I spent this entire week working through the single section Gradient and Directional Derivatives but still don’t really know what’s going on. 😒 I ended up watching all seven videos in the section and made it through two of three exercises, but struggled with and didn’t make it through the third exercise. I got a fairly decent grasp on how to solve the questions that came up in the exercises but have no real understanding of why the math works (or even what’s going on; 😔). As I’ve mentioned before, it always seems to be the case that I first figure out HOW something works before I figure out WHY it works, so I’m not too worried. It’s always frustrating feeling lost but the silver lining is that all the questions I worked through this week not only helped me practice/review derivatives and vectors but were also pretty fun (relatively speaking) to work through. 🤓
My understanding of what a (or the?) gradient is hasn’t changed from last week. I believe (but could be wrong) that you can consider the gradient to be a surface of an object in an (x, y, z) field and its slope at different points. Below are three questions I worked through where the KA answer for each begins with, “The gradient of a scaler field is all its partial derivatives put together into a vector.” My understanding of a vector comprised of partial derivatives for x, y and z is that inputting an (x, y, z) coordinate into the vector, it would tell you the degree and direction of the slope at that point. (…🤔) I don’t know if I’m making any sense right now, but here are the three questions:
Question 1
Question 2
Question 3
Although I’m not completely sure what’s going on or why these questions work, I feel pretty confident about how to answer these types of questions. I’ve gotten a lot better at solving partial derivatives and being able to think through and consider the variable that’s not being solved for as a constant.
I was stumped this week on a concept that’s called the direction derivative. Looking through my notes, it seems to be exactly what I just described in what I think the gradient is, i.e. the slope at any given point in an (x, y, z) field. I’m very confused because if that is what the directional derivative is (which its name would suggest…), then that indicates I must be wrong about what I think the gradient is. 😡 I’m sure they’re different things but I don’t know what the distinction between the two of them is. In any case, here’s a screen shot from the sixth video of the section which shows a 3D object being sliced by a plane and the calculation for the derivative at a specific point where the object is sliced:
The only difference between finding the gradient, ∇f(x, y), and the directional derivative is that you find the dot product of the gradient at a specific point and a vector. This is what I don’t understand but, even though I don’t know what’s going on, I’m glad that I at least know how to solve these questions. Here are two a examples:
Question 4
Question 5
As you can see in this last question, the way that KA solved this question is much simpler than the way I did, but I don’t know how to do it that way. I find the method which I described in question 4 above where you first find the gradient, then input a point into the gradient and dot it with a vector, to be the simplest way for me to find the solution.
Finally, I also learned this week that you can use the formal definition of a derivative, i.e. by taking the limit at a certain point as h approaches 0, to find a directional derivative. Again, I don’t know why this works or even how to do it properly, but I was able to solve these questions using the method I described above. Here are two example questions:
Question 5
Question 5
All in all, I’m somewhat disappointed with how this week went since the concepts of what I worked through didn’t exactly solidify for me. Nonetheless, like I said at the start, learning how to solve these questions is always the first hurdle for me to get past so I think I’m in a good position now to hopefully figure this all out soon enough. I’m officially >10% of the way through Multivariable Calculus (500/4,800 M.P.) and ~24% of the way through this unit, Derivatives and Multivariable Functions (500/2,100 M.P.). I’d REALLY like to get through this course by the time September rolls around, which will be the four-year mark of me working on KA. I think finishing this course off in the next three months will be tough but potentially doable. Whether I manage it or not, it’ll be good motivation to push me to do more each week going forward!