I only made it through four of six articles this week so I still didn’t make it through the first unit in MC. I only wrote out 12 pages of notes this week which seems pretty pitiful, and yet I also feel like I learned/worked through a TON of stuff. I probably could have spent more time on KA and gotten further but by the end of the week I felt like I had learned so many new concepts that I didn’t want to overload my brain with many more new ideas. At the same time I’m both happy and disappointed with how this week went. I miss the days of working through KA when each section only had two or three vids and the same number of exercises and I could get through all of it in a single day. When that was the case, I felt like I was making tons of progress each day. Now everything seems much more conceptual and even though I do feel like I’m being exposed to new, bigger ideas, I don’t feel like I’m retaining a lot of what I’m working through. Nonetheless, I know I’m making progress and am optimistic that what I’ve been working on for the past few weeks will make the next few units in MC much easier to get through. It better! 🤬

The four articles I got through this week were titled Multidimensional Graphs, Contour Maps, Parametric Functions – One Parameter, and Parametric Functions – Two Parameters. The first article I worked through, Multidimensional Graphs, explains how a function with one input (x for example) and two outputs can be represented in a (x, y, z) field where y and z represent two separate but related values for the functions. (I’m pretty sure that doesn’t make sense and/or is just straight up wrong.) It’s hard to explain, but here’s an example:

On a surface level, I understand conceptually what’s going on with this type of function, but I definitely have an intuitive grasp on it when looking at the screen shots above. Considering that I’ve never seen functions graphed this way, I’m pretty happy that I even have a weak understanding of why the function f(x) = (x², sin(x)) would be graphed the way it is as shown above. I think with more exposure to and practice with functions like this, it won’t take me too long to get a stronger intuitive grasp on how/why they’re graphed this way and how they work.

The second article, Contour Maps, only took me one day to get through and was quite straightforward. Here’s one of my notes that talks about the definition of a contour map:

Here are some screen shots from the article that paints a picture as to how and why contour maps work the way they do:

I’ve seen maps like this before depicting certain geographic locations and intuitively had a general idea of what they were representing, but this was the first time I’ve ever explicitly been shown how they work. I was glad that the concept seemed simple to understand and that I got through the article so quickly.

The third article I worked through this week, Parametric Functions – One Parameter, helped me concretize something I’ve learned in the past but wasn’t 100% sure about. As I had thought, you can graph an object’s location across time by splitting up it’s (x, y, z) coordinates into separate x(t), y(t) and z(t) functions. For example, if I was to graph a particle’s movement across a (x, y) plane, I can now visualize each axis being split into two functions, an x(t) function which would represent the particles position across the x-axis and a y(t) function which would do the same thing across the y-axis. This is referred to as there being “one input and two outputs” where the input is time and the outputs being the particle’s location on the x-axis and y-axis. Here’s an example from that article:

In this example, you can see that the output is written as a vector with the x-coordinate being (t * cos(2πt)) and the y-coordinate being (t * sin(2πt)). Working through the different values of t in the bottom of my notes above helped me visualize why the function follows the circular path that it takes. I was also happy working through this question to get a better grasp on vectors which are now becoming much less intimidating to me and are easier for me to quickly think through (relative to two months ago, anyways). But, that said, below is another parametric function that was given at the end of the article that I was barely able to understand. Grant worked through how to come up with the equation and I managed to follow along, but I wouldn’t have been able to come up with the vector on my own. Here it is:

The final article I got through this week, Parametric Functions – Two Parameters, helped me finally understand something that’s been confusing me for a long time, the idea that functions can take on more than three dimensions. I could never understand how that was possible since we live in a 3D world and that’s how we see things. What I didn’t realize is that when Grant talked about functions taking on more than three dimensions, he was implicitly saying that the function (if it were to be graphed) could be graphed on/in more than one grid/field. Here are two pages of notes I took where I finally had this realization:

In the bottom of the second page of notes above, you can see that I started talking about the “Input Space” and “Output Space”. If you have two inputs and two outputs, you could graph each on their own (x, y) plane. This also means that you need to set a range on the input. (In the case above, you can see where I wrote the range as 0 ≤ s ≤ 3 and –2 ≤ t ≤ 2 and that I graphed the range of the input space in the bottom left corner of the page.) Here are two screen shots from the article where I worked on this:

Since the output of f(s, t) has three dimensions (which you can see is written at the top of the first screen shot in vector form), the output can be graphed in an (x, y, z,) field. Here’s a video from the article that shows the transformation given the range of the input:

The last thing I worked on was another example of this same concept where a torus was graphed (or animated? Rendered?) using a function with two inputs and three outputs. Here are three screen shots from that example:

In the second screen shot you can see how the torus is broken down into two types of circles, the red one and the blue ones. In the third screen shot you can see the final equation for this particular torus in vector form. I was able to follow along with the article but I don’t have much have a grasp on how this works at this point. I can see why the equation would work, but would never be able to get there on my own. That said, I’m optimistic that I’ll be able to figure it out with some practice.

I’m confident that this coming week I’ll get through the last two articles and finally move on to the next unit, Derivatives and Multivariable Functions. With any luck, I’ll get through at least one exercise in that unit next week and FINALLY make some tangible progress through Multivariable Calculus (100/4,800 M.P.). As I said in my last two posts, I’m pretty confident that all this review I’ve been doing over the past few months is going to make the next few units WAY easier to work through that it otherwise would have been. Considering that I only have 14 more weeks to go before I hit the four-year mark (😱), I’m really hoping that’s the case so I can get through this last Calc course before reaching that milestone. 🙏🏼