Kelvin–Stokes theorem

• For a more conventional discussion see Stokes Theorem.

The Kelvin–Stokes theorem,^{[1][2][3][4][5]} also known as the curl theorem,^[6] is a theorem in vector calculus on R³. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The Kelvin–Stokes theorem is a special case of the “generalized Stokes' theorem.”^[7][8] In particular, the vector field on R³ can be considered as a 1-form in which case curl is the exterior derivative.

Theorem

Let γ : [a, b] → R² be a Piecewise smooth Jordan plane curve, that bounds the domain D ⊂ R².^{[note 1]} Suppose ψ : D → R³ is smooth, with S := ψ[D] ^{[note 2]} , and Γ is the space curve defined by Γ(t) = ψ(γ(t)).^{[note 3]} If F a smooth vector field on R³, then^[1][2][3]

\oint_\Gamma \mathbf{F}\, d\Gamma = \iint_S \nabla\times\mathbf{F}\, dS

Proof

The proof of the Theorem consists of 4 steps.^{[2][3][note 4]} We assume that the Green's theorem is known, so what is of concern is "how to boil down the three-dimensional complicated problem (Kelvin–Stokes theorem) to a two-dimensional rudimentary problem (Green's theorem)". In an ordinary, mathematicians use the differential form, especially "pull-back^{[note 4]} of differential form" is very powerful tool for this situation, however, learning differential form needs too much background knowledge. So, the proof below does not require background information on differential form, and may be helpful for understanding the notion of differential form.

First step of the proof (defining the pullback)

Define

\mathbf{P}(u,v) = (P_1(u,v), P_2(u,v))

so that P is the pull-back^{[note 4]} of F, and that P(u, v) is R²-valued function, depends on two parameter u, v. In order to do so we define P₁ and P₂ as follows.

P_1(u,v)=\left\langle \mathbf{F}(\psi(u,v)) \bigg| \frac{\partial \psi}{\partial u} \right\rangle, \qquad P_2(u,v)=\left\langle \mathbf{F}(\psi(u,v)) \bigg| \frac{\partial \psi}{\partial v} \right\rangle

Where, \langle \ |\ \rangle is the normal inner product of R³ and hereinafter, \langle \ |A|\ \rangle stands for bilinear form according to matrix A ^{[note 5]} .^{[note 6]}

Second step of the proof (first equation)

According to the definition of line integral,

\begin{align} \oint_{\Gamma} \mathbf{F} d\Gamma &=\int_a^b \left\langle (\mathbf{F}\circ \psi (t))\bigg|\frac{d\Gamma}{dt}(t) \right\rangle \, dt \\ &= \int_a^b \left\langle (\mathbf{F}\circ \psi (t))\bigg|\frac{d(\psi\circ\gamma)}{dt}(t) \right\rangle \, dt \\ &= \int_a^b \left\langle (\mathbf{F}\circ \psi (t))\bigg|(J\psi)_{\gamma(t)}\cdot \frac{d\gamma}{dt}(t) \right\rangle \, dt \end{align}

where, Jψ stands for the Jacobian matrix of ψ. Hence,^{[note 5][note 6]}

\begin{align} \left\langle (\mathbf{F}\circ \Gamma(t))\bigg|(J\psi)_{\gamma(t)}\frac{d\gamma}{dt}(t) \right\rangle &= \left\langle (\mathbf{F}\circ \Gamma (t))\bigg|(J\psi)_{\gamma(t)} \bigg|\frac{d\gamma}{dt}(t) \right\rangle \\ &= \left\langle ({}^{t}\mathbf{F}\circ \Gamma (t))\cdot(J\psi)_{\gamma(t)}\ \bigg|\ \frac{d\gamma}{dt}(t) \right\rangle \\ &= \left\langle \left( \left\langle (\mathbf{F}(\psi(\gamma(t))))\bigg|\frac{\partial\psi}{\partial u}(\gamma(t)) \right\rangle , \left\langle (\mathbf{F}(\psi(\gamma(t))))\bigg |\frac{\partial\psi}{\partial v}(\gamma(t)) \right\rangle \right) \bigg|\frac{d\gamma}{dt}(t)\right\rangle \\ &= \left\langle (P_1(u,v) , P_2(u,v))\bigg|\frac{d\gamma}{dt}(t)\right\rangle\\ &= \left\langle \mathbf{P}(u,v)\ \bigg|\frac{d\gamma}{dt}(t)\right\rangle \end{align}

