I’m happy with the progress I made this week but didn’t end up passing the unit test. I finished off the final section in Derivatives of Multivariable Functions which was about the Jacobian Matrix and actually managed to more-or-less understand what it’s all about, but then got smoked on the unit test. One good piece of news is that the unit test now has a button where I can simply “Start Over” instead of having to go through all the questions on the test and try to get them all correct or otherwise have to redo the exercises from the questions I got wrong. I started the unit test three times and only made it to the third, fifth and sixth question each time respectively before getting those questions wrong. Considering there are 21 questions in this test, I feel like it’s going to take me at least a few attempts to get through the test. Nonetheless, it was nice to finally start the test and I wasn’t too upset getting those questions wrong and having to go back and review those concepts. As I note at the bottom of this post, I went back to review curvature and actually got a much deeper understanding of how it works. (Still having a hard time understanding why it works though. 😔) So, even though I didn’t pass the test, I’m happy with where I ended this week, what I was able to learn and where I’m at heading into next week!

The second and third videos from the Jacobian section worked through a Jacobian matrix question/example and showed what the formula is, how it works and why it works. I’m happy to say I generally understand what it is and how it works, but I don’t really understand why it works… Here are a few screen shots from those videos with descriptions of what’s happening:

At the top of this image where it says “L.A. -> M.V.C.”, the was Grant just saying the Jacobian Matrix is good example of how linear algebra starts relating to multivariable calculus. Below that you can see f([y, y]) = [(x + sin(y)), (y + sin(x))]. Here, the grid of blue lines on the left represents f([y, y]).

Here you can see that the gridlines have become all squiggly. This is the graphed representation of [(x + sin(y)), (y + sin(x))].

This screen shot starts to explain the entire point of the Jac-mat, as I like to call it. It has to do with a concept known as ‘Local Linearity’. What this term means is that if you’ve transformed a matrix with a non-linear transformation, if you zoom-in at a specific point the grid at that zoomed-in point looks like a linear transformation.

This is where I start to not know what’s going on. The formula for the Jac-Mat itself (which is the matrix full of partial derivatives which you can see in image below this) is pretty easy to use, but the concept of why this matrix works is still something I don’t understand. What it seems to be saying is, “Imagine unit vectors [1, 0] and [0, 1] at a given point. (At the point (–2, 1) in this example.) After the transformation the unit vectors will move to new coordinates.” (This is also shown in the screen shot below.) I believe the reason why you use calculus and partial derivatives is because the Jac-Mat is zooming-in on a specific point/region in order to determine how the local linearity in question changes after the matrix transformation. So you’re looking at a tiny, tiny change in the matrix, just like how in calculus you’re looking at a tiny, tiny change in the slope.

This image shows the transformation of the matrix and how the local linearity gets changed from being a normal looking grid to being tilted on its side and, though it’s hard to see, expands slightly. One thing to note here is that the starting point of the coordinates in question, (–2, 1), actually gets moved to ~(–1, 0.1) but that doesn’t matter when it comes to the Jac-Mat which is only looking at how the local linearity gets altered. Like I said, I don’t understand why the Jac-Mat works and I can tell there’s still a lot I have left to learn about it, in general, but I think I may be over the hump of understanding what it is.

The last two videos and the last exercise in the Jacobian section had to do with finding the determinant of the Jac-Mat. As far as I understand, the determinant of the Jac-Mat tells you if a non-linear matrix transformation expands or contracts at a specific point after the transformation. Having worked through a bunch of determinant questions recently, I found this exercise relatively straightforward. Here’s a few example questions:

Question 1

Question 2

Question 3

After getting through that section, I started the unit test on Thursday. Here are a few questions from the test:

Question 4

I wasn’t super confident about this question as I worked through it but actually did the all of the linear algebra and calculus correct but messed up inputting the values for y and z into the top row of the matrix. 🤦🏻‍♂️ This is a good example of why I often get questions wrong, because I rush through them too quickly and make careless mistakes.

Question 5

I got this particular question correct but had already gotten a question just like this one incorrect on a previous attempt at the unit test. I had to spend a bit of time working through these types of composite, multivariable functions to remember the formula to find their derivatives. I still don’t understand why you find the gradient of the outside function but find the derivative of the inside function though. 🤔

As I mentioned at the top, I got a question on curl wrong which is the question in the screen shot just below. Again, I couldn’t remember the formula I needed. This was the question:

After getting that question wrong, I went back to rewatch one of the first videos on curl to try and better understand it. Here’s a screen shot from the video I watched:

I spent some time trying to understand why the formula is f­_y­(x, y)_­­­ –  ­f_x­(x, y) and I think I figured it out. First off, curl is a scalar value which means it’s just a single number and is simply a unit of measurement of the degree to which something – for instance a fluid – is rotating and its direction. If the curl is positive, it’s spinning counter-clockwise and if it’s negative, it’s spinning clockwise. This is just a convention that people have decided to use to describe a fluid’s motion. I spent some time trying to visualize Q and P both being positive and how the curl would work when Q was bigger than P and vice versa. I drew out this image to try to help me visualize it:

It started to make sense to me but then I thought about what it would look like if Q (a.k.a. the force [?] in the y-direction) was not a vector but illustrated on a number-line and how you could then subtract the Q number line from the P number line. This is when I started to be able to visualize the formula f­_y­(x, y)_­­ – f_x­(x, y). Here’s what I drew out:

As you can see in the bottom of my second note, it took me a minute to realize that I needed to always lineup the tips of the arrows when subtracting the two values. The red number-line indicates the value of the curl. As I already said, when it’s positive it means that the fluid would be rotating counter-clockwise and when it’s negative it means the fluid would be rotating clockwise. If I close my eyes, I can now visualize taking ∂Q/∂x and tilting on its side and then subtracting off ∂P/∂x to determine the value of the curl.

This is a pretty big post so I’m going to wrap it up quickly. I think I should be able to get through the unit test this week and finally move on from Derivatives of Multivariable Functions (1,700/2,100 M.P.). I started this unit in late May in Week 194 so it’ll definitely be nice to move on to the next unit. Hopefully it won’t take me too long to get through the test this week and I can make a decent dent in the next unit, Applications of Multivariable Derivatives! 🤞🏼