stokes' theorem

Let $\backslash Sigma$ be a smooth oriented surface in $\backslash R^3$ with boundary $\backslash partial\; \backslash Sigma\; \backslash equiv\; \backslash Gamma$. If a vector field $\backslash mathbf\{F\}(x,y,z)\; =\; (F\_x(x,\; y,\; z),\; F\_y(x,\; y,\; z),\; F\_z(x,\; y,\; z))$ is defined and has continuous first order partial derivatives in a region containing $\backslash Sigma$, then

\iint_\Sigma (\nabla \times \mathbf{F}) \cdot \mathrm{d} \mathbf{\Sigma} = \oint_{\partial\Sigma} \mathbf{F} \cdot

\mathrm{d}\mathbf{\Gamma}.

More explicitly, the equality says that

{\scriptstyle

\begin{align}

&\iint_\Sigma \left(\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z} \right)\,\mathrm{d}y\, \mathrm{d}z +\left(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\right)\, \mathrm{d}z\, \mathrm{d}x +\left (\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)\, \mathrm{d}x\, \mathrm{d}y\right) \\

& = \oint_{\partial\Sigma} \Bigl(F_x\, \mathrm{d}x+F_y\, \mathrm{d}y+F_z\, \mathrm{d}z\Bigr).

\end{align}

}

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 $\backslash R^2$.

A more detailed statement will be given for subsequent discussions.

Let $\backslash gamma:[a,b]\backslash to\backslash R^2$ be a piecewise smooth Jordan plane curve: a simple closed curve in the plane. The Jordan curve theorem implies that $\backslash gamma$ divides $\backslash R^2$ into two components, a compact one and another that is non-compact. Let $D$ denote the compact part; then $D$ is bounded by $\backslash gamma$. It now suffices to transfer this notion of boundary along a continuous map to our surface in $\backslash R^3$. But we already have such a map: the parametrization of $\backslash Sigma$.

Suppose $\backslash psi:D\backslash to\backslash R^3$ is piecewise smooth at the neighborhood of $D$, with $\backslash Sigma=\backslash psi(D)$.$\backslash Sigma=\backslash psi(D)$ represents the image set of $D$ by $\backslash psi$ If $\backslash Gamma$ is the space curve defined by $\backslash Gamma(t)=\backslash psi(\backslash gamma(t))$$\backslash Gamma$ may not be a Jordan curve if the loop $\backslash gamma$ interacts poorly with $\backslash psi$. Nonetheless, $\backslash Gamma$ is always a loop, and topologically a connected sum of countably many Jordan curves, so that the integrals are well-defined. then we call $\backslash Gamma$ the boundary of $\backslash Sigma$, written $\backslash partial\backslash Sigma$.

With the above notation, if $\backslash mathbf\{F\}$ is any smooth vector field on $\backslash R^3$, then{{Cite book |last=Stewart |first=James |author-link=James Stewart (mathematician) |url={{Google books|plainurl=yes|id=btIhvKZCkTsC|page=786 }} |title=Essential calculus: early transcendentals |publisher=Brooks/Cole |year=2010 |isbn=978-0-538-49739-8 |location=Australia; United States}}Robert Scheichl, [http://www.maths.bath.ac.uk/~masrs/ma20010/stokesproofs.pdf lecture notes] for University of Bath mathematics course$$\backslash oint\_\{\backslash partial\backslash Sigma\}\; \backslash mathbf\{F\}\backslash ,\; \backslash cdot\backslash ,\; \backslash mathrm\{d\}\{\backslash mathbf\{\backslash Gamma\}\}\; =\; \backslash iint\_\{\backslash Sigma\}\; \backslash nabla\backslash times\backslash mathbf\{F\}\backslash ,\; \backslash cdot\backslash ,\; \backslash mathrm\{d\}\backslash mathbf\{\backslash Sigma\}.$$

Here, the "$\backslash cdot$" represents the dot product in $\backslash R^3$.

