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This is the 3d version of Green's theorem, relating the surface integral of a curl vector field to a line integral around that surface's boundary.
Background
- Green's theorem
- Flux in three dimensions
- Curl in three dimensions
Not strictly required, but very helpful for a deeper understanding:
- Formal definition of curl in three dimensions
This article is for physical intuition
If you would like examples of using Stokes' theorem for computations, you can find them in the next article. Here, the goal is to present the theorem in such a way that you can get a gut feeling for what it is really saying, and why it is true.
What we're building to
- Stokes' theorem is the 3D version of Green's theorem.
- It relates the surface integral of the curl of a vector field with the line integral of that same vector field around the boundary of the surface:
is a three-dimensional vector field.
means the same thing as . It is the three-dimensional curl of , which is a vector field.
is a surface in three dimensions.
represents a function that gives unit normal vectors to .
is the boundary of
is oriented using the right-hand rule, meaning if you point the thumb of your right hand in the direction of a unit normal vector near the edge of and curl your fingers, the direction they point indicates the direction you should integrate around .
Interpreting a line integral in 3D
Let
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Think of this vector field as being the velocity vector of some gas, whooshing about through space.
Now let
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How can you interpret the line integral of
Well, first of all, this integral doesn't make sense until the curve is oriented. The differential vector
Imagine you are a bird, flying through space along the curve
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Think of each step (wing-flap?) of your motion along
Now look back at the line integral I originally asked about:
You can think of this as adding up how helpful or burdensome the wind was during your flight. It measures the tendency of the fluid flow to circulate around
Chopping up a surface
Those of you who read the Green's theorem article will find what follows very familiar.
Consider a surface
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Slice this surface in half, and name the boundaries of the two resulting pieces
The portions of
More generally, imagine slicing up
The line integrals around all of these little loops will cancel out along the slices within
Curl on each piece
The reason for chopping up
Name the boundary of this piece
.Choose some point
on the surface, inside this little loop.Let
be a unit normal vector to the surface at the point . "Pointing which way?", you might ask. Curl the fingers of your right hand around the little loop so that they align with its orientation. Stick out your thumb, and this is the direction of .Let
represent the area of this little piece (in anticipation of using an infinitesimal area for a surface integral in just a bit).
Then the line integral of
If you feel uneasy about your intuition for what curl means, or how a vector can represent rotation, consider reviewing this article on curl.
Here's the loose intuition for why this approximation works:
When we take the dot product between this curl vector and
Actually, that line integral produces a really small number (since
(For a deeper understanding of this approximation, take a look at the formal definition of curl in three dimensions.)
Surface integral of curl
Combining the ideas of the last two sections, here's what we get:
As we chop things up more and more finely, this last sum approaches the surface integral of
Putting this together, we have the following marvelous equation, known as Stokes' theorem:
Okay, a few of you may raise the objection that I started with an approximation for the line integral around each piece, and am now making a conclusion using an approximation-free equality. And you'd be right to do so!
In the article on Green's theorem, which involves a nearly identical line of reasoning, but in two dimensions, I offered a couple notes to go into the details of how the approximation disappears. If you are curious, I encourage you to go back through that same line of reasoning and think about how it works for Stokes' theorem and surface integrals.
This exercise will also be made all the more enlightening if you go in armed with the formal definition of curl in three dimensions.
Aligning orientation
Surfaces are oriented by the chosen direction for their unit normal vectors. For example, you will often see a surface oriented using outward-facing unit normal vectors (although not all surfaces have a notion of outward-facing vs. inward-facing unit normal vectors).
Curves are oriented by the chosen direction for their tangent vectors.
For Stokes' theorem to work, the orientation of the surface and its boundary must "match up" in the right way. Otherwise, the equation will be off by a factor of
If you look at the surface in such a way that the unit normal vectors are all pointed towards your face, the curve should be oriented counterclockwise.
The curve's orientation should follow the right-hand rule, in the sense that if you stick the thumb of your right hand in the direction of a unit normal vector near the edge of the surface, and curl your fingers, the direction they point on the curve should match its orientation.
When you are walking along the boundary curve with your body pointing out in the direction of the unit normal vector, you should be walking in such a way that the surface is to your left side.
Blowing bubbles
Here's something pretty awesome about Stokes' theorem: The surface itself doesn't matter, all that matters is what its boundary is.
