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Oregon State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

05:28

Felicia S.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

00:56

00:51

Heather Z.

0:00

Dungarsinh P.

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So for the final topic of this course, we're going to switch gears and talk about something that's going to see new. But it's actually going to tie in. Just about everything that we talked about and what we're going to talk about is three idea of a vector field. And so the idea of a vector field is very simple. In fact, I would say that vector fields were the most concrete things that we could talk about in this course. So I just want to think you to think about the world around you for a minute. You live on this planet, and there is all kinds of forces acting around you. There's a gravitational force pushing you towards the center of the earth. There are wind currents that air pushing you or pushing objects around you. There's electrical forces acting around you, acting on your actual just individual atoms, pushing them around. They're all these forces. They're basically nothing more than a bunch of collections of arrows. So the idea of a vector field is that in space there are he's vector fields, and a good way to think about them is just force fields where there's just a bunch of arrows everywhere. And the best way I know to think about these arrows is there, like water currents or air currents, kind of pushing through space, acting on whatever objects or in their way. And so if I'm a bird or a plane or an object or an electron, an electric field, I'm just getting pushed around by these vector fields. Now I might be moving this way or against the vector field. I can sort of move independently of the vector field, but the vector field is kind of pushing me along as I go in that direction. And if I want to move with the vector field, it's actually easier. If I want to move against the vector field, it's harder. And so we'll sort of explore that physical application of moving within a vector field. Now. There are also other things that we could talk about with vector fields to weaken. Talk about how vector fields rotate and how they kind of disperse away from sinks and from sources. So if I have, you know, a faucet in the water, there's water flowing out of of the faucet, and I can think about these rotations in these divergences from different points of these vector fields. And so we're really getting at kind of the heart of physical applications here, where you have objects. They're moving around in vector fields, and these vector fields are sort of the fundamental quantities that really dis driver universe, electric fields, magnetic fields, other force fields, wind, water, etcetera. All of it can be modeled as a vector field. And so that's what we're going to talk about. And it's really cool that the idea of a vector field is going to unify a lot of the ideas that we talked about in this class. So it's a really exciting topic. It's a really cool branching point, uh, for different fields of engineering and physics, etcetera. So I hope you enjoy it, and here we go.

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