Stokes' theorem can be viewed as a special case of the following identity:{{Cite journal |last=Pérez-Garrido |first=A. |date=2024-05-01 |title=Recovering seldom-used theorems of vector calculus and their application to problems of electromagnetism |journal=American Journal of Physics |volume=92 |issue=5 |pages=354–359 |arxiv=2312.17268 |doi=10.1119/5.0182191 |bibcode=2024AmJPh..92e.354P |issn=0002-9505}}

$$\backslash oint\_\{\backslash partial\backslash Sigma\}\; (\backslash mathbf\{F\}\backslash ,\; \backslash cdot\backslash ,\; \backslash mathrm\{d\}\{\backslash mathbf\{\backslash Gamma\}\})\backslash ,\backslash mathbf\{g\}\; =\; \backslash iint\_\{\backslash Sigma\}\backslash left[\; \backslash mathrm\{d\}\backslash mathbf\{\backslash Sigma\}\backslash cdot\backslash left(\; \backslash nabla\backslash times\backslash mathbf\{F\}-\; \backslash mathbf\{F\}\backslash times\backslash nabla\backslash right)\backslash right]\backslash mathbf\{g\},$$

where $\backslash mathbf\{g\}$ is any smooth vector or scalar field in $\backslash mathbb\{R\}^3$. When $\backslash mathbf\{g\}$ is a uniform scalar field, the standard Stokes' theorem is recovered.

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).{{Cite book |last=Colley |first=Susan Jane |author-link=Susan Jane Colley |url=https://universitytime.home.blog/wp-content/uploads/2020/04/1.vector-cal-4.pdf |title=Vector calculus |publisher=Pearson |year=2012 |isbn=978-0-321-78065-2 |edition=4th |location=Boston |pages=500–3 |oclc=732967769}} 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.

As in {{slink|#Theorem}}, we reduce the dimension by using the natural parametrization of the surface. Let {{math|ψ}} and {{mvar|γ}} be as in that section, and note that by change of variables

$$\backslash oint\_\{\backslash partial\backslash Sigma\}\{\backslash mathbf\{F\}(\backslash mathbf\{x\})\backslash cdot\backslash ,\backslash mathrm\{d\}\backslash mathbf\{\backslash Gamma\}\}$$

= \oint_{\gamma}{\mathbf{F}(\boldsymbol{\psi}(\mathbf{\gamma}))\cdot\,\mathrm{d}\boldsymbol{\psi}(\mathbf{\gamma})}

= \oint_{\gamma}{\mathbf{F}(\boldsymbol{\psi}(\mathbf{y}))\cdot J_{\mathbf{y}}(\boldsymbol{\psi})\,\mathrm{d}\gamma}

where {{mvar|J_yψ}} stands for the Jacobian matrix of {{mvar|ψ}} at {{math|1=y = γ(t)}}.

Now let {{math|{e_u, e_v}}} be an orthonormal basis in the coordinate directions of {{math|R²}}.In this article,

$$\backslash mathbf\{e\}\_u=\; \backslash begin\{bmatrix\}1\; \backslash \backslash \; 0\; \backslash end\{bmatrix\}\; ,$$

\mathbf{e}_v = \begin{bmatrix}0 \\ 1 \end{bmatrix} .

Note that, in some textbooks on vector analysis, these are assigned to different things.

For example, in some text book's notation, {{math|{e_u, e_v}}} can mean the following {{math|{t_u, t_v}}} respectively.

In this article, however, these are two completely different things.

$$\backslash mathbf\{t\}\_\{u\}\; =\backslash frac\; \{1\}\{h\_u\}\backslash frac\; \{\backslash partial\; \backslash varphi\}\{\backslash partial\; u\}\; \backslash ,\; ,$$

\mathbf{t}_{v} =\frac {1}{h_v}\frac {\partial \varphi}{\partial v}.

Here,

$$h\_u\; =\; \backslash left\backslash |\backslash frac\; \{\backslash partial\; \backslash varphi\}\{\backslash partial\; u\}\backslash right\backslash |\; ,$$

h_v = \left\|\frac {\partial \varphi}{\partial v} \right\|,

and the "$\backslash |\; \backslash cdot\; \backslash |$" represents Euclidean norm.