So, we obtain following equation

\oint_{\Gamma} \mathbf{F} d\Gamma = \oint_{\gamma} \mathbf{P} d\gamma

Third step of the proof (second equation)

First, calculate the partial derivatives, using Leibniz rule of inner product

\begin{align} \frac{\partial P_1}{\partial v} &= \left\langle \frac{\partial (\mathbf{F}\circ \psi)}{\partial v} \bigg | \frac{\partial \psi}{\partial u} \right\rangle + \left\langle \mathbf{F}\circ \psi \bigg | \frac{\partial^2 \psi}{ \partial v \partial u} \right\rangle \\ \frac{\partial P_2}{\partial u} &= \left\langle \frac{\partial (\mathbf{F}\circ \psi)}{\partial u} \bigg | \frac{\partial \psi}{\partial v} \right\rangle + \left\langle \mathbf{F}\circ \psi \bigg | \frac{\partial^2 \psi}{\partial u \partial v} \right\rangle \end{align}

So,^{[note 5] [note 6] [note 7]}

\begin{align} \frac{\partial P_1}{\partial v} - \frac{\partial P_2}{\partial u} &= \left\langle \frac{\partial (\mathbf{F}\circ \psi)}{\partial v} \bigg| \frac{\partial \psi}{\partial u} \right\rangle - \left\langle \frac{\partial (\mathbf{F}\circ \psi)}{\partial u} \bigg| \frac{\partial \psi}{\partial v} \right\rangle \\ &= \left\langle (J\mathbf{F})_{\psi(u,v)}\cdot \frac{\partial \psi}{\partial v} \bigg |\frac{\partial \psi}{\partial u} \right\rangle - \left\langle (J\mathbf{F})_{\psi(u,v)}\cdot \frac{\partial \psi}{\partial u} \bigg|\frac{\partial \psi}{\partial v} \right\rangle && \text{ chain rule}\\ &= \left\langle \frac{\partial \psi}{\partial u} \bigg|(J\mathbf{F})_{\psi(u,v)} \bigg| \frac{\partial \psi}{\partial v} \right\rangle - \left\langle \frac{\partial \psi}{\partial u} \bigg |{}^{t}(J\mathbf{F})_{\psi(u,v)} \bigg| \frac{\partial \psi}{\partial v} \right\rangle \\ &= \left\langle \frac{\partial \psi}{\partial u}\bigg |(J\mathbf{F})_{\psi(u,v)} - {}^{t}{(J\mathbf{F})}_{\psi(u,v)} \bigg| \frac{\partial \psi}{\partial v} \right\rangle \\ &= \left\langle \frac{\partial \psi}{\partial u}\bigg |\left ((J\mathbf{F})_{\psi(u,v)} - {}^t (J\mathbf{F})_{\psi(u,v)} \right )\cdot \frac{\partial \psi}{\partial v} \right\rangle \\ &= \left\langle \frac{\partial \psi}{\partial u}\bigg |(\nabla\times\mathbf{F})\times\frac{\partial \psi}{\partial v} \right\rangle && \left ( (J\mathbf{F})_{\psi(u,v)} - {}^{t} (J\mathbf{F})_{\psi(u,v)} \right ) \cdot \mathbf{x} = (\nabla\times\mathbf{F})\times \mathbf{x} \\ &=\det \left [ (\nabla\times\mathbf{F})(\psi(u,v)) \quad \frac{\partial\psi}{\partial u}(u,v) \quad \frac{\partial\psi}{\partial v}(u,v) \right ] && \text{ Scalar Triple Product} \end{align}

On the other hand, according to the definition of surface integral,

\begin{align} \iint_S (\nabla\times\mathbf{F}) \, dS &=\iint_D \left\langle (\nabla\times\mathbf{F})(\psi(u,v)) \bigg |\frac{\partial\psi}{\partial u}(u,v)\times \frac{\partial\psi}{\partial v}(u,v)\right\rangle \,du\,dv\\ &= \iint_D \det \left [ (\nabla\times\mathbf{F})(\psi(u,v)) \quad \frac{\partial\psi}{\partial u}(u,v) \quad \frac{\partial\psi}{\partial v}(u,v) \right ] \,du \,dv && \text{ scalar triple product} \end{align}

So, we obtain

\iint_S (\nabla\times\mathbf{F}) \,dS =\iint_D \left( \frac{\partial P_2}{\partial u} - \frac{\partial P_1}{\partial v} \right) \,du\,dv

Fourth step of the proof (reduction to Green's theorem)

According to the result of the second step, and according to the result of Third step, and further considering the Green's theorem, subjected equation is proved.

