Last week I spent a few days trying to get through the final exercise in the section Gradient and Directional Derivatives, so I was happy when I got through it on my first attempt this past Tuesday. I’m glad that I now have a decent understanding of HOW to solve the directional derivative but am still pretty lost on WHY it works. What makes it worse is that I spent almost half of this week reading through two articles on the directional derivative but still wasn’t able to wrap my head around what’s going on. 🤬 Regardless, I can tell I’m getting better at working with vectors and understanding partial derivatives. A few weeks ago I had no clue what a partial derivative was and was pretty intimidated just by the name so I’m definitely happy with the progress I’ve made.
After getting through the final exercise from the section Gradient and Directional Derivatives on Tuesday, I then got started the following section, Partial Derivative and Gradient (Articles). This section has five articles in it which were titled:
I got through the first article by the end of Tuesday, which was quite straightforward, and then began the next article on second partial derivatives on Wednesday. A key takeaway from that article was second partial derivative notation:
I sometimes prefer Leibniz notation when working on these questions because I literally say to myself, “a tiny, tiny change in the value of the function (which I usually now think of as z) over a tiny, tiny change in the value of x or y.” Thinking about it like this in my head simplifies it for me. That said, I also like using Lagrange notation at times just because it’s a lot more concise when writing it out.
Before going through this article, I’d worked on second partial derivatives before but only up to the second degree. In this article, there were a few practice questions which had me take the third, fourth, etc. degree partial derivative. (I’m pretty sure these are referred to as ‘higher order’ partial derivatives.) Here’s an example of a question I worked on where I had to take the fifth degree partial derivative:
The following article on gradient helped me get a clearer picture of what’s going on and how/why the gradient works. At one point Grant said, “if you walk in the direction of the gradient, you will be going straight up the hill,” which I thought was a pretty good explanation and made it easy to understand. Here are two notes I took that go through the operators used with gradients and a note on how to think about the gradient being dotted (might be the wrong word) with a function:
I finished off that third article on Thursday and then got started on the following article on directional derivatives. Like I said above, I’m pretty good at solving the questions I’ve been shown so far on the directional derivative but only because I’ve memorised the pattern/method used to find the solution, but I don’t really know how it works. Before I explain what I think is going on with the D.D. (which could be completely wrong), here are two notes I took on the notation used:
My understanding of the D.D. is that you might have, for example, a two input, one output function where the two inputs could be x and y and the output would be z. You can model these functions in 3D and go to any set of (x, y) coordinates and see what the value of the function is by looking at the value of z. You can think of a given set of (x, y) coordinates as a vector themselves which seem to often be denoted with an a with a little arrow over it (a ⃗ is the closest I can get to). I think the D.D. is looking at those (x, y) coordinates, going up or down to the z-coordinate and asking, “if I move away from these (x, y) coordinates in the direction of some vector, what is the slope of z?”.
(I could be 100% wrong about what I just wrote.)
If I’m at all correct with what I just described, when finding the slope at (x, y, z), if you’re using a vector other than a unit vector you have to either convert the vector into a unit vector or divide the dot-product of ∇f(x, y) ⋅v ⃗ by the magnitude of v ⃗. The reasoning behind this seems to have to do with Pythag’s Theorem and that if the vector is < 1 < then the D.D. (i.e. the slope) will be scaled to that value.
(I’m pretty sure very little of what I just typed makes sense or is accurate… 😡)
Here’s an example:
Writing the past few paragraphs has been pretty demoralizing. It’s frustrating that I don’t really know what’s going on. I think I’m on the right track but also think that there are some massive holes in my understanding of the directional derivative. Even though I don’t understand it, my plan is to move on and hope that I’ll get more insight a to what the D.D. is in the units to come. This has always been what happens so I’m confident that the D.D. will click for me at some point.
The next section in this unit, Derivatives of Multivariable Functions (580/2,100 M.P.), only has four videos and two exercises, of which I’ve already watched two videos and finished one exercise. ☺️ I’m hoping I can get through this section in a few days and then get started on and make some progress in the following section, Multivariable Chain Rule. I feel like it’s been a long time since I got through two sections in one week, and I think it’s unlikely that it’ll happen this week, but that’s my goal. As always, fingers crossed. 🤞🏼