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Also known as circulation theorem, curl theorem, Kelvin–Stokes theorem
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.
Article
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 Σ is an oriented surface with boundary ∂Σ, 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.
Contents
Theorem
Let
Σ
{\displaystyle \Sigma }
be a smooth oriented surface in
R
3
{\displaystyle \mathbb {R} ^{3}}
, parametrized by
Σ
(
u
,
v
)
{\displaystyle \mathbf {\Sigma } (u,v)}
, with boundary
∂
Σ
≡
Γ
{\displaystyle \partial \Sigma \equiv \Gamma }
, parametrized by
Γ
(
t
)
{\displaystyle \mathbf {\Gamma } (t)}
. If a vector field
F
(
x
,
Special case of a more general theorem
Stokes' theorem can be viewed as a special case of the following identity:
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
As in § Theorem, we reduce the dimension by using the natural parametrization of the surface. Let ψ and γ be as in that section, and note that by change of variables
can be identified with the differential 1-forms on
R
3
{\displaystyle \mathbb {R} ^{3}}
via the map
F
x
e
1
+
F
y
e
2
+
F
z
e
3
↦
F
x
d
x
+
F
y
d
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 F on an open set
U
⊆
R
3
{\displaystyle U\subseteq \mathbb {R} ^{3}}
is irrotational (lamellar vector field) if ∇ × F = 0.
This concept is very fundamental in mechanics; as we'll prove later, if F is irrotational and the domain of F is simply connected, then F is a conservative vector field.
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
U
⊆
R
3
{\displaystyle U\subseteq \mathbb {R} ^{3}}
be an open subset with a lamellar vector field F and let c0, c1: [0, 1] → U be piecewise smooth loops. If there is a function H: [0, 1] × [0, 1] → U such that
[TLH0] H is piecewise smooth,
[TLH1] H(t, 0) = c0(t) for all t ∈ [0, 1],
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
U
⊆
R
3
{\displaystyle U\subseteq \mathbb {R} ^{3}}
be an open subset, with a Lamellar vector field F and a piecewise smooth loop c0: [0, 1] → U. Fix a point p ∈ U, if there is a homotopy H: [0, 1] × [0, 1] → U such that
Above Lemma 2-2 follows from theorem 2–1. In Lemma 2-2, the existence of H satisfying [SC0] to [SC3] is crucial;the question is whether such a homotopy can be taken for arbitrary loops. If U is simply connected, such H exists. The definition of simply connected space follows:
Maxwell's equations
In the physics of electromagnetism, Stokes' theorem provides the justification for the equivalence of the differential form of the Maxwell–Faraday equation and the Maxwell–Ampère equation and the integral form of these equations. For Faraday's law, Stokes theorem is applied to the electric field,
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
R
2
{\displaystyle \mathbb {R} ^{2}}
.
A more detailed statement will be given for subsequent discussions.
Let
γ
:
[
a
,
b
]
→
R
2
{\displaystyle \gamma :[a,b]\to \mathbb {R} ^{2}}
be a piecewise smooth Jordan plane curve: a simple closed curve in the plane. The Jordan curve theorem implies that
γ
{\displaystyle \gamma }
divides
R
2
{\displaystyle \mathbb {R} ^{2}}
into two components, a compact one and another that is non-compact. Let
D
{\displaystyle D}
denote the compact part; then
D
{\displaystyle D}
is bounded by
γ
{\displaystyle \gamma }
. It now suffices to transfer this notion of boundary along a continuous map to our surface in
R
3
{\displaystyle \mathbb {R} ^{3}}
. But we already have such a map: the parametrization of
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.
Some textbooks such as Lawrence call the relationship between c0 and c1 stated in theorem 2-1 as "homotopic" and the function H: [0, 1] × [0, 1] → U as "homotopy between c0 and c1". 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).
In what follows, we abuse notation and use "
⊕
{\displaystyle \oplus }
" for concatenation of paths in the fundamental groupoid and "
⊖
{\displaystyle \ominus }
" for reversing the orientation of a path.
Let D = [0, 1] × [0, 1], and split ∂D into four line segments γj.
, so that the desired equality follows almost immediately.
Definition 2-2 (simply connected space). Let
M
⊆
R
n
{\displaystyle M\subseteq \mathbb {R} ^{n}}
be non-empty and path-connected. M is called simply connected if and only if for any continuous loop, c: [0, 1] → M there exists a continuous tubular homotopy H: [0, 1] × [0, 1] → M from c to a fixed point p ∈ c; that is,
[SC0'] H is continuous,
[SC1] H(t, 0) = c(t) for all t ∈ [0, 1],
[SC2] H(t, 1) = p for all t ∈ [0, 1],
[SC3] H(0, s) = H(1, s) = p for all s ∈ [0, 1].
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.
Fortunately, the gap in regularity is resolved by the Whitney's approximation theorem. In other words, the possibility of finding a continuous homotopy, but not being able to integrate over it, is actually eliminated with the benefit of higher mathematics. We thus obtain the following theorem.
Theorem 2-2. Let
U
⊆
R
3
{\displaystyle U\subseteq \mathbb {R} ^{3}}
be open and simply connected with an irrotational vector field F. For all piecewise smooth loops c: [0, 1] → U