Recognizing that the columns of {{math|J_yψ}} are precisely the partial derivatives of {{math|ψ}} at {{math|y}}, we can expand the previous equation in coordinates as

$$\backslash begin\{align\}$$

\oint_{\partial\Sigma}{\mathbf{F}(\mathbf{x})\cdot\,\mathrm{d}\mathbf{\Gamma}}

&= \oint_{\gamma}{\mathbf{F}(\boldsymbol{\psi}(\mathbf{y}))J_{\mathbf{y}}(\boldsymbol{\psi})\mathbf{e}_u(\mathbf{e}_u\cdot\,\mathrm{d}\mathbf{y}) + \mathbf{F}(\boldsymbol{\psi}(\mathbf{y}))J_{\mathbf{y}}(\boldsymbol{\psi})\mathbf{e}_v(\mathbf{e}_v\cdot\,\mathrm{d}\mathbf{y})} \\

&=\oint_{\gamma}{\left(\left(\mathbf{F}(\boldsymbol{\psi}(\mathbf{y}))\cdot\frac{\partial\boldsymbol{\psi}}{\partial u}(\mathbf{y})\right)\mathbf{e}_u + \left(\mathbf{F}(\boldsymbol{\psi}(\mathbf{y}))\cdot\frac{\partial\boldsymbol{\psi}}{\partial v}(\mathbf{y})\right)\mathbf{e}_v\right)\cdot\,\mathrm{d}\mathbf{y}}

\end{align}

The previous step suggests we define the function

$$\backslash mathbf\{P\}(u,v)\; =\; \backslash left(\backslash mathbf\{F\}(\backslash boldsymbol\{\backslash psi\}(u,v))\backslash cdot\backslash frac\{\backslash partial\backslash boldsymbol\{\backslash psi\}\}\{\backslash partial\; u\}(u,v)\backslash right)\backslash mathbf\{e\}\_u\; +\; \backslash left(\backslash mathbf\{F\}(\backslash boldsymbol\{\backslash psi\}(u,v))\backslash cdot\backslash frac\{\backslash partial\backslash boldsymbol\{\backslash psi\}\}\{\backslash partial\; v\}(u,v)\; \backslash right)\backslash mathbf\{e\}\_v$$

Now, if the scalar value functions $P\_u$ and $P\_v$ are defined as follows,

$$\{P\_u\}(u,v)\; =\; \backslash left(\backslash mathbf\{F\}(\backslash boldsymbol\{\backslash psi\}(u,v))\backslash cdot\backslash frac\{\backslash partial\backslash boldsymbol\{\backslash psi\}\}\{\backslash partial\; u\}(u,v)\backslash right)$$

$$\{P\_v\}(u,v)\; =\backslash left(\backslash mathbf\{F\}(\backslash boldsymbol\{\backslash psi\}(u,v))\backslash cdot\backslash frac\{\backslash partial\backslash boldsymbol\{\backslash psi\}\}\{\backslash partial\; v\}(u,v)\; \backslash right)$$

then,

$$\backslash mathbf\{P\}(u,v)\; =\; \{P\_u\}(u,v)\; \backslash mathbf\{e\}\_u\; +\; \{P\_v\}(u,v)\; \backslash mathbf\{e\}\_v\; .$$

This is the pullback of {{math|F}} along {{math|ψ}}, and, by the above, it satisfies

$$\backslash oint\_\{\backslash partial\backslash Sigma\}\{\backslash mathbf\{F\}(\backslash mathbf\{x\})\backslash cdot\backslash ,\backslash mathrm\{d\}\backslash mathbf\{l\}\}=\backslash oint\_\{\backslash gamma\}\{\backslash mathbf\{P\}(\backslash mathbf\{y\})\backslash cdot\backslash ,\backslash mathrm\{d\}\backslash mathbf\{l\}\}$$

=\oint_{\gamma}{( {P_u}(u,v) \mathbf{e}_u + {P_v}(u,v) \mathbf{e}_v)\cdot\,\mathrm{d}\mathbf{l}}

We have successfully reduced one side of Stokes' theorem to a 2-dimensional formula; we now turn to the other side.