For example, imagine a particular loop through space, and think about all the different surfaces that could have this loop as a boundary; all the different soap bubbles which could emerge from this one loop:
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For any given vector field
What this tells you is just how special curl vector fields are, since with most vector fields, the surface integral absolutely depends on the specific surface at hand. If you learned about conservative vector fields, this is analogous to path independence, and how it indicates just how special gradient vector fields are.
What if there is no boundary?
If you have a closed surface, like a sphere or a torus, then there is no boundary. This means the "line integral over the boundary" is zero, and Stokes' theorem reads as follows:
If you think back to chopping up the surface to get many tiny little line integrals, this basically says all those little line integrals cancel out with nothing left to show for their work.
Summary
- Stokes' theorem is the 3D version of Green's theorem.
The line integral
tells you how much a fluid flowing along tends to circulate around the boundary of the surface .The left-hand side surface integral can be seen as adding up all the little bits of fluid rotation on the surface
itself. The vector describes the fluid rotation at each point, and dotting it with a unit normal vector to the surface, , extracts the component of that fluid rotation which happens on the surface itself.
Log in Andrew 8 years agoPosted 8 years ago. Direct link to Andrew's post “Who is the author of thes...” Who is the author of these articles? Very well done! • (43 votes) David O'Connor 8 years agoPosted 8 years ago. Direct link to David O'Connor's post “I suspect it's Grant Sand...” I suspect it's Grant Sanderson: http://www.3blue1brown.com/about/ (38 votes) Alexander Wu 8 years agoPosted 8 years ago. Direct link to Alexander Wu's post “Does Strokes' theorem hav...” Does Strokes' theorem have something to do with Gauss' law of magnetism? I learned it as ∮B · d A = 0, but maybe it should have been ∯B · n dΣ = 0 (over S, I can't type it under the integral). • (7 votes) Paul Zander 8 years agoPosted 8 years ago. Direct link to Paul Zander's post “Yes. Good question!Strok...” Yes. Good question! (9 votes) eugene 6 years agoPosted 6 years ago. Direct link to eugene's post “I think it's crazy to say...” I think it's crazy to say that the area of a surface is the same as that of a circumference of a boundary line on the same 3D object. Like...we know this is intuitively false. So that's why I don't think that the Stoke's theorem is saying this. But I think that's the intuition and conceptual picture I (and other students maybe) have. I think there's some kind of "spcial-ness" of a boundary line and that it is not just a circumference of an irregular shaped circle round a 3D object. I wish this could be clarified. • (4 votes) joh14192 7 years agoPosted 7 years ago. Direct link to joh14192's post “I'm confused why stoke's ...” I'm confused why stoke's theorem calculates the flux and not just circulation in 3d? Because green's theorem calculates circulation so wouldn't stokes an extension of that do the same? • (3 votes) Paras Sharma 6 years agoPosted 6 years ago. Direct link to Paras Sharma's post “Hey Stokes theorem doesn'...” Hey Stokes theorem doesn't calculates the flux, it can be done by divergence theorem instead it tells us how much of the vector field is with us as we move along the closed curve C (mainly counterclockwise)..... (3 votes) khosro06001 4 years agoPosted 4 years ago. Direct link to khosro06001's post “What a wonderful expositi...” What a wonderful exposition! • (3 votes) saalimqadri.m.q.s 7 years agoPosted 7 years ago. Direct link to saalimqadri.m.q.s's post “How is ds equal to curl o...” How is ds equal to curl of partial derivatives .. • (3 votes) William Martucci 6 years agoPosted 6 years ago. Direct link to William Martucci's post “How does dΣ, a small piec...” How does dΣ, a small piece of surface area, play into evaluating overall rotation? I know that dΣ is the differential used for surface integrals, but I don't understand how it applies here. • (2 votes) Annoeska Hameete 7 years agoPosted 7 years ago. Direct link to Annoeska Hameete's post “What should I do when the...” What should I do when the direction of the unit vector and tangent vector don't match up? Devide by -1? • (2 votes) alek aleksander 6 years agoPosted 6 years ago. Direct link to alek aleksander's post “There's a catch with the ...” There's a catch with the "Walk this way around C" picture. It's wrong. :) • (1 vote)Want to join the conversation?
Strokes' theorem is very useful in solving problems relating to magnetism and electromagnetism. BTW, pure electric fields with no magnetic component are conservative fields. Maxwell's Equations contain both curl and divergence.
I saw on the book ds=dxdy/n.k^ which i also couldn't understand