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Now that we've explored Stokes' Theorem a little bit, I want to talk about the situations in wich we can use it. You'll see that this is pretty general theorem. But we do have to thing about what type of surfaces and what type of boundary are those surfaces we are actually dealing with and the case of Stokes', we need surfaces that are piecewise... piecewise-smooth piecewise-smooth surfaces so this surface right over here it is actually smooth not just even piecewise-smooth. Sounds like a very fancy term but all the smooth part means that you have just continuous derivatives and since we are talking about surfaces we're going to have continuous partial derivatives regardless of which direction you pick. So this is continuous derivatives, and another way to think about that conceptually is if you pick a direction on the surface if you say that we go in that direction, the slope in that direction changes gradually, doesn't jump around. If you pick this direction right over here, the slope is changing gradually. So we have a continuous derivative. And you're like "what does the 'piecewise' means?" Well, the piecewise actually allow us to use Stokes' Theorem with more surfaces. Because if you have a surface that looks like... Let's say a surface that looks like this. Let's say looks like a cup. So this is the opening of the top of the cup let's say that has no opening on top so we can see the backside of the cup and this is the side of the cup and this right over here is the bottom of the cup and if it was transparent we could actually see through it. So surfaces like this is not entirely smooth because it has edges. There are points right over here. So this edge right over here... If we pick this... let's say if we pick this direction to go and if we go this direction along the bottom, then right we get to the edge, all of the sudden the slope changes dramatically jumps. So the slope is not continuous at that edge. The slope jumps and we start going straight up. And so this entire surface is not smooth. But the piecewise actually give us an out. This tell us that it's okay as long as we can break the surfaces up into pieces that are smooth. And so this cup we can break it up but we were doing this wen tackling surfaces integrals we can break it up into the bottom part, which is a smooth surface, it has continuous derivative, and the sides which kind of wraps around is also is also a smooth surface so most things you'll encounter in a traditional calculus course actually do, especially surfaces, do fit piecewise-smooth. And the thing is though actually very hard to visualize. I imagine this all outer pointy fractely looking things where it's hard to break it up into pieces that are actually smooth. That's for surface part but we also have to care about the boundary, in order to apply Stokes' Theorem. And that is that right over there. The boundary needs to be a simple, which means that doesn't cross itself, a simple closed piecewise-smooth boundary. So once again: simple and closed that just means so this is not a simple boundary. If it is really crossing itself or intersecting itself, although you can break it up into to tow simple boundaries. But something like this is a simple boundary. So that is a simple boundary right over there. It also have to be closed wich really means that just loops in on itself. You just have something like that. It actually has to close and actually has to loops in on itself. In order to use Stokes' Theorem and once again it has to be piecewise-smooth but now we are talking about a path or a line or curve like this and a piecewise-smooth just means that you can break it up into sections were derivatives are continuous. The way I've drawm this one, this one and this one, the slope is changing gradually. So over there the slope is like that. It is changing gradually as we go around this path. Something that is not smooth, a path that is not smooth might look something like this. Might look something like that. And the places that this aren't smooth are at the edges: not smooth there, not smooth there and not smooth there. But we have to be simple-closed and this is simple and closed. And it's not smooth but it is piecewise-smooth. We can break it up into this section of the path. Which is that line right over there is smooth, that line over there is smooth, that line is smooth, and that line is smooth. And we've done that when evaluating in line integrals. We broke it up into smooth segments that we can then use to actually compute line integral. So if you find... if you have a boundary where the... if you have a surface that is piecewise-smooth and its boundary is a simple-closed piecewise-smooth boundary, you're good to go.