First, calculate the partial derivatives appearing in Green's theorem, via the product rule:

$$\backslash begin\{align\}$$

\frac{\partial P_u}{\partial v} &= \frac{\partial (\mathbf{F}\circ \boldsymbol{\psi})}{\partial v}\cdot\frac{\partial \boldsymbol\psi}{\partial u} + (\mathbf{F}\circ \boldsymbol\psi) \cdot\frac{\partial^2 \boldsymbol\psi}{\partial v \, \partial u} \\[5pt]

\frac{\partial P_v}{\partial u} &= \frac{\partial (\mathbf{F}\circ \boldsymbol{\psi})}{\partial u}\cdot\frac{\partial \boldsymbol\psi}{\partial v} + (\mathbf{F}\circ \boldsymbol\psi) \cdot\frac{\partial^2 \boldsymbol\psi}{\partial u \, \partial v}

\end{align}

Conveniently, the second term vanishes in the difference, by equality of mixed partials. So,

For all $\backslash textbf\{a\},\; \backslash textbf\{b\}\; \backslash in\; \backslash mathbb\{R\}^\{n\}$, for all $A\; ;\; n\backslash times\; n$ square matrix, $\backslash textbf\{a\}\backslash cdot\; A\; \backslash textbf\{b\}\; =\; \backslash textbf\{a\}^\backslash mathsf\{T\}A\; \backslash textbf\{b\}$ and therefore $\backslash textbf\{a\}\backslash cdot\; A\; \backslash textbf\{b\}\; =\; \backslash textbf\{b\}\; \backslash cdot\; A^\backslash mathsf\{T\}\; \backslash textbf\{a\}$.

$$\backslash begin\{align\}$$

\frac{\partial P_v}{\partial u} - \frac{\partial P_u}{\partial v}

&= \frac{\partial (\mathbf{F}\circ \boldsymbol\psi)}{\partial u}\cdot\frac{\partial \boldsymbol\psi}{\partial v} - \frac{\partial (\mathbf{F}\circ \boldsymbol\psi)}{\partial v}\cdot\frac{\partial \boldsymbol\psi}{\partial u} \\[5pt]

&= \frac{\partial \boldsymbol\psi}{\partial v}\cdot(J_{\boldsymbol\psi(u,v)}\mathbf{F})\frac{\partial \boldsymbol\psi}{\partial u} - \frac{\partial \boldsymbol\psi}{\partial u}\cdot(J_{\boldsymbol\psi(u,v)}\mathbf{F})\frac{\partial \boldsymbol\psi}{\partial v} && \text{(chain rule)}\\[5pt]

&= \frac{\partial \boldsymbol\psi}{\partial v}\cdot\left(J_{\boldsymbol\psi(u,v)}\mathbf{F}-{(J_{\boldsymbol\psi(u,v)}\mathbf{F})}^{\mathsf{T}}\right)\frac{\partial \boldsymbol\psi}{\partial u}

\end{align}

But now consider the matrix in that quadratic form—that is, $J\_\{\backslash boldsymbol\backslash psi(u,v)\}\backslash mathbf\{F\}-(J\_\{\backslash boldsymbol\backslash psi(u,v)\}\backslash mathbf\{F\})^\{\backslash mathsf\{T\}\}$. We claim this matrix in fact describes a cross product.

Here the superscript "$\{\}^\{\backslash mathsf\{T\}\}$" represents the transposition of matrices.

