Kelvin-Stokes Theorem

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Theorem

Let $S$ be some orientable smooth surface with boundary in $\R^3$.

Let $\mathbf F:\R^3 \to \R^3$ be a vector-valued function with Euclidean coordinate expression:

$\mathbf F = f_1 \mathbf i + f_2 \mathbf j + f_3 \mathbf k$

where $f_i: \R^3 \to \R$.

Then:

$\ds \oint_{\partial S} f_1 \rd x + f_2 \rd y + f_3 \rd z = \iint_S \paren {\nabla \times \mathbf F} \cdot \mathbf n \rd A$

where $\mathbf n$ is the unit normal to $S$ and $\d A$ is the area element on the surface.


Proof

Let $\mathbf r:\R^2 \to \R^3, \map {\mathbf r} {s, t}$ be a smooth parametrization of $S$ from some region $R$ in the $st$-plane, so that:

$\map {\mathbf r} R = S$

and:

$\map {\mathbf r} {\partial R} = \partial S$

First, we convert the left hand side into a line integral:

\(\ds \oint_{\partial S} f_1 \rd x + f_2 \rd y + f_3 \rd z\) \(=\) \(\ds \oint_{\partial S} \mathbf F \cdot \rd \mathbf r\)
\(\ds \) \(=\) \(\ds \oint_{\partial R} \mathbf F \cdot \frac {\partial \mathbf r} {\partial s} \rd s + \mathbf F \cdot \frac {\partial \mathbf r} {\partial t} \rd t\)

so that if we define:

$\mathbf G = \paren {G_1, G_2} = \paren {\mathbf F \cdot \dfrac {\partial \mathbf r} {\partial s}, \mathbf F \cdot \dfrac {\partial \mathbf r} {\partial t} }$

then:

$\ds \int_{\partial S} \mathbf F \cdot \rd \mathbf r = \int_{\partial R} \mathbf G \cdot \rd \mathbf s$

where $\mathbf s$ is the position vector in the $s t$-plane.

Note that:

\(\ds \frac {\partial G_2} {\partial s}\) \(=\) \(\ds \mathbf F \cdot \map {\frac \partial {\partial s} } {\frac {\partial \mathbf r} {\partial t} } + \frac {\partial \mathbf r} {\partial t} \cdot \frac {\partial \mathbf F} {\partial s}\) Derivative of Dot Product of Vector-Valued Functions
\(\ds \frac {\partial G_1} {\partial t}\) \(=\) \(\ds \mathbf F \cdot \map {\frac \partial {\partial t} } {\frac {\partial \mathbf r} {\partial s} } + \frac {\partial \mathbf r} {\partial s} \cdot \frac {\partial \mathbf F} {\partial t}\) Derivative of Dot Product of Vector-Valued Functions
\(\ds \frac {\partial G_2} {\partial s} - \frac {\partial G_1} {\partial t}\) \(=\) \(\ds \mathbf F \cdot \map {\frac \partial {\partial s} } {\frac {\partial \mathbf r} {\partial t} } + \frac {\partial \mathbf r} {\partial t} \cdot \frac {\partial \mathbf F} {\partial s}\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \mathbf F \cdot \map {\frac \partial {\partial t} } {\frac {\partial \mathbf r} {\partial s} } - \frac {\partial \mathbf r} {\partial s} \cdot \frac {\partial \mathbf F} {\partial t}\)
\(\ds \) \(=\) \(\ds \frac {\partial \mathbf r} {\partial t} \cdot \frac {\partial \mathbf F} {\partial s} - \frac {\partial \mathbf r} {\partial s} \cdot \frac {\partial \mathbf F} {\partial t}\) Symmetry of Second Derivatives


We turn now to the expression on the right hand side and write it in terms of $s$ and $t$:

\(\ds \) \(\) \(\ds \iint_S \paren {\nabla \times \mathbf F} \cdot \mathbf n \rd A\)
\(\ds \) \(=\) \(\ds \iint_R \paren {\nabla \times \mathbf F \cdot \paren {\frac {\partial \mathbf r} {\partial s} \times \frac {\partial \mathbf r} {\partial t} } } \rd s \rd t\)

Let us investigate the integrand:

