I had another solid week working through the first unit in Multivariable Calculus. Looking back through my notebook, I didn’t write many notes but I definitely got a better understanding of where MC is heading and on what I think are some of the fundamental concepts that will be used. I ended up watching the 12 remaining videos in the unit and finished off two of the eight articles. There were a number of things that came up that were tricky to think through but nothing that I was 100%, completely lost trying to understand. Even though I’m disappointed to still only be 2% of the way through the course, all in all I’m happy with how this week went as I think I got a better understanding of some of the tools/techniques that will be key in MC.

I spent Tuesday, Wednesday and Thursday watching the 12 videos and began the first article on Friday. Typically I write these posts out in chronological order but I’m going to switch it up this time and first go through some of the definitions that came up in the first article:

Although I didn’t get to these definitions until Friday, the very first note I wrote down this week was: 

• “I always thought of ‘f(x) = x’ as simply ‘y = x’. I never understood why you don’t simply use ‘y’ in place of ‘f(x)’.” 

Turns out this was a prescient thought. I understood before that f(x) stood for ‘function-of-x’ but I see now that the reason you use f(x) instead of simply using y is because it’s more general and can be applied with multivariable functions but still serve the same purpose. 

Along those lines, here’s the formal name/definition for the variable(s) used in functions and the for the output:

With those definitions in mind, the 12 vids I watched this week for the most part dealt with 3D objects. For example, here’s an image of a 3D object that looks like a donut but in math is referred to as a torus:

The function for this particular torus is this:

After having only used single-variable functions for the past ~3.5 years, this type of function is completely unintuitive to me but I’m slowly starting to make sense of it. The rows of the output of the function (which you can see is written as a vector) correspond with the object’s (x, y, z) coordinates with top row representing the x-coordinate, middle row representing the y-coordinate and bottom row representing the z-coordinate. To get a better grasp on how these functions work, in the video Grant sets one of the variables, t and s, to 0 at different times and then solves each row of the output space. Here’s an image of the two circles that are created when he sets each variable to 0:

The red circle is what’s created when s = 0 and the blue circle is what’s created when t = 0. Here are the notes I took when I worked through setting t to 0 and then solving each row in the output space:

Since the second row in the output space equals 0 when t = 0, this means that when t = 0 the vector has no y-dimension. On my third page of notes you can see where I worked through s being equal to {0, π/2, π, 3π/2} with the resulting coordinates being (4, 0, 0), (3, 0, 1), (2, 0, 0) and (3, 0, –1) which denote the top, sides and bottom points on the blue circle. Working through this video, I can see how multivariable functions and vectors come together to describe 3D objects and I can understand how/why multivariable calculus will factor into these types of questions.

On Wednesday I made it to the section titled Visualizing Vector-Valued Functions which mostly dealt with vector fields. I’ve learned about vector fields in the past but it had always been in 2D whereas this week they were introduced to me in 3D, as well. I remembered vector fields being a bunch of little arrows pointing in different directions depending on what the overarching function was. I always imagined that if you dropped a marble on a vector field it was start swirling around the (x, y) planes following the directions of the little arrows. It turns out that 3D vector fields are essentially the same thing, albeit a little more complicated given that there’s a third dimension. Here’s a screen shot from one of the vids which shows an example of a 3D vector field:

Depending on what the function is, the vectors at each point in a vector field often have different magnitudes. If every arrow was proportionate to its magnitude the graph would be unreadable since they’d all be overlapping, so to get around this what you do is draw each arrow with a size somewhere between 0.1 and 1 unit in length (or somewhere a long those lines) and/or colour coordinate each arrow to reflect the magnitude with purple being the shortest and red being the longest.

One final thing that I made a note of was that Grant said that in single-variable calculus it was common and useful to think of the derivative of a function as the line tangent to that function at a given point and the integral of a function to be the signed area between the function and the x-axis. BUT, he said that going into multivariable calculus this isn’t always the best was to think about it. I’m interested to see what that means exactly and why that’s the case, but also a bit worried that it’s going to be very confusing and take me awhile to intuitively understand what’s going on. I do think, however, that when I figure it all out, it will make derivatives and integrals in single-variable calculus seem very straightforward. 🤞🏼

I said last week that I thought it would take me two-ish weeks to get through this unit. I have six articles left to get through so I think it’s possible that I could finish it off this coming week. The first two articles I worked through this week were pretty lengthy though so depending on how long the remaining ones are, it could still be tricky. In any case, even though I haven’t earned any M.P. in Multivariable Calculus (100/4,800 M.P.) in a LONG time, I’m starting to make some decent progress through the course which I’m happy about. And like I said last week, I think I’m doing a good job reviewing and getting a better grasp on these fundamental concepts which I’m sure will make things easier when I finally get through this first unit. 😤