I didn’t make as much progress this week as I hoped I would but I’m still happy with the effort I put in and with what I was able to learn. I ended up watching six videos, reading two articles and finishing off two exercises. The majority of my week was spent working on understanding the operator curl and trying to memorize the notation/formula and understand how and why it works. Part of my week was also spent working through an exercise which had me thinking through the differences between the gradient, determinant and curl, along with memorizing the notation for the other two operators, as well. I got started on the next section, Laplacian, and made it through all four videos in that section but didn’t take any notes on it so I’m not going to talk about it this week. All in all, I can tell I’m getting a stronger intuitive understanding of vectors and their associated operators, but it’s hard for me to tell how close I am to fully understanding how they work. As frustrating as it is, I’m still just as fired up as ever to keep going! 💪🏼 😤
I started my week on the nineth video from Curl which walked through an example of curl and how to calculate it using the curl formula. Grant talked about both the determinant and cross product operators in this video which I didn’t know the difference between (which is embarrassing considering how long I’ve been studying this). I asked ChatGPT to explain it to me and this is what it said:
The concept of what the two operators are and do seems pretty clear to me now, but I still find it the big picture of what’s happening confusing to think through. Nonetheless, here’s a few screen shots I took from the video I mentioned and the notes I took while working through the question presented in that video:
At the very end of the week I read through two articles on curl and in the second article found a pretty good explanation for why the formula works. Here’s a screen shot of that explanation and my note below it on it’s formula and notation:
The first exercise I worked through this week was titled Finding Curl in 3D. As you’d imagine, the questions in it asked me to use the formula above to try to calculate the curl of a given 3D vector field. The math wasn’t too difficult to work through but there were so many parts to each question that it took me a few attempts to get through the exercise. That was probably for the best, however, as it helped me memorize the formula for the cross product of a 3×3 matrix. Here are two questions I worked through from that exercise:
Question 1
Question 2
After getting through that exercise, I had one final exercise left to get through to finish off the section. The exercise was titled Symbols Practice: The Gradient and it was all about reading the notation for gradient, the determinant and curl of 2D and 3D vector fields. After a few days, I managed to get through it but even by the end of it I wasn’t really able to understand the questions that well. Here are five of the types of questions that came up:
Question 3
Question 4
Question 5
Question 6
Question 7
By the end of the exercise, I started to be able to visualize 2D vector fields where the determinant would equal a z-value which would look like a surface, and 3D vector fields where the curl would represent vectors coming off of vectors. (If that makes sense…) I also started to understand that a scalar field is like a heat map and a vector field is like fluid swirling around in different directions at different speeds. I still find it all very confusing though so I’m not trying to pretend like I know what I’m talking about. I do think I’m starting to slowly figure it out through.
I again used ChatGPT and asked it what the difference between a scalar field and a vector field is and this is what it said:
I also wrote down the notation for the different operators here:
Lastly, in the articles on curl I also was shown a simple example of curl in 2D and how it can be thought of as a stick swirling around in water. I found this example very illustrative of how/why curl works and thought it was worth sharing here:
At the end of last week I said that my goal was to get through Derivatives of Multivariable Functions (1,460/2,100 M.P.) by the end of this coming week but I don’t think that’s going to happen. I only have five videos and three exercises left to get through in this unit but I think the unit test is probably going to take me at least two attempts if not more. I feel like it’s been a LONG time since I’ve done a unit test so, in a weird type of way, it’ll be kind of nice to work through one again. There are 21 questions in the test so I’m sure it’ll be pretty tough. Regardless, even if I don’t get through it by the end of next week, my goal is to at least get through the videos and exercises so I can start it. Who knows, maybe I’ll begin my next post by saying I crushed the test and am on to the next unit, Applications of Multivariable Derivatives (0/500 M.P.)! Probably not but, as usual, fingers crossed! 🤞🏼