Kelvin-Stokes Theorem

(Redirected from Classical Stokes' Theorem)

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

The Kelvin-Stokes Theorem is also known as the Classical Stokes' Theorem.

It is also known as just Stokes's Theorem, or Stokes' Theorem.

Source of Name

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