The Idea Behind Stokes’ Theorem
Math Insight
Stokes' theorem is a generalization of Green's theorem1 from circulation in a
planar region to circulation along a surface2.
Green's theorem states that, given a continuously differentiable two-dimensional
vector field F, the integral of the “microscopic circulation” of F over the
region D inside a simple3 closed4 curve C is equal to the total
circulation5 of F around C, as suggested by the equation
We often write that C=∂D as fancy notation meaning simply that C is the
boundary of D. Green's theorem requires that C=∂D.
The “microscopic circulation” in Green's theorem is captured by the curl6 of the
vector field and is illustrated by the green circles in the below figure.
Green's theorem applies only to two-dimensional vector fields and to regions in
the two-dimensional plane. Stokes' theorem generalizes Green's theorem to
three dimensions. For starters, let's take our above picture and simply embed it in
three dimensions. Then, our curve C becomes a curve in the xy-plane, and our
region D becomes a surface S in the xy-plane whose boundary is the curve C.
Even though S is now a surface, we still use the same notation as ∂ for the
boundary. The boundary ∂S of the surface S is a closed curve, and we require
that ∂S=C.
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The next question is what the microscopic circulation along a surface should be.
For Green's theorem, we found that
“microscopic circulation”=(curlF)⋅k,
(where k is the unit-vector in the z-direction). We wanted the component of the
curl7 in the k direction because this corresponded to microscopic circulation in
the xy-plane. Similarly, for a surface, we will want the microscopic circulation
along the surace. This corresponds to the component of the curl that
is perpendicular to the surface, i.e,
“microscopic circulation”=(curlF)⋅n,
where n is a unit normal vector8 to the surface. You can see this using the righthand rule. If you point the thumb of your right hand perpendicular to a surface,
your fingers will curl in a direction corresponding to circulation parallel to the
surface.
In summary, to go from Green's theorem to Stoke's theorem, we've made two
changes. First, we've changed the line integral living in two dimensions (Green's
theorem) to a line integral living in three dimensions (Stokes' theorem). Second,
we changed the double integral of curlF⋅k over a region D in the plane (Green's
theorem) to a surface integral of curlF⋅n over a surface floating in space (Stokes'
theorem). The required relationship between the curve C and the
surface S (Stokes' theorem) is identical to the relationship between the
curve C and the region D (Green's theorem): the curve C must be the
boundary ∂D of the region or the boundary ∂S of the surface.
We write Stokes' theorem as:
(Recall that a surface integral9 of a vector field is the integral of the component of
the vector field perpendicular to the surface.) We see that the integral on the right
is the surface integral of the vector field curlF. Stokes theorem says the surface
integral of curlF over a surface S (i.e., ∬ScurlF⋅dS) is the circulation of F around
the boundary of the surface (i.e., ∫CF⋅ds where C=∂S ).
Once we have Stokes' theorem, we can see that the surface integral of curlF is a
special integral. The integral cannot change if we can change the surface S to
any surface as long as the boundary of S is still the curve C. It cannot change
Source URL: http://mathinsight.org/stokes_theorem_idea
Saylor URL: http://www.saylor.org/courses/ma103/
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because it still must be equal to ∫CF⋅ds, which doesn't change if we don't
change C. (In analogy of how the gradient ∇f is a path-independent vector field10,
you could say that curlF is “surface independent” vector field, but we don't usually
use that term.)
For example, staring with a planar surface such as sketched above, we see that
the surface S doesn't have to be the flat surface inside C. We can bend and
stretch S, and the above formula is still true. In the below applet, you can move
the blue point on the slider to change the surface S. For any of those surfaces,
the integral of the “microscopic circulation” curlF over that surface will be the total
circulation ∫CF⋅ds of F around the curve C (shown in red). The important
restriction is that the boundary of the surface S is still the curve C.
Macroscopic and microscopic circulation in three dimensions. The relationship between the
macroscopic circulation of a vector field F around a curve and the microscopic circulation
of F (illustrated by small green circles) along a surface in three dimensions must hold for any
surface whose boundary is the curve. No matter which surface you choose, the total microscopic
circulation of F along the surface must equal the circulation of F around the curve, as long as the
vector field F is defined everywhere on the surface.
Stokes' theorem allows us to do even more. We don't have to leave the
curve C sitting in the xy-plane. We can twist and turn C as well. If S is a surface
whose boundary is C (i.e., if C=∂S), it is still true that
Source URL: http://mathinsight.org/stokes_theorem_idea
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
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Changing surfaces in Stokes' theorem. Since Stokes' theorem states that the integral of the
microscopic circulation curlFover a surface is equal to the circulation of F around the surface
boundary (in red), changing the surface without changing the boundary cannot change the
surface integral of curlF. On the other hand, if you change the boundary then both the surface
integral and the line integral defining the circulation around the boundary do change, though they
are still equal to each other.
Note that moving the blue dot on the slider does not change the value of either
integral in the above formulas. Since the curve C does not change, the left line
integral doesn't change, which means the value of the right surface integral
cannot change. On the other hand, moving the green dot on the slider does
change the values of the integrals since the curve C changes. The important
point is that, even in this case, the left line integral and the right surface integral
are always equal.
There is one more subtlety that you have to get correct, or else you may be off by
a sign. You need to orient the surface and boundary properly11.
You can read some examples here12.
Source URL: http://mathinsight.org/stokes_theorem_idea
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
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Notes and Links
1. http://mathinsight.org/greens_theorem_idea
2. http://mathinsight.org/parametrized_surface_introduction
3. http://mathinsight.org/simple_curve_definition
4. http://mathinsight.org/closed_curve_definition
5. http://mathinsight.org/line_integral_circulation
6. http://mathinsight.org/curl_idea
7. http://mathinsight.org/curl_components
8. http://mathinsight.org/parametrized_surface_normal_vector
9. http://mathinsight.org/surface_integral_vector_field_introduction
10. http://mathinsight.org/conservative_vector_field_introduction
11. http://mathinsight.org/stokes_theorem_orientation
12. http://mathinsight.org/stokes_theorem_examples
Source URL: http://mathinsight.org/stokes_theorem_idea
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 5 of 5