Vector fields, introduction | Multivariable calculus | Khan Academy

Hello everyone!

So, in this video, I'm going to introduce Vector Fields. Now, these are concepts that come up all the time in multivariable calculus, and that's probably because they come up all the time in physics. You know, it comes up with fluid flow, with electrodynamics; you see them all over the place.

What a vector field is, is it's pretty much a way of visualizing functions that have the same number of dimensions in their input as in their output. So here, I'm going to write a function that's got a two-dimensional input, X and Y, and then its output is going to be a two-dimensional vector. Each of the components will somehow depend on X and Y.

I'll make the first one Y cubed minus 9Y, and then the second component, the Y component of the output, will be X cubed minus 9X. I made them symmetric here, looking kind of similar; they don't have to be. I'm just kind of a sucker for symmetry.

If you imagine trying to visualize a function like this with, I don't know, like a graph, it would be really hard because you have two dimensions in the input and two dimensions in the output. So, you'd have to somehow visualize this thing in four dimensions. Instead, what we do is look only in the input space. That means we look only in the XY plane.

So, I'll draw these coordinate axes and just mark it up. This here is our X axis; this here is our Y axis. For each individual input point, like let's say (1, 2), so let's say we go to (1, 2). I'm going to consider the vector that it outputs and attach that vector to the point.

So, let's walk through an example of what I mean by that. If we actually evaluate F at (1, 2)—X is equal to 1, Y is equal to 2—we plug in 2 cubed, whoops, 2 cubed minus 9 times 2 up here in the X component, and then 1 cubed minus 9 times Y, 9 times 1, excuse me, down in the Y component.

2 cubed is 8; 9 times 2 is 18, so 8 minus 18 is -10. Now, first, imagine that this was—if we just drew this vector where we count, starting from the origin, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10—so, it's going to have this as its component.

And the Y component, 1, 2, 3, 4, 5, 6, 7, 8—it's going to actually go off the screen; it's a very, very large vector, so it's something here, and it ends up having to go off the screen.

But the nice thing about vectors is it doesn't matter where they start. So, instead, we can start it here, and we still want it to have that -10 X component and the -8 as its Y component there. So, this is a really big vector, and a plan with the vector field is to do this at not just (1, 2) but at a whole bunch of different points and see what vector is attached to them.

If we drew them all according to their size, this would be a real mess; there'd be markings all over the place. You'd have, you know, this one might have some huge vector attached to it, and this one would have some huge vector attached to it, and it would get really, really messy.

Instead, what we do—so, I'm going to just clear up the board here—we scale them down. This is common; you'll scale them down so that you're kind of lying about what the vectors themselves are, but you get a much better feel for what each thing corresponds to.

Another thing about this drawing that's not entirely faithful to the original function that we have is that all of these vectors are the same length. You know, I made this one just kind of the same unit, this one the same unit, and over here, they all just have the same length. Even though, in reality, the length of the vectors output by this function can be wildly different, this is kind of common practice when vector fields are drawn or when some kind of software is drawing them for you.

There are ways of getting around them. This one way is to just use colors with your vector. So, I'll switch over to a different vector field here, and here, color is used to kind of give a hint of length. It still looks organized because all of them have the same lengths, but the difference is that red and warmer colors are supposed to indicate this is a very long vector somehow, and then blue would indicate that it's very short.

Another thing you can do is scale them to be roughly proportional to what they should be. So, notice all the blue vectors scaled way down to basically be zero; red vectors kind of stayed the same size. Even though, in reality, this might be representing a function where the true vector here should be really long or the true vector here should be kind of medium length, it's still common for people to just shrink them down so it's a reasonable thing to view.

In the next video, I'm going to talk about fluid flow, a context in which vector fields come up all the time, and it's also a pretty good way to get a feel for a random vector field that you look at to understand what it's all about.