Application for conservative force and scalar potential

In this section, we will discuss the lamellar vector field based on Kelvin–Stokes theorem.

First, we define the notarization map, \theta_:[0,1]\to[a,b] as follows.

\theta_=s(b-a)+a

Above-mentioned \theta_ is a strongly increasing function that, for all piece wise smooth paths c:[a,b] → R³, for all smooth vector field F, domain of which includes c (image of [a,b] under c.), following equation is satisfied.

\int_c \mathbf{F}\ dc\ =\int_{c\circ\theta_}\ \mathbf{F}\ d(c\circ\theta_)

So, we can unify the domain of the curve from the beginning to [0,1].

The Lamellar vector field

In mechanics a lamellar vector field is called a conservative force; in fluid dynamics, it is called a Vortex-free vector field. So, lamellar vector field, conservative force, and vortex-free vector field are the same notion.

Helmholtz's theorems

In this section, we will introduce a theorem that is derived from the Kelvin–Stokes theorem and characterizes vortex-free vector fields. In fluid dynamics it is called Helmholtz's theorems,.^{[note 8]}

That theorem is also important in the area of Homotopy theorem.^[7]

Some textbooks such as Lawrence^[7] call the relationship between c₀ and c₁ stated in Theorem 2-1 as “homotope”and the function H : [0, 1] × [0, 1] → U as “homotopy between c₀ and c₁”.

However, “homotope” or “homotopy” in above-mentioned sense are different toward (stronger than) typical definitions of “homotope” or “homotopy”.^{[note 9]}

So there are no appropriate terminology which can discriminate between homotopy in typical sense and sense of Theorem 2-1. So, in this article, to discriminate between them, we say “Theorem 2-1 sense homotopy as tube-like-homotopy and, we say “Theorem 2-1 sense homotope” as tube-like homotope.^{[note 10]}

Proof of the theorem

The definitions of γ₁, ..., γ₄

Hereinafter, the ⊕ stands for joining paths ^{[note 11]} the \ominus stands for backwards of curve ^{[note 12]}

Let D = [0, 1] × [0, 1]. By our assumption, c₁ and c₂ are piecewise smooth homotopic, there are the piecewise smooth homogony H : D → M

\begin{align} \begin{cases} \gamma_1:[0, 1]\to D \\ \gamma_1(t) := (t,0) \end{cases}, \qquad &\begin{cases}\gamma_2:[0,1] \to D \\ \gamma_2(s) := (1, s) \end{cases} \\ \begin{cases} \gamma_3:[0, 1] \to D \\ \gamma_3(t) := (-t+0+1, 1)\end{cases}, \qquad &\begin{cases}\gamma_4:[0,1] \to D \\ \gamma_4(s) := (0, 1-s)\end{cases} \end{align}

\gamma(t):= (\gamma_1 \oplus \gamma_2 \oplus \gamma_3 \oplus \gamma_4)(t)

\Gamma_i(t):= H(\gamma_{i}(t)), \qquad i=1, 2, 3, 4

\Gamma(t):= H(\gamma(t)) =(\Gamma_1 \oplus \Gamma_2 \oplus \Gamma_3 \oplus \Gamma_4)(t)

And, let S be the image of D under H. Then,

\oint_{\Gamma} \mathbf{F}\, d\Gamma = \iint_S \nabla\times\mathbf{F}\, dS

will be obvious according to the Theorem 1 and, F is Lamellar vector field that, right side of that equation is zero, so,

\oint_{\Gamma} \mathbf{F}\, d\Gamma =0

Here,

\oint_{\Gamma} \mathbf{F}\, d\Gamma =\sum_{i=1}^{4} \oint_{\Gamma_i} \mathbf{F} d\Gamma ^{[note 11]}

and, H is Tubeler-Homotopy that,

\Gamma_{2}(s)= {\Gamma}_{4}(1-s)=\ominus{\Gamma}_{4}(s)

that, line integral along \Gamma_{2}(s) and line integral along \Gamma_{4}(s) are compensated each other^{[note 12]} so,

\oint_ \mathbf{F} d\Gamma +\oint_{\Gamma_3} \mathbf{F} d\Gamma =0

On the other hand,

c_{1}(t)=H(t,0)=H({\gamma}_{1}(t))={\Gamma}_{1}(t)

c_{2}(t)=H(t,1)=H(\ominus{\gamma}_{3}(t))=\ominus{\Gamma}_{3}(t)

that, subjected equation is proved.