To be precise, let $A=(A\_\{ij\})\_\{ij\}$ be an arbitrary {{math|3 × 3}} matrix and let

$$\backslash mathbf\{a\}=\; \backslash begin\{bmatrix\}a\_1\; \backslash \backslash \; a\_2\; \backslash \backslash \; a\_3\backslash end\{bmatrix\}\; =\; \backslash begin\{bmatrix\}A\_\{32\}-A\_\{23\}\; \backslash \backslash \; A\_\{13\}-A\_\{31\}\; \backslash \backslash \; A\_\{21\}-A\_\{12\}\backslash end\{bmatrix\}$$

Note that {{math|x ↦ a × x}} is linear, so it is determined by its action on basis elements. But by direct calculation

$$\backslash begin\{align\}$$

\left(A-A^{\mathsf{T}}\right)\mathbf{e}_1 &= \begin{bmatrix} 0 \\ a_3 \\ -a_2 \end{bmatrix} = \mathbf{a}\times\mathbf{e}_1\\

\left(A-A^{\mathsf{T}}\right)\mathbf{e}_2 &= \begin{bmatrix} -a_3 \\ 0 \\ a_1 \end{bmatrix} = \mathbf{a}\times\mathbf{e}_2\\

\left(A-A^{\mathsf{T}}\right)\mathbf{e}_3 &= \begin{bmatrix} a_2 \\ -a_1 \\ 0 \end{bmatrix} = \mathbf{a}\times\mathbf{e}_3

\end{align}

Here, {{math|{e₁, e₂, e₃}}} represents an orthonormal basis in the coordinate directions of $\backslash R^3$.In this article,

$$\backslash mathbf\{e\}\_1=\; \backslash begin\{bmatrix\}1\; \backslash \backslash \; 0\; \backslash \backslash \; 0\backslash end\{bmatrix\}\; ,$$

\mathbf{e}_2 = \begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix} ,

\mathbf{e}_3 = \begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix} .

Note that, in some textbooks on vector analysis, these are assigned to different things.

Thus {{math|1=(A − A{{sup|T}})x = a × x}} for any {{math|x}}.

Substituting $\{(J\_\{\backslash boldsymbol\backslash psi(u,v)\}\backslash mathbf\{F\})\}$ for {{mvar|A}}, we obtain

$$\backslash left(\{(J\_\{\backslash boldsymbol\backslash psi(u,v)\}\backslash mathbf\{F\})\}\; -\; \{(J\_\{\backslash boldsymbol\backslash psi(u,v)\}\backslash mathbf\{F\})\}^\{\backslash mathsf\{T\}\}\; \backslash right)\; \backslash mathbf\{x\}\; =(\backslash nabla\backslash times\backslash mathbf\{F\})\backslash times\; \backslash mathbf\{x\},\; \backslash quad\; \backslash text\{for\; all\}\backslash ,\; \backslash mathbf\{x\}\backslash in\backslash R^\{3\}$$

We can now recognize the difference of partials as a (scalar) triple product:

$$\backslash begin\{align\}$$

\frac{\partial P_v}{\partial u} - \frac{\partial P_u}{\partial v}

&= \frac{\partial \boldsymbol\psi}{\partial v}\cdot(\nabla\times\mathbf{F}) \times \frac{\partial \boldsymbol\psi}{\partial u} = (\nabla\times\mathbf{F})\cdot \frac{\partial \boldsymbol\psi}{\partial u} \times \frac{\partial \boldsymbol\psi}{\partial v}

\end{align}

On the other hand, the definition of a surface integral also includes a triple product—the very same one!

$$\backslash begin\{align\}$$

\iint_\Sigma (\nabla\times\mathbf{F})\cdot \, d\mathbf{\Sigma}

&=\iint_D {(\nabla\times\mathbf{F})(\boldsymbol\psi(u,v))\cdot\frac{\partial \boldsymbol\psi}{\partial u}(u,v)\times \frac{\partial \boldsymbol\psi}{\partial v}(u,v)\,\mathrm{d}u\,\mathrm{d}v}

\end{align}

So, we obtain

$$\backslash iint\_\backslash Sigma\; (\backslash nabla\backslash times\backslash mathbf\{F\})\backslash cdot\; \backslash ,\backslash mathrm\{d\}\backslash mathbf\{\backslash Sigma\; \}\; =\; \backslash iint\_D\; \backslash left(\; \backslash frac\{\backslash partial\; P\_v\}\{\backslash partial\; u\}\; -\; \backslash frac\{\backslash partial\; P\_u\}\{\backslash partial\; v\}\; \backslash right)\; \backslash ,\backslash mathrm\{d\}u\backslash ,\backslash mathrm\{d\}v$$