\(\ds \) \(\) \(\ds \nabla \times \mathbf F \cdot \paren {\frac {\partial \mathbf r} {\partial s} \times \frac {\partial \mathbf r} {\partial t} }\)
\(\ds \) \(=\) \(\ds \paren {\paren {\dfrac {\partial f_3} {\partial y} - \dfrac {\partial f_2} {\partial z} } \mathbf i + \paren {\dfrac {\partial f_1} {\partial z} - \dfrac {\partial f_3} {\partial x} } \mathbf j + \paren {\dfrac {\partial f_2} {\partial x} - \dfrac {\partial f_1} {\partial y} } \mathbf k}\) Definition of Curl Operator
\(\ds \) \(\) \(\, \ds \cdot \, \) \(\ds \paren {\paren {\dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t} - \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t} } \mathbf i + \paren {\dfrac {\partial z} {\partial s} \dfrac {\partial x} {\partial t} - \dfrac {\partial x} {\partial s} \dfrac {\partial z} {\partial t} } \mathbf j + \paren {\dfrac {\partial x} {\partial s} \dfrac {\partial y} {\partial t} - \dfrac {\partial y} {\partial s} \dfrac {\partial x} {\partial t} } \mathbf k}\) Definition of Cross Product
\(\ds \) \(=\) \(\ds \paren {\dfrac {\partial f_3} {\partial y} - \dfrac {\partial f_2} {\partial z} } \paren {\dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t} - \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {\dfrac {\partial f_1} {\partial z} - \dfrac {\partial f_3} {\partial x} } \paren {\dfrac {\partial z} {\partial s} \dfrac {\partial x} {\partial t} - \dfrac {\partial x} {\partial s} \dfrac {\partial z} {\partial t} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {\dfrac {\partial f_2} {\partial x} - \dfrac {\partial f_1} {\partial y} } \paren {\dfrac {\partial x} {\partial s} \dfrac {\partial y} {\partial t} - \dfrac {\partial y} {\partial s} \dfrac {\partial x} {\partial t} }\) Definition of Dot Product
\(\ds \) \(=\) \(\ds \dfrac {\partial f_3} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t} - \dfrac {\partial f_3} {\partial y} \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t} - \dfrac {\partial f_2} {\partial z} \dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t} + \dfrac {\partial f_2} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {\partial f_1} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial x} {\partial t} - \dfrac {\partial f_1} {\partial z} \dfrac {\partial x} {\partial s} \dfrac {\partial z} {\partial t} - \dfrac {\partial f_3} {\partial x} \dfrac {\partial z} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_3} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial z} {\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {\partial f_2} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial y} {\partial t} - \dfrac {\partial f_2} {\partial x} \dfrac {\partial y} {\partial s} \dfrac {\partial x} {\partial t} - \dfrac {\partial f_1} {\partial y} \dfrac {\partial x} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial x} {\partial t}\)
\(\ds \) \(=\) \(\ds \dfrac {\partial f_1} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial x} {\partial t} - \dfrac {\partial f_1} {\partial z} \dfrac {\partial x} {\partial s} \dfrac {\partial z} {\partial t} - \dfrac {\partial f_1} {\partial y} \dfrac {\partial x} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial x} {\partial t}\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \dfrac {\partial f_2} {\partial z} \dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t} + \dfrac {\partial f_2} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_2} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial y} {\partial t} - \dfrac {\partial f_2} {\partial x} \dfrac {\partial y} {\partial s} \dfrac {\partial x} {\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {\partial f_3} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t} - \dfrac {\partial f_3} {\partial y} \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t} - \dfrac {\partial f_3} {\partial x} \dfrac {\partial z} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_3} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial z} {\partial t}\) grouping the terms that reference like components of $\mathbf F$ together
\(\ds \) \(=\) \(\ds \dfrac {\partial f_1} {\partial z} \dfrac {\partial z} {\partial s} \dfrac{\partial x}{\partial t} - \dfrac {\partial f_1} {\partial z} \dfrac {\partial x} {\partial s} \dfrac{\partial z}{\partial t} - \dfrac {\partial f_1} {\partial y} \dfrac {\partial x} {\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac {\partial y} {\partial s} \dfrac{\partial x}{\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {\dfrac {\partial f_1} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial x} {\partial t} - \dfrac {\partial f_1} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial x} {\partial t} }\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \dfrac {\partial f_2} {\partial z} \dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t} + \dfrac {\partial f_2} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_2} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial y} {\partial t} - \dfrac {\partial f_2} {\partial x} \dfrac {\partial y} {\partial s} \dfrac {\partial x} {\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {\dfrac {\partial f_2} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial y} {\partial t} - \dfrac {\partial f_2} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial y} {\partial t} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {\partial f_3} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t} - \dfrac {\partial f_3} {\partial y} \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t} - \dfrac {\partial f_3} {\partial x} \dfrac {\partial z} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_3} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial z} {\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {\dfrac {\partial f_3} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial z} {\partial t} - \dfrac {\partial f_3} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial z} {\partial t} }\) introducing terms that themselves sum to zero,

