I started this week by getting crushed pretty badly by the Integrating Multivariable Functions unit test. After going back and reviewing a bunch of the concepts however, I finished the week getting seven correct in a row and feeling like I had a decent understanding of what I was doing. It was helpful going back and reviewing certain concepts for a number of reasons. For one, some of the things I reviewed were a LOT easier to grasp coming back to them the second time around. Also, reviewing certain concepts helped make other things clear that I was struggling to understand. All in all, it was good practice conceptualizing what was happening with the bigger picture with some of the math, plus I got a lot of practice using vector multiplication, parameter substitution, derivatives and integrals, and of course algebra, trig, and arithmetic.

Here are eight questions from the unit test I worked through and a brief explanation of what’s going on beneath each one:

Question 1

I think it would take a cryptologist to decipher my notes from this question, at least on the right side of the page. This question took me a good 15 to 20 minutes to think through. In retrospect, it seems pretty easy but at the time I found it hard to understand what the bounds of each integral were referring to. After a few minutes it dawned on me that I needed to draw an (x, y) plane and put the upper bounds of the inner integrals in terms of x (as in y = (something) * x) and then I could just draw the lines accordingly. Once I did that and then created the triangle shape, I knew the y bounds would be from 0 to 2 and the x bounds would be from the lower line, y = x/2, to the upper line, y = 2 – x/2.

Question 2

This was a question I initially got wrong because I couldn’t remember what was going on. I had to go back and review a video on what a line integral in a scaler field was all about. After watching that video, I remembered that a line integral is like a meandering curtain that has different heights as you go along it. The heights are denoted with f(x, y) (a.k.a. z). To find the curve you have to take the parametrization of the line, α(t), and find it’s derivative (i.e. the slope of the curve at every single tiny, tiny point along the curve) and then square the x- and y-components, sum them, and find their root. (This is the same thing as (a² + b²)^1/2 = c but it’s more like (x² + y²)^1/2 = hypotenuse which equals the tiny, tiny bit of the curve of the line at that point.) You then multiply those functions together to get the height of essentially a line at every point along the curve and then sum all those line with the integral.

(That may not have made sense technically but hopefully it’s an indication that I understand it conceptually – I think.)

Question 3

I’m not going to explain this question it’s literally the same thing as the last question.

Question 4

I vaguely remembered what this question was asking me to do when I first looked at it. It took me a minute but I remembered that where it said f = ∇F, that was saying that I needed to find the antiderivative of f since the question was asking for F on its own. Although I was correct about that, I got this question wrong because I forgot how to work through the antiderivative and why you have to add the extra H(y) and G(x) functions at the end of each antiderivative and how they factor into the final solution. I rewatched a video on it and, as you can see from the explanation in my notes, was able to figure out why the math works and understand how to do it. Boom.

Question 5

I’m sad to say that I still don’t understand why this question works. 😔 I definitely understand how to solve these questions as the pattern to solve it seems very similar if not identical to solving line integrals, but I don’t understand the whole ‘scaling factor’ using the Jacobian part. First of all, I don’t understand why the Jacobian is a scaling factor in the first place, and second, I don’t understand why you’d need to use it to scale from Cartesian to Spherical coordinates. 😒

Question 6

This question was a pretty straightforward integration question which I didn’t find difficult and I don’t think needs much explanation. The only reason why I made a note to add it was because I had gotten a bunch of questions wrong before this one and wanted to remind myself that I at least knew how to solve SOME of the questions. 😂

Question 7

I was pretty impressed and happy with myself that I was able to get this question correct on the first attempt. I had reached the correct answer in about 10 minutes but it took me another 10 – 15 minutes of thinking it through to try to 100% understand why my answer was correct. I could think through the vector coordinates of the sphere and knew that theta was the angle around the z-axis, phi was the angle up and down the z-axis and that p was the radius, AND I knew why and could visualize how the cos and sin functions worked for each component. I was able to think through that the normal vector would be coming below the (x, y) plane and therefore z would be negative which is why I very quickly knew that the second answer was correct, but I couldn’t figure out how to do the math to get to that solution. As you can see, I eventually figured it out but it took me awhile. This was one of the very few instances where I actually knew why the solution to a question was correct before understanding how it worked, which was very weird.

Question 8

My notes here are pretty messy, but you can see that this was just another line integral question. I worked through it very quickly and fully understood why each step of the process worked and, most importantly, could visualize the steps. I was pumped being able to understand what was going on and found it very satisfying that I could answer this question relatively easily.

As I mentioned at the beginning, I ended the week getting seven questions correct in a row. There are 16 questions on the test meaning that if I’m able to get nine more questions correct in row, I’ll pass the unit test and will officially be through Integrating Multivariable Functions (1,280/1,600 M.P.). Even if I get a question wrong and have to restart, I’m pretty optimistic that I’ll get through the test at some point this coming week. As always, my fingers are crossed. 🤞🏼 🤞🏼 🤞🏼