I had a pretty abysmal week. I started the week reviewing implicit differentiation and after spending ~2 days watching videos, I didn’t really get a better understanding of what, how, or why implicit differentiation works. 😒 The rest of my week was spent reading three articles, all of which had to do with using integrals to find the arc length of parametric functions. I’m happy to say that by the end of the week I at least got a decent understanding of the formula for arc length. Even still, I’m disappointed that I didn’t make more progress than that. If I had to give myself a grade for this week for my effort, I’d probably give myself a C or a C+. I’ve definitely had worse weeks but it certainly could have been better.
When I first learned about implicit differentiation, I never really understood what was going on. This week wasn’t much different. My general understanding of implicit differentiation is that you can have some function f(x) = a(x)b where you can then say that y = a(x)b and then somehow find the derivative with respect to x on both sides leaving you with:
- d/dx[y] = d/dx[a(x)b]
- y’(x) = a * b(x)b – 1
I don’t even know if what I just wrote is correct or makes any sense at all. I’m pretty sure that when using implicit differentiation, you somehow find the derivative of y and find how it equals the derivative of x. I don’t really get it, but for a minute I thought it was essentially the same thing as finding the gradient of a 3D function. After working through some of the questions in the videos I watched and trying to find the gradient of the given functions, it seemed clear that implicit differentiation and gradient are not the same thing, although I think they’re closely related. I believe that implicit differentiation just tells you the slope along some (x, y) function whereas the gradient tells you the slope in the z-dimension. (I could be completely wrong about this.)
On Wednesday I got back to the articles from the section Line Integrals for Scaler Functions (Articles). I had made it through the first article last week so I started on the second article on Wednesday. This screen shot sums up what that article was all about:
The practice questions in this article were all pretty straightforward, although I don’t fully understand why the math works. Here’s an example question which shows how the calculus and algebra is pretty simple:
Question 1
What confuses me is the notation. I still don’t completely understand what dx represents. I think it must mean a tiny, tiny change in the x-direction but I’m not really sure.
The second article I worked through (third in the section) was titled Arc Length of Parametric Curves and took what I worked through in the previous article and added a bit more complexity to it. I believe the gist of what the article says is that in a parametric function, say f(x, y) = something, the x and y components are actually functions themselves of another variable (often t for time). I believe this is what literally defines a parametric equation, that there are multiple functions leading to one output. (I.e. the function could/should be thought of as f(x(t), y(t)) = something.) The arc length formula must therefore be thought of as:
- ∫ ((x’(t)*dt)2 + (y’(t)*dt)2)1/2
- ∫ ((x’(t))2 + (y’(t))2)1/2 dt
This is because dx/dt = x’(t) and you can do some hand-wavey math (as Sal says) and multiply both sides of the equation by dt to leave you with dx = x’(t)dt and the same for dy = y’(t)dt. Here are two practice questions from the article that work through this equation and give a decent explanation of how/why it works:
Question 2
Question 3
I got both of these questions wrong initially because I screwed up the algebra both times. For that reason, I was happy to work through the questions in this article just to practice factoring and simplifying expressions.
I started and finished the last article I worked through this week on Saturday. It went over the notation that’s used in line integrals and was pretty straightforward. The gist of this article is that when solving line integral problems instead of using set bounds like a and b, sometimes a placeholder variable C is used to say something like, “this integral will be calculated along a segment of the curve, C.” Also, instead of using √((x’(t))2 + (y’(t))2) dt, you can denote that expression with ds. Here are some screen shots and my notes from that article:
Like I’ve said throughout this post, I don’t really understand what’s going on but I can tell I’m getting closer to figuring it all out and am kind of excited to see where this is all going. This coming week I’d like to get through the last article, Line Integrals in a Scaler Field, and then get started on the next section, Line Integrals in Vector Fields. I watched the first three videos in this section two weeks ago, but there are nine videos and four exercises in total so I’ll still have my work cut out for me. If I can get through that section by the end of the week, I’ll be ~25% of the way through Integrating Multivariable Functions (80/1,600 M.P.) which would somewhat make up for this lacklustre week. As always, fingers crossed I can make it happen! 🤞🏼