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Multivariable calculus
Course: Multivariable calculus > Unit 5
Lesson 5: Stokes' theorem- Stokes' theorem intuition
- Green's and Stokes' theorem relationship
- Orienting boundary with surface
- Orientation and stokes
- Orientations and boundaries
- Conditions for stokes theorem
- Stokes example part 1
- Stokes example part 2
- Stokes example part 3
- Stokes example part 4
- Stokes' theorem
- Evaluating line integral directly - part 1
- Evaluating line integral directly - part 2
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Orientation and stokes
Determining the proper orientation of a boundary given the orientation of the normal vector. Created by Sal Khan.
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Why is Sal drawing 3D bodies? I thought Stokes theorem deals with 2D surfaces in 3D space.(12 votes)
You are correct in that it applies to surfaces in 3D. However, Sal isn't drawing "volumes." He is simply sketching a surface (in purple) and its boundary (in blue). It sort of looks like a 3D body, but it's not filled in.(29 votes)
for the end of the video, I've learned in physics and in a course of mechanics whats called the rule of the right hand (or whatever it's called in english) for the momentum . anyway, your thumb is the direction of the normal vector and the four fingers (when you twist your wrist) show the way you "walk" to. can i apply this rule here or is it just a coincidence that the two rules match?(8 votes)
Yes, you can apply the rule here. Counterclockwise direction is usually considered to be positive.(5 votes)
Can we not just use the right hand rule?(6 votes)
A lot of people have been asking this question. Yes, you can just use the right-hand rule. I strongly prefer it, for it is very intuitive.(6 votes)
- Am I becoming delirious from studying for my final or is this little man really walking the wrong way on this video to be keeping the surface to his left?(2 votes)
- Try your keyboard and a pencil. First walk along with your pencil around your keyboard(the side with your keys) with the keyboard on the left of your pencil (make sure you know what you define as the left side of your pencil). Now take the backside of your keyboard ( the side with no keys and do the same). Counterclockwise in the first case, it becomes clockwise in the second.
Visualizing it is cumbersome, so I just use the right hand rule. :)(2 votes)
- Can't we use the right-hand rule? It seems it works perfectly.(2 votes)
- plz tell me about volume integrals ....to find volume integrals,..(1 vote)
A volume integral is simply a triple integral. For example, think of some mass density, p, which is a function of x, y and z - let p = xyz.
Now, say you want to find the total amount of mass in a cubic region bounded by the origin, and a point at (2,4,6). You can draw this as a cuboid in the xyz plane.
You can set up a volume integral of p with respect to x, y and z, with 3 limits for x (0 to 2), y (0 to 4) and z (0 to 6).
Now, you solve the integral in any order - for simplicity integrate with respect to (wrt) x first (ie. (x^2)/2), then you can factor out that x term leaving only yz inside the integral - you can calculate the limits of the x term outside of the integral. Then solve for y, factor out, and finally solve for z.
If you'd like to give the above example a try, let p = 5xyz, and use the same cuboid (0,0,0) to (2,4,6), and you should find that the volume integral solves to 1440kg.(2 votes)
The little guy's head should point to the normal of where he is. The first example, where the normal is pointing up, by the time the point moves to where the little guy is, it should be twisted to pointing outside, i.e. downward. Same for the second example, the normal is pointing inside.
The bottle cap example, are we twisting the bottle or the cap? If we hold the cap stationary and twist the bottle instead, we get the other result. After applying the right-hand rule, we come to the conclusion that the bottle stays and cap turns.
Thanks for the video!(1 vote)- there is also the right hand trick:
if you imagine that you grab the surface with your four fingers oriented in the direction of the rotation, your thumb indicate the normal vector orientation! ;P(1 vote) - Enjoyed the bottle analogy. Lefty loosy righty tighty(1 vote)
How does orientation relate to physics ?(1 vote)
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.