Stokes' theorem, also known as the Kelvin–Stokes theorem, is a theorem in vector calculus that relates the behavior of a vector field along the edge of a surface to the behavior of its curl on the surface itself. In its usual three-dimensional form, it says that the total circulation of a vector field around a closed curve is equal to the total curl of the field through a surface bounded by that curve.
If is an oriented surface with boundary , Stokes' theorem is commonly written as
Here the left side is the line integral of the vector field around the boundary curve, while the right side is the surface integral of its curl over the surface. Informally, the theorem says that adding up the local rotation of a vector field across a surface gives the net circulation around its edge.
The theorem is also called the fundamental theorem for curls, the curl theorem, or the rotor theorem. It is a special case of the generalized Stokes theorem. In the language of differential forms, the vector field corresponds to a 1-form and its curl corresponds to the exterior derivative of that form.
Theorem
Let be a smooth oriented surface in , parametrized by , with boundary , parametrized by . If a vector field
has continuous first-order partial derivatives in , then
with the shorthands for the line element and the surface element
where is the vector orthogonal to the surface at the point .
The equality can be expressed in terms of differential forms, with being the wedge product and the exterior derivative:
The main challenge in a precise statement of Stokes' theorem is in defining the notion of a boundary. Surfaces such as the Koch snowflake, for example, are well-known not to exhibit a Riemann-integrable boundary, and the notion of surface measure in Lebesgue theory cannot be defined for a non-Lipschitz surface. One (advanced) technique is to pass to a weak formulation and then apply the machinery of geometric measure theory; for that approach see the coarea formula. In this article, we instead use a more elementary definition, based on the fact that a boundary can be discerned for full-dimensional subsets of .
A more detailed statement will be given for subsequent discussions. Let be a piecewise smooth Jordan plane curve: a simple closed curve in the plane. The Jordan curve theorem implies that divides into two components, a compact one and another that is non-compact. Let denote the compact part; then is bounded by . It now suffices to transfer this notion of boundary along a continuous map to our surface in . But we already have such a map: the parametrization of .
Suppose is piecewise smooth at the neighborhood of From the definition of , is obviously a bounded closed set in . "A neighborhood of D" means "an open set in that contains D." , with . represents the image set of by If is the space curve defined by may not be a Jordan curve if the loop interacts poorly with . Nonetheless, is always a loop or loops, and topologically a connected sum of countably many Jordan curves, so that the integrals are well-defined. then we call the boundary of , written Even if we consider two-dimensional polar coordinates, then if we let ∂Σ = Γ, then ∂Σ can clearly contain a topological interior point of Σ. However, in orientable combinatorial manifolds, this is not such a serious problem, since the line integrals over the interior points cancel out: in an orientable manifold, the line integral over ∂Σ coincides with the line integral over the true boundary.An even more important example is that this theorem also applies to manifolds without a topological boundary, such as the sphere or the torus. In such cases, if ∂Σ = Γ, then ∂Σ is completely contained within Σ. However, in such cases, the resulting line integrals over Γ completely cancel each other out. This allows us to assert that the surface integral of the curl of a vector field on a manifold without a boundary is zero.
With the above notation, if is any smooth vector field on , thenRobert Scheichl, lecture notes for University of Bath mathematics course
Here, the "" represents the dot product in .
Special case of a more general theorem
Stokes' theorem can be viewed as a special case of the following identity:
where is any smooth vector or scalar field in . When is a uniform scalar field, the standard Stokes' theorem is recovered.
Proof The proof of the theorem consists of 4 steps. We assume Green's theorem, so what is of concern is how to boil down the three-dimensional complicated problem (Stokes' theorem) to a two-dimensional rudimentary problem (Green's theorem). When proving this theorem, mathematicians normally deduce it as a special case of a more general result, which is stated in terms of differential forms, and proved using more sophisticated machinery. While powerful, these techniques require substantial background, so the proof below avoids them, and does not presuppose any knowledge beyond a familiarity with basic vector calculus and linear algebra. At the end of this section, a short alternative proof of Stokes' theorem is given, as a corollary of the generalized Stokes' theorem.
Elementary proof First step of the elementary proof (parametrization of integral) As in , we reduce the dimension by using the natural parametrization of the surface. Let and be as in that section, and note that by change of variables
where stands for the Jacobian matrix of at .
Now let be an orthonormal basis in the coordinate directions of .In this article,
Note that, in some textbooks on vector analysis, these are assigned to different things. For example, in some text book's notation, can mean the following respectively. In this article, however, these are two completely different things.
Here,
and the "" represents Euclidean norm.
Recognizing that the columns of are precisely the partial derivatives of at , we can expand the previous equation in coordinates as
Second step in the elementary proof (defining the pullback) The previous step suggests we define the function
Now, if the scalar value functions and are defined as follows,
then,
This is the pullback of along , and, by the above, it satisfies
We have successfully reduced one side of Stokes' theorem to a 2-dimensional formula; we now turn to the other side.
