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Video transcript
I've restated Stokes' theorem. And what I want to do in this video is make sure that we get our orientation right. Because when we think about a normal vector to a surface there are actually two normal vectors. There-- based on the way I've drawn it right over here, there could be the one that might pop outward, like this. Or there might be the one that pops inward, just like that. Both of those would be normal to the surface right there. And also, when we think about a path that goes around the boundary of a surface, there's two ways to think about that path. We could be going-- based on how I've oriented it right now, we could go in a counterclockwise orientation, or direction. Or we could go in a clockwise orientation, or direction. So in order to make sure we're using Stokes' theorem correctly, we need to make sure we understand which each convention it is using. And the way we think about it is, whatever the normal direction we pick-- and so let's say we pick this normal direction right over here, the one I am drawing in yellow. So if we pick this as our normal vector. So we're essentially saying maybe that's the top, one way of doing it is, that's the top of our surface, then the positive orientation that we need to traverse the path in is the one that if your head was pointed in the direction of the normal vector, and you were to walk along that path, the inside, or the surface itself would be to your left. And so, if my head is pointed in the direction of the normal vector-- so this is me right over here-- my head is pointed in the direction of the normal vector-- I'm wearing a big arrow hat right over there-- and if I'm walking around the boundary, the actual surface needs to be to my left. So I need to be-- this is me walking right over here-- I need to be walking in the counterclockwise direction just like that. Then that's the convention that we use when we're thinking about Stokes' theorem. If oriented this thing differently, or if we said that no, no, no, no, no, this is not the normal vector. This is not the, essentially, the top that we want to pick. If we wanted to pick it the other way, if we wanted to go in that direction, If we wanted that to be our normal vector, in order to be consistent, we would have to now do the opposite. I would now have to have my head going in that direction. And then I would have to walk, once again, and this might be a little bit harder to visualize. I would have to walk in the direction that the surface is to my left. And now, in this situation, instead of the surface looking like a hill to me, the surface would look like some type of a bowl, or some type of a valley or something like that to me. And the way that I would have to do it now, and it's a little bit hard to visualize the upside down Sal, but the upside down Sal would have to walk in this direction in order for the bowl, or the dip, to be to my left. So that's just important to keep this in mind in order for this to be consistent with this right over here. Put your head in the direction of the normal vector. Or you can kind of view that as the top of the direction that the top of the surface is going in. And then the contour, or the direction that you would have to traverse the boundary in order for this to be true, is the direction with which the surface is to your left.