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.