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Studying for a test? Prepare with these 11 lessons on Green's, Stokes', and the divergence theorems.

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# Orienting boundary with surface

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.