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:
- $F = f_1 \mathbf i + f_2 \mathbf j + f_3 \mathbf k$
where $f_i: \R^3 \to \R$.
Then:
- $\displaystyle \oint_{\partial S} f_1 \rd x + f_2 \rd y + f_3 \rd z = \iint_S \paren {\nabla \times \mathbf F} \cdot \mathbf n \rd A$
where $\mathbf n$ is the unit normal to $S$ and $\d A$ is the area element on the surface.
Proof
Let $\mathbf r:\R^2 \to \R^3, \mathbf r \paren {s, t}$ be a smooth parametrization of $S$ from some region $R$ in the $st$-plane, so that:
- $\mathbf r \paren R = S$
and:
- $\mathbf r \paren {\partial R} = \partial S$
First, we convert the left hand side into a line integral:
| \(\displaystyle \oint_{\partial S} f_1 \rd x + f_2 \rd y + f_3 \rd z\) | \(=\) | \(\displaystyle \oint_{\partial S} \mathbf F \cdot \rd \mathbf r\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \oint_{\partial R} \mathbf F \cdot \frac{\partial \mathbf r} {\partial s} \rd s + \mathbf F \cdot \frac{\partial \mathbf r} {\partial t} \rd t\) |
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:
- $\displaystyle \int_{\partial S} \mathbf F \cdot \rd \mathbf r = \int_{\partial R} \mathbf G \cdot \rd \mathbf s$
where $\mathbf s$ is the position vector in the $s t$-plane.
Note that:
| \(\displaystyle \frac {\partial G_2} {\partial s}\) | \(=\) | \(\displaystyle \mathbf F \cdot \frac{\partial} {\partial s} \paren {\frac {\partial \mathbf r} {\partial t} } + \frac {\partial \mathbf r} {\partial t} \cdot \frac{\partial \mathbf F}{\partial s}\) | Derivative of Dot Product of Vector-Valued Functions | ||||||||||
| \(\displaystyle \frac {\partial G_1} {\partial t}\) | \(=\) | \(\displaystyle \mathbf F \cdot \frac {\partial} {\partial t} \paren {\frac {\partial \mathbf r} {\partial s} } + \frac {\partial \mathbf r} {\partial s} \cdot \frac {\partial \mathbf F} {\partial t}\) | Derivative of Dot Product of Vector-Valued Functions | ||||||||||
| \(\displaystyle \frac {\partial G_2} {\partial s} - \frac {\partial G_1} {\partial t}\) | \(=\) | \(\displaystyle \mathbf F \cdot \frac {\partial} {\partial s} \paren {\frac {\partial \mathbf r} {\partial t} } + \frac {\partial \mathbf r} {\partial t} \cdot \frac {\partial \mathbf F} {\partial s}\) | |||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle - \, \) | \(\displaystyle \mathbf F \cdot \frac {\partial} {\partial t} \paren {\frac {\partial \mathbf r} {\partial s} } - \frac {\partial \mathbf r} {\partial s} \cdot \frac {\partial \mathbf F} {\partial t}\) | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\partial \mathbf r} {\partial t} \cdot \frac {\partial \mathbf F} {\partial s} - \frac {\partial \mathbf r} {\partial s} \cdot \frac {\partial \mathbf F} {\partial t}\) | Symmetry of Second Derivatives |
We turn now to the right-hand expression and write it in terms of $s$ and $t$:
- $\displaystyle \iint_S \paren {\nabla \times \mathbf F} \cdot \mathbf n \rd A$
| \(\displaystyle \) | \(=\) | \(\, \displaystyle \displaystyle \ \ \iint_R \Biggl( \, \) | \(\displaystyle \nabla \times \mathbf F \cdot \paren {\frac {\partial \mathbf r} {\partial s} \times \frac {\partial \mathbf r} {\partial t} } \Biggr) \rd s \rd t\) | ||||||||||
| \(\displaystyle \) | \(=\) | \(\, \displaystyle \displaystyle \ \ \iint_R \Biggl( \, \) | \(\displaystyle \paren {\dfrac {\partial f_3} {\partial y} - \dfrac {\partial f_2} {\partial z} } \mathbf i + \paren {\dfrac {\partial f_1} {\partial z} - \dfrac {\partial f_3} {\partial x} } \mathbf j + \paren {\dfrac {\partial f_2} {\partial x} - \dfrac {\partial f_1} {\partial y} } \mathbf k \Biggr)\) | Definition:Curl Operator | |||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \cdot \Biggl( \, \) | \(\displaystyle \paren {\dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t} - \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t} } \mathbf i\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \paren {\dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t} - \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t} } \mathbf j\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \paren {\dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t} - \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t} } \mathbf k \Biggr) \rd s \rd t\) | Definition:Cross Product | |||||||||
| \(\displaystyle \) | \(=\) | \(\, \displaystyle \ \ \iint_R \Biggl( \, \) | \(\displaystyle \paren {\dfrac {\partial f_3} {\partial y} - \dfrac {\partial f_2} {\partial z} } \paren {\dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t} - \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t} }\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \paren {\dfrac {\partial f_1} {\partial z} - \dfrac {\partial f_3} {\partial x} } \paren {\dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t} - \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t} }\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \paren {\dfrac {\partial f_2} {\partial x} - \dfrac {\partial f_1} {\partial y} } \paren {\dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t} - \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t} } \Biggr) \rd s \rd t\) | Definition:Dot Product | |||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \iint_R \Biggl( \, \) | \(\displaystyle \dfrac {\partial f_3} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t} - \dfrac {\partial f_3} {\partial y} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t} - \dfrac {\partial f_2} {\partial z} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t} + \dfrac {\partial f_2} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \dfrac {\partial f_1} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t} - \dfrac {\partial f_1} {\partial z} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t} - \dfrac {\partial f_3} {\partial x} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_3} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \dfrac {\partial f_2} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t} - \dfrac {\partial f_2} {\partial x} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t} - \dfrac {\partial f_1} {\partial y} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t} \Biggr) \rd s \rd t\) | FOIL | |||||||||
