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Video transcript
I've rewritten Stokes' theorem right over here. What I want to focus on in this video is the question of orientation because there are two different orientations for our boundary curve. We could go in that direction like that, or we could go in the opposite direction. We could be going like that. And there are also two different orientations for this normal vector. The normal vector might pop out like that, or it could actually go into the surface like that. So we have to make sure that our orientations are consistent, and what I want to do is give you two different ways of thinking about it. And you might think of others, but these are the ones that work for me. In order for Stokes' theorem to hold, we have to make sure that we're not actually picking the negative of one or the other orientations. And so the easiest way for me to remember-- it is if our normal vector is-- let's say, it goes in that direction. And if you have some hypothetical person traversing the boundary of our surface, and their direction is pointed. And their head is pointed in the same direction as the normal vector-- so this is the normal vector. So their head is pointed in the exact same direction as the normal vector-- or you could say maybe their body or, really, their head-- So that's them. Then the direction that you would have to actually traverse the boundary is the direction that would allow this person to keep the surface to their left. So over here, he would have to go in this direction in order to keep the surface to his left. So he would have to go just like that. If we oriented the surface differently-- so let me redraw the surface right over here and draw similar surface. So if we had a surface-- so this surface looks very similar. This is a very similar looking surface that I'm drawing right over here just to give a idea of some of the contours. But if we said that the normal vector for this surface, if we orient it in the opposite way-- so if we said that the normal vector here was actually pointing downward like that, then we would have to, in order for Stokes' theorem to hold, we would have to traverse the boundary in a different direction because, once again, if I draw my little character right over here, his head is pointed in the direction of the normal vector. He is now upside down. So let me draw him. So this is him running right over here. I could draw a better job. This is him running right over here. Now, in order to keep-- and from his point of view, this would kind of look like a some type of a pool or a ditch of some kind. It would actually go down. Here, it looks like a hill to him. But since he's upside down, in order for him to keep the boundary to his left, he would have to now go in the other direction. So depending on the orientation of your normal vector, which is really the orientation of your actual surface, will dictate how you need to traverse the path. Now, another way to think about it-- and this idea was introduced by one of the viewers on YouTube, but it's a valid way of thinking about it-- is to imagine that the surface is a bottle cap. And so let me draw some type of a bottle over here. So I'll draw. Let me draw a bottle. You could imagine some type of a glass soda bottle. So what we really care about is the cap of the bottle-- so make it feel like it's glass. So there, that's our bottle, And let me draw its cap. Let me draw the cap of the bottle because that's what we care about. We can kind of imagine that being the surface. So this is the cap of our bottle, and you just need to think about, well, which way would I have to twist the cap in order to make the cap move up, in order to take the cap off. And you could think of the normal vector as the direction that the cap would move, and the twisting is the direction that you would have to traverse the path. So you would have to twist the bottle that way, or you could think about the other way. If you twisted the bottle the other way, then the cap would move down. So the normal vector is the direction that the cap would move, and the direction that you would traverse the boundary is how you would actually twist it. So either of these are ways of thinking about it, but they're important to keep in mind, especially once the shapes start getting a little bit more convoluted and oriented in strange ways.