I had a VERY productive week on KA and learned a lot but once again didn’t make much progress earning Mastery Points. Most disappointingly, I still didn’t make it through the Line Integrals in Vector Fields section. 😔 I did, however, read four of the five articles from the following section, Line Integrals in Vector Fields (Articles), which helped clear up a lot of the ambiguity I had regarding scaler and vector fields. I also used ChatGPT a lot this week which made a big difference helping me understand the difference between the two types of fields. So, even though I’m disappointed that after three (or four?) weeks I STILL haven’t made it through the four exercises from Line Integrals in Vector Fields (😒), I’m really happy with what I know now compared to the start of the week. I feel like I have a way better understanding of what’s going on so hopefully I’ll finally be able to start making quicker progress through this unit. 🤞🏼🤞🏼🤞🏼
I started this week by reading the first article from Line Integrals in Vector Fields (Articles) which was, funny enough, titled Line Integrals in a Vector Field. This is where the difference between scaler and vector fields started to make more sense for me. Here’s a screen shot from that article and my notes:
(It turns out I wasn’t quite right in my note but I was on the right track.)
On Wednesday I started the next article, Fundamental Theorem of Line Integrals, but for some reason before I started the article I asked CGPT to explain to me when a vector field is NOT conservative. I can’t remember why I decided to do this at this point, but my thinking was that knowing when a vector field was NOT conservative would help me get a grasp on when and why a vector field IS conservative. Here’s what CGPT told me:
This explanation really helped me wrap my head around what makes a vector field conservative, mainly that the force of the field needs to stay constant no matter what path is taken from point A to point B. (I had one more conversation with CGPT which is posted below that more-or-less solidified my understanding of the difference between a conservative and non-conservative vector field.)
The second article kept the insights on scaler vs vector fields coming. This article dealt primarily with the formulas for line integrals in each type of field. Below are two screen shots from the article with the first showing the formula for a line integral in a vector field and the second being for a line integral in a scaler field. The formulas look very similar to each other, but what stands out to me that most notably differentiates the two is that the formula for a line integral in a vector field incorporates the gradient, ∇F:
Below are my notes from this article. My plan for this post was to only add the first photo that’s below but I’m too lazy to rewrite my notes from the second and third photos so I apologize for how messy they are:
The fog around conservative vector fields was starting to lift for me at this point, but it was the following conversation I had with CGPT where it all (I think) became clear to me:
Last week I explained that I visualize the difference between a scaler and vector field as being a room that has different temperatures throughout it (temperature being a scaler field) and a fan blowing the air around it (the air being a vector field). The difference between a conservative vector field and a non-conservative vector field would be the difference between gravity, which is uniform and therefore is the same throughout the room (a conservative vector field) and the air blowing around the room which would be going in different directions with different forces at different points throughout the room (a non-conservative field).
Boom. Figured it out.
I didn’t take many notes from the third article, Conservative Vector Fields, but did take a screen shot of the notation used for closed loop integrals:
I got through the third article pretty quickly on Thursday and then finally returned to the second exercise from the previous section. After messing up a few of the questions, I finally ended up passing that exercise on Thursday. Below are two example questions from that exercise. The gist of what’s going on in both questions is there’s some vector field, f, and you’re asked to find out what the line integral is from t = a to t = b. You have to use the fundamental theorem of line integrals which I now know is that if you have a conservative vector field, the line integral across some curve is equal to the antiderivative of the end point minus the starting point, a.k.a. a∫b f ds = ∇F(b) – ∇F(a). (I think…) One part that confused me, however, is that the questions didn’t explicitly say if the vector field was conservative or not. I’m not sure if there’s a way to determine if the field, f, is conservative or not (I’m sure there is), but by the end of the exercise I just started assuming that was true for each question. Anyways, here are two example questions from the exercise:
Question 1
Question 2
On Friday I started the fourth article which was titled Flux in Two Dimensions. I had no idea what “flux” was so before I got into the article, the first thing I did was ask CGPT what it was. This was its response:
This answer made the KA article much easier to understand. Something that definitively became clear to me this week was that asking CGPT to give me an eli5 explanation of a new topic before delving into it makes the concept a lot easier to understand. (I feel like this could potentially speed up my progress on KA quite a bit!) Here are a few screen shots from the article that cover what flux is and how/why it works:
I still find flux very confusing (that algebra/vector calculus in the image just above is fairly difficult for me just to follow along with) but I think I have a decent general idea of what’s happening. It’s something like there’s a screen over a window and flux calculates how much air if being blown through a specific section (or loop) on the screen. If I’m right about that, even though I find it confusing, I think that having that low resolution idea of what it is will make it a lot easier to figure out moving forward. CGPT I think could be a game changer for getting that type of low resolution understanding of a certain concepts before going deeper into them.
The last thing I did this week was try to get through the final exercise from the previous section. I got lit up trying to work through a few questions on it but got a decent grasp on how to answer the questions by the end. Below is an example question of one of the questions. To answer this question, I had to review integration by parts but was happy that I was able to derive the formula from memory. This is probably obvious, but I don’t know what is going on in this questions or why the math works but somewhat understand the process of how to solve them. Here’s the question:
Question 2
I’m looking forward to this coming week as I feel like I actually have a decent understanding of the differences between scaler fields, vector fields, and understanding what path independence is. Having got through the second exercise this week, I’m now 20% of the way through Integrating Multivariable Functions (320/1,600 M.P.). Hopefully I can make it through that final exercise this coming week and get up to 25% of the way through the unit and FINALLY move on from that section. If I can also finish off the fifth article in Line Integrals in Vector Fields (Articles) then I’ll be starting on the next section titled Double Integrals which has six videos and four exercises. I definitely think it’s unlikely, but my goal for this week is to get through that entire section. It’s a tall order but I would be PUMPED if I could do it, especially given how slow my progress has been lately.
PLEASE math gods, let me make some actual progress this coming week!! 🙏🏼 🙏🏼 🙏🏼