in order to complete the parts of the derivatives of $\mathbf F$ in a later step

\(\ds \) \(=\) \(\ds \dfrac {\partial f_1} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_1} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial x} {\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {- \dfrac {\partial f_1} {\partial z} \dfrac {\partial x} {\partial s} \dfrac {\partial z} {\partial t} - \dfrac {\partial f_1} {\partial y} \dfrac {\partial x} {\partial s} \dfrac {\partial y} {\partial t} - \dfrac {\partial f_1} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial x} {\partial t} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {\partial f_2} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_2} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_2} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial y} {\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {- \dfrac {\partial f_2} {\partial z} \dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t} - \dfrac {\partial f_2} {\partial x} \dfrac {\partial y} {\partial s} \dfrac {\partial x} {\partial t} - \dfrac {\partial f_2} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial y} {\partial t} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {\partial f_3} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t} + \dfrac {\partial f_3} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial z} {\partial t} + \dfrac {\partial f_3} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial z} {\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {- \dfrac {\partial f_3} {\partial y} \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t} - \dfrac {\partial f_3} {\partial x} \dfrac {\partial z} {\partial s} \dfrac {\partial x} {\partial t} - \dfrac {\partial f_3} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial z} {\partial t} }\) grouping terms by sign,

but keeping terms that reference like components of $\mathbf F$ together

\(\ds \) \(=\) \(\ds \dfrac {\partial f_1} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_1} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial x} {\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {\partial f_2} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_2} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_2} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial y} {\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {\partial f_3} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t} + \dfrac {\partial f_3} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial z} {\partial t} + \dfrac {\partial f_3} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial z} {\partial t}\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \leftparen {\dfrac {\partial f_1} {\partial z} \dfrac {\partial x} {\partial s} \dfrac {\partial z} {\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac {\partial x} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_1} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial x} {\partial t} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {\partial f_2} {\partial z} \dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t} + \dfrac {\partial f_2} {\partial x} \dfrac {\partial y} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_2} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial y} {\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \rightparen {\dfrac {\partial f_3} {\partial y} \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_3} {\partial x} \dfrac {\partial z} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_3} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial z} {\partial t} }\) grouping all negative terms together and factoring out the negative sign
\(\ds \) \(=\) \(\ds \dfrac {\partial f_1} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_1} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial x} {\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {\partial f_2} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_2} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_2} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {\partial f_3} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial z} {\partial t} + \dfrac {\partial f_3} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t} + \dfrac {\partial f_3} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial z} {\partial t}\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \leftparen {\dfrac {\partial f_1} {\partial x} \dfrac {\partial x} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac {\partial x} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_1} {\partial z} \dfrac {\partial x} {\partial s} \dfrac {\partial z} {\partial t} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {\partial f_2} {\partial x} \dfrac {\partial y} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_2} {\partial y} \dfrac {\partial y} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_2} {\partial z} \dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \rightparen {\dfrac {\partial f_3} {\partial x} \dfrac {\partial z} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_3} {\partial y} \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_3} {\partial z} \dfrac {\partial z} {\partial s} \dfrac {\partial z} {\partial t} }\) rearranging into groups,

in order to clarify use of the Chain Rule for Real-Valued Functions in the next step

\(\ds \) \(=\) \(\ds \paren {\dfrac {\partial f_1} {\partial s} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_2} {\partial s} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_3} {\partial s} \dfrac {\partial z} {\partial t} }\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {\dfrac {\partial f_1} {\partial t} \dfrac {\partial x} {\partial t} + \dfrac {\partial f_2} {\partial t} \dfrac {\partial y} {\partial t} + \dfrac {\partial f_3} {\partial t} \dfrac {\partial z} {\partial t} }\) Chain Rule for Real-Valued Functions
\(\ds \) \(=\) \(\ds \dfrac {\partial \mathbf F} {\partial s} \cdot \dfrac {\partial \mathbf r} {\partial t} - \dfrac {\partial \mathbf F} {\partial t} \cdot \dfrac {\partial \mathbf r} {\partial s}\) Definition:Dot Product
\(\ds \) \(=\) \(\ds \frac {\partial G_2} {\partial s} - \frac {\partial G_1} {\partial t}\)

That is:

$\ds \iint_S \paren {\nabla \times \mathbf F} \cdot \mathbf n \rd A = \iint_R \paren {\frac {\partial G_2} {\partial s} - \frac {\partial G_1} {\partial t} } \rd s \rd t$


By Green's Theorem, this can be written as:

$\ds \int_{\partial R} \mathbf G \cdot \rd \mathbf s$

Hence both sides of the theorem equation are equal.

$\blacksquare$


Also known as

Also known as the Classical Stokes' Theorem.


Source of Name

This entry was named for Lord Kelvin and George Gabriel Stokes.


Sources