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 $P\_u(u,v),\; P\_v(u,v)$ defined on D;

$$\backslash oint\_\{\backslash gamma\}\{(\; \{P\_u\}(u,v)\; \backslash mathbf\{e\}\_u\; +\; \{P\_v\}(u,v)\; \backslash mathbf\{e\}\_v)\backslash cdot\backslash ,\backslash mathrm\{d\}\backslash mathbf\{l\}\}$$

= \iint_D \left( \frac{\partial P_v}{\partial u} - \frac{\partial P_u}{\partial v} \right) \,\mathrm{d}u\,\mathrm{d}v

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.

The functions $\backslash R^3\backslash to\backslash R^3$ can be identified with the differential 1-forms on $\backslash R^3$ via the map

$$F\_x\backslash mathbf\{e\}\_1+F\_y\backslash mathbf\{e\}\_2+F\_z\backslash mathbf\{e\}\_3\; \backslash mapsto\; F\_x\backslash ,\backslash mathrm\{d\}x\; +\; F\_y\backslash ,\backslash mathrm\{d\}y\; +\; F\_z\backslash ,\backslash mathrm\{d\}z\; .$$

Write the differential 1-form associated to a function {{math|F}} as {{math|ω_F}}. Then one can calculate that

$$\backslash star\backslash omega\_\{\backslash nabla\backslash times\backslash mathbf\{F\}\}=\backslash mathrm\{d\}\backslash omega\_\{\backslash mathbf\{F\}\},$$

where {{math|★}} is the Hodge star and $\backslash mathrm\{d\}$ is the exterior derivative. Thus, by generalized Stokes' theorem,{{Cite book |last=Edwards |first=Harold M. |url={{Google books|plainurl=yes|v9mFWa_SKGIC}} |title=Advanced calculus: a differential forms approach |publisher=Birkhäuser |year=1994 |isbn=978-0-8176-3707-1 |edition=3rd |location=Boston}}

$$\backslash oint\_\{\backslash partial\backslash Sigma\}\{\backslash mathbf\{F\}\backslash cdot\backslash ,\backslash mathrm\{d\}\backslash mathbf\{\backslash gamma\}\}$$

=\oint_{\partial\Sigma}{\omega_{\mathbf{F}}}

=\int_{\Sigma}{\mathrm{d}\omega_{\mathbf{F}}}

=\int_{\Sigma}{\star\omega_{\nabla\times\mathbf{F}}}

=\iint_{\Sigma}{\nabla\times\mathbf{F}\cdot\,\mathrm{d}\mathbf{\Sigma}}

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 {{math|F}} on an open $U\backslash subseteq\backslash R^3$ is irrotational (lamellar vector field) if {{math|1=∇ × F = 0}}.

This concept is very fundamental in mechanics; as we'll prove later, if {{math|F}} is irrotational and the domain of {{math|F}} is simply connected, then {{math|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).{{rp|142}} Let $U\backslash subseteq\backslash R^3$ be an open subset with a lamellar vector field {{math|F}} and let {{math|c₀, c₁: [0, 1] → U}} be piecewise smooth loops. If there is a function {{math|H: [0, 1] × [0, 1] → U}} such that

• [TLH0] {{mvar|H}} is piecewise smooth,
• [TLH1] {{math|1=H(t, 0) = c₀(t)}} for all {{math|t ∈ [0, 1]}},
• [TLH2] {{math|1=H(t, 1) = c₁(t)}} for all {{math|t ∈ [0, 1]}},
• [TLH3] {{math|1=H(0, s) = H(1, s)}} for all {{math|s ∈ [0, 1]}}.

