So… I still didn’t pass the Integrating Multivariable Functions unit test this week. 😣

It’s starting to get a but ridiculous. I’ve been working on it for four weeks. I didn’t put enough effort in this week which is what makes it so much worse. I’ve been slacking pretty hard (I’ve also been working on another project that I’m pretty excited about, but I don’t want that project to take away from my effort here on KA) and so I really need to turn things around. I think I studied for at least five hours this week but it wouldn’t have been much more than that, if at all. I did have some insights come to me this week which I was happy about (you’ll see in the first question below that I asked CGPT about line integrals and the answer finally helped make them more clear to me), but that still doesn’t change the fact that I should have made it through the test by this point and it’s 100% due to a lack of effort that’s stopped me from doing so. I feel guilty and I feel like I’m letting myself down… 😒

Nonetheless, here are four questions I worked through this week with explanations for each:

Question 1

So, I don’t really understand how this question works. Before I explain what I think is happening, here’s an explanation of conservative line integrals that CGPT gave me that gives a bit more incite into line integrals:

So, based on this explanation, it seems to me that the first question above is essentially the same thing as calculating a single variable integral where the aim is to calculate the area under a curve, except that in the question above the area being calculated is 3D and the line integral takes the shape of half of an ellipse. If this is true, then this part I understand (at least conceptually). What I definitely don’t understand is how to tell if the curve Φ(x, y) is a curve in a scaler field or a vector field. Maybe I’m supposed to assume I can use the method from above to calculate a line integral unless it’s explicitly stated that the gradient of the curve is not continuous? But I’m not sure…

Question 2

I was happy that I got this question correct, but I didn’t know what I was doing as I worked through it. I remembered that the area of the transformed surface would need to be calculated by finding the partial derivatives in the x and y directions (or, in this case, the u and v directions) and then by finding the function for the magnitude of their cross product which tells you the tiny, tiny parallelogram created by the partial derivatives at any given point along the surface. (That was a confusing sentence but it’s possible that it made sense.) You then just have to throw that function (in this case, sin(u)) into a double integral, solve it, and boom goes the dynamite. 🧨

Question 3

I skipped a few steps in my notes from this question, but all I had to do was find the partial derivatives of P and Q which both equalled e^y. I didn’t really know (and still don’t) whether that made the function f(x, y) conservative or not, but I took a stab at it and guessed that it was which ended up being correct. I vaguely remembered that conservative fields have an equal gradient no matter which direction you travel from some arbitrary point a to point b, so I assumed that if the partial derivatives were the same then that likely meant it was conservative.

Question 4

This question is essentially the exact same as the second question, but I was still pretty happy with myself that I got it correct. For some reason I was very confused working through this type of question this time around and couldn’t understand what I needed to put into the double integral. I knew that I had to put in the scaler (21)^1/2 but I wasn’t sure what OG function I needed to put in there. I thought I needed to put in some function like f(x, y, z), but put it in as f(T(u, v)), and then multiply it by |T_u X T_v| = (21)^1/2, but I didn’t see the function f(x, y, z). I figured that since the question said the OG plane was from –1 < u < 1 and 0 < v < 1 then that meant the surface was just flat and so there was no OG function to put into the double integral. So, assuming that was true (which I still don’t know if it is?), I assumed I could just put in the scaler into the double integral and solve which turned out to be correct.

(I don’t know if any of what I just made sense. I think I’m on the right track with all of this but am still really confused… 😔)

I ended this week having gotten 7/16 questions correct in a row which means if I can manage to get the next nine questions correct, I’ll have passed the test. 🥵 I’m PRAYING to the math gods that I can do it. 🙏🏼 🙏🏼 🙏🏼 I’ve been working on Integrating Multivariable Functions (1,280/1,600 M.P.) for almost five months now, but somehow it’s felt even longer than that. I need to get through this stupid test to get some momentum back and feel like I’m actually making some progress. 😤 To get through the test, my goal this week is to work on KA for at least 1.25 hours a day. I think if I can manage that I should be able to make it through the test. Even if I don’t – god forbid – I’ll at least be back on track with my overall effort.