This week was a bit like the opposite of last week. I actually made it through quite a few videos and a handful of exercises but I felt like I didn’t learn very much. 😒 I was introduced to a concept known as “path independence” and spent the majority of the week trying to wrap my head around it but didn’t really make any progress… I’ll explain my vague understanding of what it is below but the silver lining is that, trying to understand path independence, I was able to get a stronger intuitive grasp on the difference between scaler fields and vector fields. Part of me is disappointed with how this week went but I put in a decent effort and I think that understanding the difference between scaler and vector fields is probably fundamental to vector calc so, all in all, this week could have been worse. 🤷🏻‍♂️

Thinking through the difference between scaler and vector fields, I now visualize a box that has different temperatures inside of it and circulating airflow. The temperature inside the box would be denoted with little coloured dots (a scaler field) but the airflow would be measured with little arrows at different points (a vector field – I like to imagine there’s a fan inside the box bowing the air around). To be honest, I don’t remember which video I watched that spurred me to look into it, but I asked ChatGPT what the difference between the two was. Here’s what it said:

As you can see from ChatGPT’s response, scaler fields represent points in space, which I believe can also represent 3D objects but I think are more often used to represent values such as temperature. (I have no idea if that’s true though.)

I started off watching the third and fourth videos this week from the section Line Integrals in Vector Fields which were titled Parameterization of a Reverse Path and Scaler Field Line Integral Independent of Path Direction, respectively. The first of the two videos explained the math behind changing the direction of some hypothetical object moving along a line (I think Sal used the example of a particle) in an (x, y) plane. Here’s a screen shot from that video:

The left side of the screen shot shows the OG path that the particle took where its x and y coordinates are functions of t, time. The particle starts off at t = a and moves along the line up to the ending position at t = b. The right side of the image shows the particle going in the opposite direction and the math used to reverse the path. Long story short, the way you do this is by instead of inputting t = (“a” or “b” or whatever) into the x(t) and y(t) functions (which together make up the position function f(x, y) which the screen shot doesn’t indicate), you input t = (a + b – t). I’m not going to explain it myself, but Sal works through the algebra in the bottom right which proves why this would reverse the path of the particle.

The point of the fourth video (as far as I understood it) was to explain that the value of a line integral (which can be thought of as a squiggly fence or curtain that has different heights across it) doesn’t change if you reverse the bounds. (In layman’s terms this means that if you were measuring the area of the curtain from one end to the other but then switched the start and end points from which you were measuring from, it wouldn’t make a difference to the overall area of the curtain.) Here are a bunch of screen shots from that video and some explanations as to what’s going on in each one:

This was the setup for the question where Sal asked if the value of the line integral would change if he were to switch directions of measurement. (The –C at the bottom of the second integral from the left is what denotes “negative curve”, a.k.a. going in the opposite direction along the curve/line.)

There’s a lot of algebra going on here, but this is where Sal begins the proof by inputting (a + b – t) into the line integral formula. What he’s trying to do (which is shown in the next three screen shots) is prove that the formula with (a + b – t) is equal to the OG formula where the variable is simply t.  

Here Sal substitutes (u) in for (a + b – t). I have a bit of a hard time following along with what’s going on, especially since I never really understood u-sub, but my understanding is that since he’s replacing (a + b – t) with u he has to replace dt with du which means he has to find the derivative of u with respect to t. He does this close to the top of the image which results in –du = dt. In the second integral from the bottom, you can see that Sal placed the (–) in front of the integral. I kind of forget why but since there’s a negative sign outside the integral, you can actually switch the bounds if you take the negative sign away. You can see that’s what Sal did in the bottom integral.

To be honest I don’t really understand what’s going on here, but in this image Sal shows that the formula now with u in it is the exact same as the OG formula except that the bounds have been switched. Since the two formulas equal each other (and since u = t for some reason? 🤔) this proves that the integrals equal each other and therefore it doesn’t matter what direction the integral is being measured from. 

The final four videos in the section introduced and worked through the concept I mentioned at the start, path independence. Like I said, I don’t understand what it is so I asked ChatGPT and was given this response:

As you’d expect based on my ELI5 prompt, this answer makes it seem simpler than it is. What this response doesn’t explain is that path independence has something to do with determining whether the gradient in the vector field is equal to the gradient of the scaler field which… doesn’t make any sense to me… 😔 I’m not going explain it, because I can’t.

The main concept of the seventh video (which used path independence) was something like, “if you take a path along some curve, C₁, in a conservative vector field (which has something to do with path independence), and you take a reverse path along another curve, –C₂, you get back to the start.” Here are two screen shots from that vid:

Although I don’t really understand it, one thing worth mentioning is that the integral symbol with circle in the centre denotes a “closed loop” which is what this video was referring to. 

The final video in the section was an example of using path independence to solve a line integral problem. Like I’ve said a million times already, I don’t understand why it works or what’s going on but here’s a screen shot from that video and a page of my notes which sums up what I think is happening:

I started the second exercise in the section on Saturday but got lit up and had no idea what was going on in the questions. After working through seven or eight questions and not getting very far, I decided to try the following exercise to see if I would have any more luck with that one. It turned out that the third exercise was WAY easier than the second exercise. Below are three example questions from that exercise. The first question was my first attempt at the exercise. I had no clue what I was going but ended up getting it correct which was a bit shockeding. Here are the questions:

Question 1

Question 2

Question 3

As you can see, the math really isn’t too difficult. What’s hard is understanding what is going on and why the math works the way it does. I’m hoping I’m close to understanding what is happening but I have a feeling I’m a long way off from understanding why the math works. 😡

I’m bummed that I didn’t accomplish the goal I set for myself this week of getting through this section. I only have two exercises left so I think I should have no problem finishing it off next week, however. I also just realized that the section after is titled Line Integrals in Vector Fields (Articles) and has five articles which look like they all have to do with everything I’ve been working through in this current section. Hopefully the articles will help me get a understanding of what path independence is all about. 🙏🏼 I’m now a whopping 15% of the way through Integrating Multivariable Functions (240/1,600 M.P.). If I don’t start making quicker progress soon, there’s no chance I’m going to be able to get through this unit AND the next by the end of December in order for me to wrap up Multivariable Calculus before the New Year. It won’t be the end of the world if that ends up happening but I’ll still be disappointed. 😠