I first started Multivariable Calculus in Week 185 and made it through 2% of the course by the end of that week. Until this week, in terms of achieving Mastery Points and bumping up that 2%, I hadn’t made any more “progress” until this week. I am now officially 5% of the way through the course! 😤 I finished the final two articles I needed to get through in the first unit which were titled Visualizing Vector-Valued Functions and Transformations, and which were both pretty straightforward. I got started on the following unit, Derivatives and Multivariable Functions, on Thursday and really enjoyed (relatively speaking) the first four videos I watched and the two exercises I made it through, all of which had to do with a concept known as partial derivatives. I’m 10/10 happy with the progress I made this week and feel like I took a pretty big step forward!

I didn’t take any notes on the second article I worked through this week, Transformations, because it all seemed pretty straightforward. The first article, Visualizing Vector-Valued Functions, was also essentially review but I did take some notes that I thought were worth posting:

As I mentioned at the end of the second note, knowing that you can write matrix vectors in the form of unit vector notation (i.e. i-hat (î), j-hat (ĵ), and k-hat (k̂)) will make things easier going forward since I don’t know how to write matrix vector notation in these posts. The two forms of notation makes me think of switching between Leibniz and LaGrange notation or degrees and radians. It’ll probably take me awhile before I can seamlessly switch between matrix vector notation and unit vector notation without any friction in my mind, so to speak.

Here’s an example of how to convert a matrix vector into a unit vector:

Lastly, after studying vectors for the last 6 or 7 weeks, the info in the following screenshot wasn’t anything new for me, but I thought it was hilarious that Grant used the images to denote vectors and wanted to post it:

As I mentioned, I got started on the second unit, Derivatives and Multivariable Functions, on Thursday. The first section of the unit, Partial Derivatives, is made up of four videos and three exercises and I made it through everything this week but the last exercise. I was a bit intimidated by the title of the section, but it turns out the concept of partial derivatives is relatively easy to understand. To put it simply, If you’re working with a 3D graph that has a 2D input and a 1D output (i.e. the inputs are graphed on the (x, y) plane and output graphed in the z-dimension), a partial derivative is the SLOPE of a SLICE of the graph. Here are three screen shots from the video where Grant explains how it works:

As far as I understand it, essentially what you’re doing is slicing the 3D object, then picking a point on the (x, y) plane and then finding the slope of the graph at the (x, y, z) point in either the x-direction or y-direction. How to do this turns out to be pretty easy. You simply treat either x or y as a constant (which is what “slices” the graph) and then find the derivative of the function and input a given set of (x, y) coordinates. Here’s how it works:

(Also, the notation used for “partial derivative” is the symbol for the Roman letter “d” which is this little guy – ∂.) 

As you can see, at least in this first example, taking a partial derivative is as easy as taking a regular derivative as long as you can think through one of the variables taking on the value of a constant. I’m not going to go through the math, but here are five examples of partial derivatives I worked through this week:

Question 1

Question 2

Question 3

Question 4

Question 5

Like I said at the start, I’m very happy with how things went this week. Based on how much review I’ve done over the past 8 weeks and how this first week went, I’m optimistic that I’ll get through Derivatives of Multivariable Functions (260/2,100 M.P.) fairly quickly. I’m guessing it’ll get tougher as I move forward so that could obviously change, but for now I’m feeling pretty good! 💪🏼