I had a fairly productive week working on KA but didn’t reach my goal of making it through the unit test. I got through both the Flux in 3D and Flux in 3D (Articles) sections and then started the unit test, but I only made it through a single question on the test. 😅 I got a much stronger understanding of what flux is and a decent grasp on how to calculate it, but I definitely don’t understand why the math used to calculate flux works. Each of the three articles I worked through gave examples of flux and how it’s related to physics. There was a lot of linear algebra involved. I was frustrated trying to put all the pieces together but realized that flux will be WAY easier to understand once I go through the linear algebra playlist and physics course when I (eventually) get there. So, as bummed as I am about my weak understanding of flux, I’m trying not to beat myself up too bad about it. And in a glass half-full kind of way, it means that when I do get to linear algebra and physics, I’ll likely get through the sections that have to do with flux WAY faster than I would have thanks to the work I’m doing now. 🤷🏻‍♂️ 💪🏼

The first section I worked through this week, Flux in 3D, only contained three videos. The series of videos all worked through a single question which outlined the steps to calculate flux. After watching the videos, I felt like I didn’t understand it well enough to make notes on what I watched and decided to move onto the next section, Flux in 3D (Articles), hoping that I’d better understand the videos after going through the articles. I made it through the three articles in Flux in 3D (Articles) by the end of Thursday and did go back to the videos, but still felt like I didn’t understand the videos well enough to make notes on them. 😒 As I just mentioned in my intro, that’s when I realized it may not be until I work through the linear algebra and physics sections of KA that I’ll be able to explain what was going on in those three videos.

The first article in the Flux in 3D (Articles) section talked about what’s called a ‘Unit Normal Vector’ which you need to use to calculate flux. The second article went on to give a more general description of flux, describing what it is and how to calculate it. The order of the two articles seemed weird to me, like they should have been reversed. Below are a bunch of screen shots from the second article which explain flux and a brief note I took at the end which gives a layman’s explanation of the formula. You’ll also notice close to the start of the screen shots that I asked CGPT what flux was which definitely helped me better understand it:

(I should mention here that the screen shot just above is the one helps me understand what’s going on with flux the most. My understanding is that, when zooming into the surface of an object to look at a tiny, tiny patch of surface area, the flow of the ‘water’ (as I like to think of it) going through the surface at that point can be thought of as a tilted rectangular prism, a.k.a. a parallelepiped. The volume of a parallelepiped is base * height where the base is the magnitude of the cross product of the partial derivatives of the parameters (lol confusing enough?). The height is calculated by finding the dot product of the fluid, F(x₀, y₀, z₀), with the unit normal vector, n̂. I think of n̂ as a flag pole of unit length 1 and the dot product being a shadow being cast from the parallelepiped at exactly 90 degrees onto the flag pole so that the shadow of the parallelepiped shows the exact height of the parallelepiped against the flag pole. That probably sounds weird, but visualizing it that way helps me to simplify what’s going on.)

So, like I said, I don’t have a great understanding of what’s going on. I know that flux calculates how much fluid is travelling through a surface area but couldn’t tell you how or why the math works.

Here are a bunch of screen shots from the second article and my notes working through the example it gives on how to calculate a normal unit vector:

The steps shown in my notes are:

1. Parameterize the surface area and put it into a vector valued function
2. Find the partial derivatives of the vector valued
3. Take the cross product of the partial derivatives
   1. (You can see that I also inputted the point in question here, v(1, –2).)
4. Find the magnitude of the cross product of the partial derivatives
5. To find the unit normal vector, divide the cross product of the partial derivatives by the magnitude of the cross product of the partial derivatives
6. This step shows the formula/equation/vector valued function for a unit normal vector at any (t, s) point on the surface.

Lastly, here are a TON of screen shots from the third article and my notes working through the last bit of the example:

It’s interesting to me that at the end of the question, Sal (or Grant?) just states that you need a computer to solve the rest of it. This makes me think that I’m starting to reach a point in math where we set up the equations and just let computers deal with the computation. I can see why we’d need to understand how and why the math works in order to do this. If it’s true that I’m reaching a limit of what humans typically do, plus considering that I’m nearing the end of the MATH section of KA, it makes me wonder if I’m pretty close to learning as much about math as the average person who studies math would typically learn. 🤔

So, after 14 weeks of working through Integrating Multivariable Functions (1,280/1,600 M.P.), I’m hoping I can finish the unit test this coming week and FINALLY move on to the fifth and final unit in Multivariable Calculus. The last unit is titled Green’s, Stokes’, and the Divergence Theorems which seems like a pretty weird title to me. It’s only 600 M.P. but that’s deceiving as there are a TON of videos and articles in it but only a handful of exercises. It would be great to get through that unit and then the course challenge before the spring. That way I could potentially get through the rest of the MATH section of KA before next September, i.e. five years of me studying on KA. Regardless, hopefully it won’t take me another five years before I can get through the PHYSICS, CHEMISTRY, and BIOLOGY sections of KA. But if it does, studying STEM for an even decade would actually be pretty satisfying. 🤓