yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Computing the partial derivative of a vector-valued function


2m read
·Nov 11, 2024

Hello everyone. It's what I'd like to do here, and in the following few videos, is talk about how you take the partial derivative of vector-valued functions.

So the kind of thing I have in mind there will be a function with a multiple variable input. So this specific example has a two-variable input T and s. You could think of that as a two-dimensional space, as the input are just two separate numbers, and its output will be three-dimensional. The first component is T squared minus s squared. The Y component will be s times T, and that Z component will be T times s squared minus s times T squared minus s times T squared.

And the way that you compute a partial derivative of a guy like this is actually relatively straightforward. It's, if you were to just guess what it might mean, you’d probably guess right: it will look like the partial of V with respect to one of its input variables, and I'll choose T with respect to T. You just do it component-wise, which means you look at each component and you do the partial derivative to that because each component is just a normal scalar-valued function.

So you go up to the top one, and you say T squared looks like a variable as far as T is concerned, and its derivative is 2T. But s squared looks like a constant, so its derivative is zero. s times T, when s is s, looks like a constant, and when T looks like a variable, it has a derivative of s. Then T times s squared, when T is the variable and s is the constant, it just looks like that constant, which is s squared minus s times T squared.

So now, the derivative of T squared is 2T, and that constant s stays in. So that’s 2 times s times T, and that’s how you compute it probably relatively straightforward. The way you do it with respect to s is very similar. But where this gets fun and where this gets cool is how you interpret the partial derivative, right?

How you interpret this value that we just found, and what that means, depends a lot on how you actually visualize the function. So what I'll go ahead and do in the next video, and in the next few ones, is talk about visualizing this function. It'll be as a parametric surface in three-dimensional space; that's why I've got my graph or program out here. I think you'll find there's actually a very satisfying understanding of what this value means.

More Articles

View All
Lance Romance | Wicked Tuna
Who does this man? Is that Bubba? He’s got to learn to reel it without reeling it like that. Who did it? Lance romance? Really, Lance? Come here. It’s week nine, and Lance is still making rookie mistakes. I want Lance to learn these things because if he …
How Trees Bend the Laws of Physics
Sometimes the simplest questions have the most amazing answers. Like how can trees be so tall? It’s a question that doesn’t even seem like it needs an answer. Trees just are tall. Some of them are over 100 meters. Why should there be a height limit? I’ll…
Give Society What It Doesn't Know How to Get
You’re not going to get rich renting out your time, but you say that you will get rich by giving society what it wants but does not yet know how to get at scale. That’s right. So essentially, I could… We talked about before, money is IOU’s from society sa…
Common denominators: 3/5 and 7/2 | Math | 4th grade | Khan Academy
Rewrite each fraction with a denominator of 10. We have two fractions: 3 fifths and 7 halves, and we want to take their denominators of five and two and change them to be a common denominator of 10. Let’s start with 3 fifths. We can look at this visuall…
Mariya Nurislamova, Founder of Scentbird at the Female Founders Conference
Really bright and sunny today. I can’t unsee the slides, but I guess that’s okay. Hi everyone, my name is Maria. For the past four and a half years, I’ve been building a company called Sunbird. Sunbird is a fragrance subscription service, and we help peop…
Peer Into a Fallen Battleship at Pearl Harbor | National Geographic
Here we are at the number one guns of the USS Arizona. Oftentimes diving on the USS Arizona, we come across artifacts like this shoe or boot sole. It’s artifacts like this that remind us of the human connection of the ship and those who lost their lives h…