Then,

$$\backslash int\_\{c\_0\}\; \backslash mathbf\{F\}\; \backslash ,\; \backslash mathrm\{d\}c\_0=\backslash int\_\{c\_1\}\; \backslash mathbf\{F\}\; \backslash ,\; \backslash mathrm\{d\}c\_1$$

Some textbooks such as Lawrence call the relationship between {{math|c₀}} and {{math|c₁}} stated in theorem 2-1 as "homotopic" and the function {{math|H: [0, 1] × [0, 1] → U}} as "homotopy between {{math|c₀}} and {{math|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).{{refn|group="note"|name="TLH"|There do exist textbooks that use the terms "homotopy" and "homotopic" in the sense of Theorem 2-1. Indeed, this is very convenient for the specific problem of conservative forces. However, both uses of homotopy appear sufficiently frequently that some sort of terminology is necessary to disambiguate, and the term "tubular homotopy" adopted here serves well enough for that end.}}

File:Domain of singular 2 cube.svg

In what follows, we abuse notation and use "$\backslash oplus$" for concatenation of paths in the fundamental groupoid and "$\backslash ominus$" for reversing the orientation of a path.

Let {{math|1=D = [0, 1] × [0, 1]}}, and split {{math|∂D}} into four line segments {{math|γ{{sub|j}}}}.

$$\backslash begin\{align\}$$

\gamma_1:[0,1] \to D;\quad&\gamma_1(t) = (t, 0) \\

\gamma_2:[0,1] \to D;\quad&\gamma_2(s) = (1, s) \\

\gamma_3:[0,1] \to D;\quad&\gamma_3(t) = (1-t, 1) \\

\gamma_4:[0,1] \to D;\quad&\gamma_4(s) = (0, 1-s)

\end{align}

so that $$\backslash partial\; D\; =\; \backslash gamma\_1\; \backslash oplus\; \backslash gamma\_2\; \backslash oplus\; \backslash gamma\_3\; \backslash oplus\; \backslash gamma\_4$$

By our assumption that {{math|c₀}} and {{math|c₁}} are piecewise smooth homotopic, there is a piecewise smooth homotopy {{math|H: D → M}}

$$\backslash begin\{align\}$$

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

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

\end{align}

Let {{mvar|S}} be the image of {{mvar|D}} under {{mvar|H}}. That

$$\backslash iint\_S\; \backslash nabla\backslash times\backslash mathbf\{F\}\backslash ,\; \backslash mathrm\{d\}S\; =\; \backslash oint\_\backslash Gamma\; \backslash mathbf\{F\}\backslash ,\; \backslash mathrm\{d\}\backslash Gamma$$

follows immediately from Stokes' theorem. {{math|F}} is lamellar, so the left side vanishes, i.e.

$$0=\backslash oint\_\backslash Gamma\; \backslash mathbf\{F\}\backslash ,\; \backslash mathrm\{d\}\backslash Gamma\; =\; \backslash sum\_\{i=1\}^4\; \backslash oint\_\{\backslash Gamma\_i\}\; \backslash mathbf\{F\}\; \backslash ,\; \backslash mathrm\{d\}\backslash Gamma$$

As {{mvar|H}} is tubular(satisfying [TLH3]),$\backslash Gamma\_2\; =\; \backslash ominus\; \backslash Gamma\_4$ and $\backslash Gamma\_2\; =\; \backslash ominus\; \backslash Gamma\_4$. Thus the line integrals along {{math|Γ₂(s)}} and {{math|Γ₄(s)}} cancel, leaving

$$0=\backslash oint\_\{\backslash Gamma\_1\}\; \backslash mathbf\{F\}\; \backslash ,\; \backslash mathrm\{d\}\backslash Gamma\; +\backslash oint\_\{\backslash Gamma\_3\}\; \backslash mathbf\{F\}\; \backslash ,\; \backslash mathrm\{d\}\backslash Gamma$$

On the other hand, {{math|1=c{{sub|1}} = Γ{{sub|1}}}}, $c\_3\; =\; \backslash ominus\; \backslash Gamma\_3$, so that the desired equality follows almost immediately.

