I’m happy with how my week went on KA but at the same time it wasn’t that productive. I made it through ten articles in Integrating Multivariable Functions, the fourth unit of Multivariable Calculus, and got a decent handle on much of the notation that I’ve been struggling with lately which was great, but I’m disappointed I didn’t make it through more articles. I’m fairly sure I studied for at least five hours but I’m also sure that I could have put more effort in and made it further than I did. So, all in all, it wasn’t a bad week but, in the big picture, it was a bit disappointing considering how close I am to getting through all of this. (A lot of my time was spent working on CK though, so it’s not like I was just sitting around twiddling my thumbs.)

Here are some screenshots from the articles I worked through that mostly talk about the notation I was able to more-or-less figure out:

Article 1 – Arc Length of Parametric Curves

This helped me remember/understand that a parametric curve is a function where the x- and y-components of the curve are both functions of t, and the reason why you need this type of function (the “parametric function” if that’s what it’s called?) is so that you can draw functions where the curve goes all over the place—as is shown in the second screenshot above. (This is unlike a “normal” f(x) function where the function cannot zig-zag across the domain, i.e. it can’t have two outputs on the range for a single input on the domain.)

Article 2 – Notation for Integrating long a Curve

This was a helpful reminder of what’s happening at the infinitesimal/derivative level, although I’m sure that’s not the proper way to say that. It all comes back to Pythag’s theorem of (a² + b²)^1/2 = c which is equivalent to ((dx)² + (dy)²)^1/2 = “the tiny, tiny length of the curve” which, when you sum them all together with the integral, gives you the length of the curve. It was also a helpful reminder to read that ((dx)² + (dy)²)^1/2 gets short-formed to ds. 

Article 3 – Line Integrals in a Scalar Field

The first screenshot above helped me remember that this is more-or-less a base * height equation where _c∫ ds is the base and f(x, y) is the height. The second screenshot shows that you use the same setup when calculating a vector valued function with the main thing to remember being that ds = |r ⃗’(t)|dt.

Article 4 – Fundamental Theorem of Line Integrals

This helped me understand that what the equation _a∫^b∇ f(r ⃗(t)) ⋅ r ⃗’(t)dt does is tell you the sum of all the tiny, tiny changes in the slope along the top of a “fence”. But, you can also just find the height of the end and start points of the fence, f(r ⃗(b)) and f(r ⃗(a)) respectively, and find their difference which will give you the same solution as _a∫^b∇ f(r ⃗(t)) ⋅ r ⃗’(t)dt. Therefore:

• _a∫^b∇ f(r ⃗(t)) ⋅ r ⃗’(t)dt = f(r ⃗(b)) ­ f(r ⃗(a))

Article 5 – Line Integrals in a Vector Field

The metaphor I think of here is that this integral, _a∫^b∇ F(r ⃗(t)) ⋅ r ⃗’(t)dt, calculates how much the wind is blowing in the same direction as a cart on a path and summing up how much the wind pushed the cart in the same direction as it was going along the path (or how much is pushed it in the opposite direction).

Here’s a page from my notes where I wrote out all the notation I just talked about and made a note on how I think about/visualize what’s happening:

Article 6 – Flux in Two Dimensions

As you can see in the first screenshot above, this states that “Flux” is the flow of a vector field out of a region or across a line. As you can also see, the formula for Flux is _c∮ F ⋅ n̂ ds where F is the vector field, n̂ is the unit normal vectors all along the curve, ds is the tiny, tiny line segments along the curve, and _c∮ is the sum of all those tiny, tiny line segments. The second screenshot shows what’s happening at the derivative level. I think I understand what’s going on with the equation/concept and can pretty much visualize it which, I must say, is a pretty big relief. 😮‍💨

Article 7 – Constructing a Unit Normal Vector to Curve

This was the last article I made it through which shows the steps to figure out a unit normal vector (or unit normal function?) on a 2d curve. Step one is to find the tangent vector by taking the derivative of the parametric curve vector/function/thing. Step 2 is to take the tangent vector’s reciprocal by swapping the components in the vector and changing the (+)/(–) sign on one of them (which I think of as basically analogous to what you do when finding the reciprocal slope of m in y = mx + b). And step 3 is to normalize the reciprocal tangent vector by dividing its components by the magnitude of itself which you do by putting the components into a version of Pythag’s theorem. (That probably doesn’t make sense, but hopefully you can figure out what I’m talking about from the screenshots. 🙃)

Aaand that was it for this week. Like I said, I probably should have gotten through more, but I did wrap my head around a bunch of the integral notation which I was struggling with for the past few weeks, so I’ve at least got that going for me which is nice. My hope was that I was going to get through all 24 articles, so I clearly fell short of that, BUT maybe I can get it done this coming week. Considering I’m SO CLOSE to getting through this goal (which I’ve been working towards for almost 6.5 years at this point, but who’s counting), I’m pretty motivated to get through them, but we’ll see if that motivation actually turns into a productive weeks. As usual, fingers crossed that it will so I can finish this off before I turn 92. 🤞🏼👴🏻