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Now that we've explored Stokes' Theorem a little bit, I want to talk about

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the situations in wich we can use it. You'll see that this is pretty general

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theorem. But we do have to thing about what type of surfaces

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and what type of boundary are those surfaces we are actually dealing with

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and the case of Stokes', we need

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surfaces that are piecewise...

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piecewise-smooth

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piecewise-smooth surfaces so this surface right over here

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it is actually smooth not just even piecewise-smooth.

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Sounds like a very fancy term but all the smooth part means

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that you have just continuous derivatives

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and since we are talking about surfaces we're going to have continuous

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partial derivatives regardless of which direction you pick.

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So this is continuous derivatives,

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and another way to think about that conceptually is if you pick a direction

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on the surface if you say that we go in that direction, the slope in that

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direction changes gradually, doesn't jump around. If you pick this direction right over here,

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the slope is changing gradually. So we have a continuous

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derivative. And you're like "what does the 'piecewise' means?"

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Well, the piecewise actually allow us to use Stokes' Theorem with more surfaces.

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Because if you have a surface that looks like... Let's say a surface that looks like

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this. Let's say looks like a cup. So this is the opening of the top of the cup

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let's say that has no opening on top so we can see the backside of the cup

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and this is the side of the cup and this right over here is the bottom

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of the cup and if it was transparent we could actually see through it. So

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surfaces like this is not entirely smooth because it has

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edges. There are points right over here. So this edge

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right over here... If we pick this... let's say if we pick this direction to go and if we go

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this direction along the bottom, then right we get to the edge, all of the sudden the slope changes dramatically

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jumps. So the slope is not continuous at that edge. The slope jumps

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and we start going straight up. And so this entire

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surface is not smooth. But the piecewise actually give us

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an out. This tell us that it's okay as long as we can break

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the surfaces up into pieces that are smooth. And so

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this cup we can break it up but we were doing this wen tackling surfaces integrals

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we can break it up into the bottom part, which is a smooth

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surface, it has continuous derivative, and the sides

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which kind of wraps around is also

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is also a smooth surface

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so most things you'll encounter in a traditional calculus course

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actually do, especially surfaces, do fit piecewise-smooth. And the thing is

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though actually very hard to visualize. I imagine this all outer pointy

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fractely looking things where it's hard to break it up into pieces that are

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actually smooth. That's for surface part but we also

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have to care about the boundary, in order to apply Stokes' Theorem. And that is that

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right over there. The boundary needs to be a simple,

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which means that doesn't cross itself, a simple closed

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piecewise-smooth boundary. So once

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again: simple and closed that just means so this is not a

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simple boundary. If it is really crossing itself or intersecting itself, although you can break it up

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into to tow simple boundaries. But something like this is a simple

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boundary. So that is a simple boundary right over there. It also have to be closed

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wich really means that just loops in on itself. You just have something like that. It actually has to

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close and actually has to loops in on itself. In order to use Stokes' Theorem and once again

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it has to be piecewise-smooth but now we are talking about a path or a

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line or curve like this and a piecewise-smooth just means that you can break it up

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into sections were derivatives are continuous. The way I've drawm

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this one, this one and this one, the slope is changing gradually. So over there

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the slope is like that. It is changing gradually as we go around

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this path. Something that is not smooth, a path that is not

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smooth might look something like this.

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Might look something like that. And the places that this aren't smooth are

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at the edges: not smooth there, not smooth there and not smooth there.

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But we have to be simple-closed and this is simple and closed. And it's not

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smooth but it is piecewise-smooth. We can break it up into

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this section of the path. Which is that line right over there is smooth,

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that line over there is smooth, that line is smooth, and that line is

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smooth. And we've done that when evaluating in line integrals. We broke it up into

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smooth segments that we can then use to actually compute line

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integral. So if you find... if you have a boundary where the...

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if you have a surface that is piecewise-smooth and its

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boundary is a simple-closed piecewise-smooth

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boundary, you're good to go.