Third step of the elementary proof (second equation) First, calculate the partial derivatives appearing in Green's theorem, via the product rule:
Conveniently, the second term vanishes in the difference, by equality of mixed partials. So,For all , for all square matrix, and therefore .
But now consider the matrix in that quadratic form—that is, . We claim this matrix in fact describes a cross product. Here the superscript "" represents the transposition of matrices.
To be precise, let be an arbitrary matrix and let
Note that is linear, so it is determined by its action on basis elements. But by direct calculation
Here, represents an orthonormal basis in the coordinate directions of .In this article,
Note that, in some textbooks on vector analysis, these are assigned to different things.
Thus for any .
Substituting for , we obtain
We can now recognize the difference of partials as a (scalar) triple product:
On the other hand, the definition of a surface integral also includes a triple product—the very same one!
So, we obtain
Fourth step of the elementary proof (reduction to Green's theorem) Combining the second and third steps and then applying Green's theorem completes the proof. Green's theorem asserts the following: for any region D bounded by the Jordans closed curve γ and two scalar-valued smooth functions defined on D;
We can substitute the conclusion of STEP2 into the left-hand side of Green's theorem above, and substitute the conclusion of STEP3 into the right-hand side. Q.E.D.
Proof via differential forms The functions can be identified with the differential 1-forms on via the map
Write the differential 1-form associated to a function as . Then one can calculate that
where is the Hodge star and is the exterior derivative. Thus, by generalized Stokes' theorem,
History
The theorem first appears in a letter from William Thomson to George Stokes dated 2 July 1850. Stokes set it as question 8 on the 1854 Smith's Prize examination at Cambridge, which led to the result bearing his name.
Applications
Irrotational fields In this section, we will discuss the irrotational field (lamellar vector field) based on Stokes' theorem.
Definition 2-1 (irrotational field). A smooth vector field on an open set is irrotational (lamellar vector field) if .
This concept is very fundamental in mechanics; as we'll prove later, if is irrotational and the domain of is simply connected, then is a conservative vector field.
Helmholtz's theorem In this section, we will introduce a theorem that is derived from Stokes' theorem and characterizes vortex-free vector fields. In classical mechanics and fluid dynamics it is called Helmholtz's theorem.
Theorem 2-1 (Helmholtz's theorem in fluid dynamics). Let be an open subset with a lamellar vector field and let be piecewise smooth loops. If there is a function such that [TLH0] is piecewise smooth, [TLH1] for all , [TLH2] for all , [TLH3] for all . Then,
Some textbooks such as Lawrence call the relationship between and stated in theorem 2-1 as "homotopic" and the function as "homotopy between and ". However, "homotopic" or "homotopy" in above-mentioned sense are different (stronger than) typical definitions of "homotopic" or "homotopy"; the latter omit condition [TLH3]. So from now on we refer to homotopy (homotope) in the sense of theorem 2-1 as a tubular homotopy (resp. tubular-homotopic).
Proof of Helmholtz's theorem
In what follows, we abuse notation and use "" for concatenation of paths in the fundamental groupoid and "" for reversing the orientation of a path.
Let , and split into four line segments .
so that
By our assumption that and are piecewise smooth homotopic, there is a piecewise smooth homotopy
Let be the image of under . That
follows immediately from Stokes' theorem. is lamellar, so the left side vanishes, i.e.
As is tubular(satisfying [TLH3]), and . Thus the line integrals along and cancel, leaving
On the other hand, , , so that the desired equality follows almost immediately.
Conservative forces Above Helmholtz's theorem gives an explanation as to why the work done by a conservative force in changing an object's position is path independent. First, we introduce the Lemma 2-2, which is a corollary of and a special case of Helmholtz's theorem.
Lemma 2-2. Let be an open subset, with a Lamellar vector field and a piecewise smooth loop . Fix a point , if there is a homotopy such that [SC0] is piecewise smooth, [SC1] for all , [SC2] for all , [SC3] for all . Then,
Above Lemma 2-2 follows from theorem 2–1. In Lemma 2-2, the existence of satisfying [SC0] to [SC3] is crucial;the question is whether such a homotopy can be taken for arbitrary loops. If is simply connected, such exists. The definition of simply connected space follows:
Definition 2-2 (simply connected space). Let be non-empty and path-connected. is called simply connected if and only if for any continuous loop, there exists a continuous tubular homotopy from to a fixed point ; that is, [SC0'] is continuous, [SC1] for all , [SC2] for all , [SC3] for all .
The claim that "for a conservative force, the work done in changing an object's position is path independent" might seem to follow immediately if the M is simply connected. However, recall that simple-connection only guarantees the existence of a continuous homotopy satisfying [SC1-3]; we seek a piecewise smooth homotopy satisfying those conditions instead.