Application for conservative force

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, obviously follows from Theorem 2-1. In Lemma 2-2, the existence of H satisfying [SC0] to [SC3]" is crucial. It is a well-known fact that, if U is simply connected, such H exists. The definition of Simply connected space follows:

You will find that, the [SC1] to [SC3] of both Lemma 2-2 and Definition 2-2 is same.

So, someone may think that, the issue, "when the Conservative Force, the work done in changing an object's position is path independent" is elucidated. However there are very large gap between following two.

• There are continuous H such that it satisfies [SC1] to [SC3]
• There are piecewise smooth H such that it satisfies [SC1] to [SC3]

To fill that gap, the deep knowledge of Homotopy Theorem is required. For example, to fill the gap, following resources may be helpful for you.

• Lee teaches Whitney approximation theorem (^[8] page 136) and "How to use that theorem to this isuue" (^[8] page 421).
• More general statements appear in^[9] (see Theorems 7 and 8).

Considering above-mentioned fact and Lemma 2-2, we will obtain following theorem. That theorem is anser for subjecting issue.

Kelvin–Stokes theorem on singular 2-cube and cube subdivisionable sphere

Singular 2-cube and boundary

Given D:=\times, we define the notarization map of sngler two cube {\theta}_{D}:{I}^{2}\to D

{\theta}_{D}({u}_{1},{u}_{2})= \left( \begin{array}{c} {u}_{1}({b}_{1}-{a}_{1}) +{a}_{1}\\ {u}_{2}({b}_{2}-{a}_{2}) +{a}_{2} \end{array} \right)

here, the I:=[0,1] and I² stands for {I}^{2}=I\times I.

Above-mentioned is strongly increase function (that means det(J(\theta_{D})_{({u}_{1},{u}_{2})})>0 (for all ({u}_{1},{u}_{2})\in\mathbb{R}^{3} ) that, following lemma is satisfied.

Above-mentioned lemma is obverse that, we neglects the proof. Acceding to the above-mentioned lemma, hereinafter, we consider that, domain of all singular 2-cube are notarized (that means, hereinafter, we consider that domain of all singular 2-cube are from the beginning, I².

In order to facilitate the discussion of boundary, we define \delta_:\mathbb{R}^k \to \mathbb{R}^{k+1} by

\delta_(t_1,\cdots,t_k):= (t_1,\cdots,t_{j-1},c,t_{j+1},\cdots,t_k)

γ₁, ..., γ₄ are the one-dimensional edges of the image of I².Hereinafter, the ⊕ stands for joining paths^{[note 11]} and,　 the \ominus stands for backwards of curve .^{[note 12]}

\begin{align} \begin{cases} \gamma_{1}:[0, 1]\to {I}^{2} \\ \gamma_{1}(t) := {\delta}_(t)=(t,0) \end{cases}, \qquad &\begin{cases}\gamma_{2}:[0,1] \to { I }^{2} \\ \gamma_{2}(t) :={\delta}_(t)=(1, t) \end{cases} \\ \begin{cases} \gamma_{3}:[0, 1] \to {I}^{2} \\ \gamma_{3}(t) := \ominus{\delta}_(t)= (-t+0+1, 1)\end{cases}, \qquad &\begin{cases}\gamma_{4}:[0,1] \to {I}^{2} \\ \gamma_{4}(t) := \ominus{\delta}_(t) =(0, 1-t)\end{cases} \end{align}

\gamma(t):= (\gamma_{1} \oplus \gamma_{2} \oplus \gamma_{3} \oplus \gamma_{4})(t)

\Gamma_{i}(t):= \varphi(\gamma_{i}(t)), \qquad i=1, 2, 3, 4

\Gamma(t):= \varphi(\gamma(t)) =(\Gamma_{1} \oplus \Gamma_{2} \oplus \Gamma_{3} \oplus \Gamma_{4})(t)

Cube subdivision

The definition of the boundary of the Definitions 3-3 is apparently depends on the cube subdevision. However, considering the following fact, the boundary is not depends on the cube subdevision.

So, considering the above-mentioned fact, following "Definition3-4" is well-defined.

Notes and references

Notes

References