix equals the area of the parallelogram that you can create with the two vectors. As you’ll see at the end of this post, I took me awhile but I was able to figure it out which seems like a much bigger milestone to me than if I’d simply made it through the final video and the two exercises at the end of Curl. So, for the third of fourth week in a row, I’m really happy with the work I did and what I was able to learn this week. I definitely feel like I’ve been picking up momentum lately which has been nice! 💪🏼

I spent Monday and Tuesday watching the fifth and sixth videos from Divergence and also reading an article that also happened to be titled Divergence but was in a different section. The videos and article really helped me understand what divergence in a vector a field really is. Here are four pages of the notes I took about it:

I think these notes do a pretty good job of summing up what divergence is (at least to the degree that understand it) so I’m not going to try to go into any more detail.

On Wednesday I got started on the section Curl. For some reason I was worried that it was going to be really difficult to understand but, so far, it hasn’t been too bad. Here’s a screen shot from the first vid in the section that I’m sure won’t make sense but I’ll explain below:

In the video the little blue dots actually move around the screen which have the look of some sort of fluid-like motion. In the center right of the screen where there’s a white circle with a plus sign in it, the arrows point around the circle in a counter clockwise rotation. This is what’s known as positive curl. At the top of the screen, you can see the same type of thing but there’s a circle with a minus sign in it and the arrows point clockwise which means there’s negative curl in that region. At the origin you can see that the arrows point in from the first and third quadrants. At this point there’s what’s known as zero-curl since the fluid is not rotating one way or the other.

Here are three example questions I worked on from the first exercise in this section. The first example shows zero-curl, the second shows negative curl and the third shows positive curl:

Question 1

Question 2

Question 3

At the end of Wednesday I also had a fairly big insight into how these types of vectors and partial derivatives work and, more importantly, why they work:

I’m still not 100% sure if this is correct, but as the week continued it seemed more and more likely that I was correct about this based on the other videos and questions I watched and worked through. The videos where there were blue dots floating around the screen helped me to understand that there’s a difference between the static xy-plane and the coordinates of each individual dot. Like I said, I’m not completely sure if I have this correct but I think I could be.

I don’t understand why the formula for curl for a 2D vector field works but the formula itself is very easy to memorize. Since I don’t understand why it works, I’m not going to try and explain it yet, but here’s a screen shot from the video I watched where Grant worked through how to derive the formula and then my note of the formula itself:

Here are two questions I worked through from the second exercise in this section titled Finding Curl in 2D:

Question 4

Question 5

As you can see, the questions from this exercise also gave me some good practice/review on working through partial derivatives and using derivative rules. It took me a couple of attempts to get through this exercise but only because I was rushing through the questions and made careless mistakes. It was very satisfying for me to be able to work through the calc and algebra without having any hesitations from not knowing what to do.

There were five videos on 3D curl at the end of Curl, four of which I watched before the end of the week. At the beginning of the fifth video Grant talks about how to calculate the determinant of a 3×3 matrix. I vaguely remembered the determinant of a 2×2 matrix could be thought of as the area of the parallelogram that you can create by sliding the two vectors across each other, but I had no idea why the formula, det(v ⃗) = |ad – bc|, worked. I asked ChatGPT why it worked but still couldn’t figure it out. I then Googled it and found a diagram that made the whole thing much easier to understand. I copied out that diagram and can now understand why the determinant of a 2×2 matrix equals the area of the parallelogram, but still can’t visualize/think through the entire formula in my head. Anyways, here’s how it all played out:

Heading into Week 204, I’m optimistic that I’ll be close to finishing off this unit, Derivatives of Multivariable Functions (1,300/2,100 M.P.), by the end of the week. I have nine videos, two articles and five exercises left to get through. Two of the exercises are from Curl and the rest are split between two sections titled Laplacian and Jacobian, both of which sound pretty weird to me. 🤔 It’d be GREAT if I could get through everything by the end of the week, but I think that’s a bit unlikely. Nonetheless, my goal is to get through the entire unit, including the Course Challenge, by the end of Week 205. If I have as good of a week this week as I’ve had the past three weeks, I think it could be possible!