I spent this entire week trying to understand curvature but haven’t really figured it out. I think I understand what it is (a measurement for how much a function curves – no surprise there given the name), but I don’t fully understand how to calculate it or why the formula works the way it does. There were nine videos in the Curvature section that I watched, five of which went through the formula and which I ended up watching twice. There were no exercises in this section which was surprising and a bit annoying as I’m assuming I would have been able to get a better understanding of why curvature works if I had to work through some questions and figuring out how to solve them. In any case, I still made a bit of progress figuring it all out which you can see below. Also, I can’t believe that I’m in Week 200… 😳

This is the first note I took which was from the first video in the series titled Curvature Intuition:

I feel like this note makes curvature seem pretty straightforward. To reiterate the note, curvature on a function can be thought of as the size of a circle created at any given point along the function if the circle’s curve at that point perfectly matched the curve of the function. A way to express the value of the function’s curvature is 1/R where R is the radius of the circle that would be created based off the curvature at that point in question. If the radius is small, then curvature is large. If the radius is large, then curvature is small. It just dawned on me that one way to think about it is that curvature, K, and the radius of curvature, R, are inversely correlated.

Here’s a note I took where I tried to work through an example in one of the videos:

This is where I started to get confused on how to calculate curvature. In layman’s terms, here’s what I think each of the steps above are expressing:

1. Here’s a circle.
2. If you find the derivative of that circle, you can input any value into t and it will output the line tangent to the circle at that point, t. Note that the derivative is in vector form.
3. (This is where I start to not really understand what’s happening.) To figure out the curvature, you need to find the “unit vector tangent function”. This (maybe but I’m not sure) is because curvature equals 1/R and the derivative will likely output a numerator other than 1 so you need to divide the derivative of the unit tangent vector function by the magnitude of itself to get it to equal 1. (I don’t know if what I just wrote is true…)
4. In this question, because we’re finding the curvature for a perfect circle, the magnitude of the derivative IS the radius of the circle which is why T(t) = s ⃗’(t)/R. In all functions other than a circle, the magnitude of the derivative will not simplify to R so you need to then use Pythag’s theorem to find the absolute value of s ⃗’(t). In those instances, dividing s ⃗’(t) by the absolute value of s ⃗’(t) will result in T(t) = 1 which (I believe) is what you need to be to find the curvature, K = 1/R.

I don’t know if what I just wrote is accurate or correct, but here’s a page from my notes that gives a bit more detail/insight into what curvature is:

Reading through the note above now, I don’t know if the second part of it makes sense. I don’t understand how curvature is K = 1/R but also equals dT/ds. I think that means that the ratio between 1/R is the same ratio as a tiny, tiny change in the unit tangent vector over a tiny, tiny change in the arc-length of the function. 🤔

I don’t understand this… 😡

Since I don’t understand what’s going on, I’m not going to bother trying to explain the notes below but they’re the notes I took on how to derive the formula for curvature:

On a mostly unrelated side note, I had an insight come to me this week which I already understood but for some reason the concept became more clear to me. The reason why they’re called para-metric equations (I think) is because “para” means multiple which implies that a single function (say s ⃗ (t), an object’s position across time) can equal a vector where the x-position and y-position are their own functions. The SINGLE parametric equation is actually TWO functions combined/layered onto/into each other. Here’s the ntoe I took:

What I think when when I see s ⃗(t) = [x(t), y(t)] is, “the function s ⃗(t) IS the two functions x(t) and y(t)”. For whatever reason this concept became easier for me to wrap my head around this week, so I’ve at least got that going for me which is nice.

My plan for the upcoming week is to rewatch the three example videos in the Curvature section which cover two examples where Grant finds the curvature of a helix and a cycloid. Hopefully writing out those examples will help me figure out how/why curvature works. I’m also hoping I can get through the following section, Partial Derivatives of Vector-Valued Functions, which sounds like it could be kind of fun. I’m hoping to get through this entire unit, Derivatives of Multivariable Functions (820/2,100 M.P.), at least before the end of July so it’d be key to get through the latter section this coming week. Plus if I get through it, I’m sure I’ll feel a lot better about my grasp on math compared to how I felt at the end of this past week. 🤦🏻‍♂️