3d vector field example | Multivariable calculus | Khan Academy

So in the last video, I talked about three-dimensional vector fields, and I finished things off with this sort of identity function example where at an input point (X, Y, Z), the output vector is also (X, Y, Z). Here, I want to go through a slightly more intricate example. So I'll go ahead and get rid of this vector field.

In this example, the X component of the output will be Y * Z, the Y component of the output will be X * Z, and the Z component of the output will be X * Y. So I'll just show this vector field, and then we can start to get a feel for how the function that I just wrote relates to the vectors that you're seeing.

You see some of the vectors are kind of pointing away from the origin, and some of them are pointing in towards the origin. So how can we understand this vector field in terms of the function itself? A good step is to just zero in on one of the components. In this case, I'll choose the Z component of the output, which is made up of X * Y, and kind of start to understand what that should be.

The Z component will represent how much the vectors point up or down. This is the Z-axis, and the XY plane here. I'll point the Z-axis straight in our face. This is the X-axis, and this is the Y-axis. The values of X and Y are going to completely determine that Z component.

So I'm going to go off to the side here and just draw myself a little XY plane for reference. This is my X value; this is my Y value. I want to understand the meaning of the term X * Y. When both X and Y are positive, their product is positive, and when both of them are negative, their product is also positive. If X is negative and Y is positive, their product is negative, but if X is positive and Y is negative, the product is also negative.

So what this should mean in terms of our vector field is that when we're in this first quadrant, vectors tend to point up in the Z direction. Same over here in the third quadrant, but over in the other two, they should tend to point down. So let's focus in on that first quadrant and try to look at what's going on.

We see, like, this vector here applies. This vector and all of them generally point upwards; they have a positive Z component. So that seems in line with what we were predicting, whereas over here, which corresponds to the fourth quadrant of the XY plane, the Z component of each vector tends to be down. You know this, and they're doing other things in terms of the X and Y component. It's not just Z component action, but right now we're just focusing on up and down.

If you look over in the third quadrant, they tend to be pointing up, and that corresponds to the fact that X * Y will be positive. You look, and it all starts to align that way. Because I chose a rather symmetric function, you could imagine doing this where you analyze also the Y component here; you analyze the X component in terms of Y and Z, and it's actually going to look very similar for understanding when the X component of a vector tends to be positive, like up here, or when it turns out to be negative, like over here.

And same with when the Y component of a vector tends to be negative or if the Y component tends to be positive. Overall, it's a very complicated image to look at, but you can slowly piece by piece get a feel for it. Just like with two-dimensional vector fields, a kind of neat thing to do is imagine that this represents a fluid flow.

So imagine maybe air around you flowing in towards the origin here, flowing out away from the origin there. You know, it could kind of be rotating around here. Later on in multivariable calculus, you'll learn about various ways that you can study just the function itself and just the variables to get a feel for how that fluid itself would behave.

Even though it's a very complicated thing to think about, it's even complicated to draw or use graphing software with. But just with analytic tools, you can get very powerful results, and these kinds of things come up in physics all the time because you're thinking in three-dimensional space. It doesn't just have to be fluid flow; it could be a force field, like an electric force field or a gravitational force field, where each vector tells you how a particle tends to get pushed.

As we continue on with multivariable calculus, you'll get to see lots more examples. But hopefully, this helps get a small feel for how you can go piece by piece and understand something that's kind of a complicated expression.