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 $U\backslash subseteq\backslash R^3$ be an open subset, with a Lamellar vector field {{math|F}} and a piecewise smooth loop {{math|c₀: [0, 1] → U}}. Fix a point {{math|p ∈ U}}, if there is a homotopy {{math|H: [0, 1] × [0, 1] → U}} such that

• [SC0] {{mvar|H}} is piecewise smooth,
• [SC1] {{math|1= H(t, 0) = c₀(t)}} for all {{math|t ∈ [0, 1]}},
• [SC2] {{math|1= H(t, 1) = p}} for all {{math|t ∈ [0, 1]}},
• [SC3] {{math|1= H(0, s) = H(1, s) = p}} for all {{math|s ∈ [0, 1]}}.

Then,

$$\backslash int\_\{c\_0\}\; \backslash mathbf\{F\}\; \backslash ,\; \backslash mathrm\{d\}c\_0=0$$

Above Lemma 2-2 follows from theorem 2–1. In Lemma 2-2, the existence of {{mvar|H}} satisfying [SC0] to [SC3] is crucial;the question is whether such a homotopy can be taken for arbitrary loops. If {{mvar|U}} is simply connected, such {{mvar|H}} exists. The definition of simply connected space follows:

Definition 2-2 (simply connected space). Let $M\backslash subseteq\backslash R^n$ be non-empty and path-connected. {{mvar|M}} is called simply connected if and only if for any continuous loop, {{math|c: [0, 1] → M}} there exists a continuous tubular homotopy {{math|H: [0, 1] × [0, 1] → M}} from {{mvar|c}} to a fixed point {{math|p ∈ c}}; that is,

• [SC0'] {{mvar|H}} is continuous,
• [SC1] {{math|1= H(t, 0) = c(t)}} for all {{math|t ∈ [0, 1]}},
• [SC2] {{math|1= H(t, 1) = p}} for all {{math|t ∈ [0, 1]}},
• [SC3] {{math|1= H(0, s) = H(1, s) = p}} for all {{math|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.{{rp|136,421}}{{Cite journal |last=Pontryagin |first=L. S. |date=1959 |title=Smooth manifolds and their applications in homotopy theory |url=https://people.math.rochester.edu/faculty/doug/otherpapers/pont4.pdf |journal=American Mathematical Society Translations |series=Series 2 |language=en |location=Providence, Rhode Island |publisher=American Mathematical Society |volume=11 |pages=1–114 |doi=10.1090/trans2/011/01 |isbn=978-0-8218-1711-7 |mr=0115178 |translator-last1=Hilton |translator-first1=P. J.}} See theorems 7 & 8. 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 $U\backslash subseteq\backslash R^3$ be open and simply connected with an irrotational vector field {{math|F}}. For all piecewise smooth loops {{math|c: [0, 1] → U}}

$$\backslash int\_\{c\_0\}\; \backslash mathbf\{F\}\; \backslash ,\; \backslash mathrm\{d\}c\_0\; =\; 0$$

{{See also|Maxwell's equations#Circulation and curl}}

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, $\backslash mathbf\{E\}$:

$$\backslash oint\_\{\backslash partial\backslash Sigma\}\; \backslash mathbf\{E\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash boldsymbol\{l\}=\; \backslash iint\_\backslash Sigma\; \backslash mathbf\{\backslash nabla\}\backslash times\; \backslash mathbf\{E\}\; \backslash cdot\; \backslash mathrm\{d\}\; \backslash mathbf\{S\}\; .$$

For Ampère's law, Stokes' theorem is applied to the magnetic field, $\backslash mathbf\{B\}$:

$$\backslash oint\_\{\backslash partial\backslash Sigma\}\; \backslash mathbf\{B\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash boldsymbol\{l\}=\; \backslash iint\_\backslash Sigma\; \backslash mathbf\{\backslash nabla\}\backslash times\; \backslash mathbf\{B\}\; \backslash cdot\; \backslash mathrm\{d\}\; \backslash mathbf\{S\}\; .$$

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Category:Electromagnetism

Category:Mechanics

Category:Vectors (mathematics and physics)

Category:Vector calculus

Category:Theorems in calculus