| \(\displaystyle \) | \(=\) | \(\, \displaystyle \ \ \iint_R \Biggl( \, \) | \(\displaystyle \dfrac {\partial f_1} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t} - \dfrac {\partial f_1} {\partial z} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t} - \dfrac {\partial f_1} {\partial y} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ - \, \) | \(\displaystyle \dfrac {\partial f_2} {\partial z} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t} + \dfrac {\partial f_2} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_2} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t} - \dfrac {\partial f_2} {\partial x} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \dfrac {\partial f_3} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t} - \dfrac {\partial f_3} {\partial y} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t} - \dfrac {\partial f_3} {\partial x} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_3} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t} \Biggr) \rd s \rd t\) | grouping the terms that reference like components of $\mathbf F$ together | |||||||||
| \(\displaystyle \) | \(=\) | \(\, \displaystyle \ \ \iint_R \Biggl( \, \) | \(\displaystyle \dfrac {\partial f_1} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t} - \dfrac {\partial f_1} {\partial z} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t} - \dfrac {\partial f_1} {\partial y} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \paren {\dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t} - \dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t} }\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ - \, \) | \(\displaystyle \dfrac {\partial f_2} {\partial z} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t} + \dfrac {\partial f_2} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_2} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t} - \dfrac {\partial f_2} {\partial x} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \paren {\dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t} - \dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t} }\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \dfrac {\partial f_3} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t} - \dfrac {\partial f_3} {\partial y} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t} - \dfrac {\partial f_3} {\partial x} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_3} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \paren {\dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t} - \dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t} } \Biggr) \rd s \rd t\) | introducing terms that themselves sum to zero,
in order to complete the parts of the derivatives of $\mathbf F$ in a later step |
|||||||||
| \(\displaystyle \) | \(=\) | \(\, \displaystyle \ \ \iint_R \Biggl( \, \) | \(\displaystyle \dfrac {\partial f_1} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \paren {- \dfrac {\partial f_1} {\partial z} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t} - \dfrac {\partial f_1} {\partial y} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t} - \dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t} }\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \dfrac {\partial f_2} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_2} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \paren {- \dfrac {\partial f_2} {\partial z} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t} - \dfrac {\partial f_2} {\partial x} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t} - \dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t} }\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \dfrac {\partial f_3} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t} + \dfrac {\partial f_3} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t} + \dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \paren {- \dfrac {\partial f_3} {\partial y} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t} - \dfrac {\partial f_3} {\partial x} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t} - \dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t} } \Biggr) \rd s \rd t\) | grouping terms by sign,
but keeping terms that reference like components of $\mathbf F$ together |
|||||||||
| \(\displaystyle \) | \(=\) | \(\, \displaystyle \ \ \iint_R \Biggl( \, \) | \(\displaystyle \dfrac {\partial f_1} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \dfrac {\partial f_2} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_2} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \dfrac {\partial f_3} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t} + \dfrac {\partial f_3} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t} + \dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ - \Biggl( \, \) | \(\displaystyle \dfrac {\partial f_1} {\partial z} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \dfrac {\partial f_2} {\partial z} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t} + \dfrac {\partial f_2} {\partial x} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \dfrac {\partial f_3} {\partial y} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_3} {\partial x} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t} \Biggr) \Biggr) \rd s \rd t\) | grouping all negative terms together and factoring out the negative sign | |||||||||
| \(\displaystyle \) | \(=\) | \(\, \displaystyle \iint_R \Biggl( \Biggl( \, \) | \(\displaystyle \dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_1} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \dfrac {\partial f_2} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_2} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \dfrac {\partial f_3} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t} + \dfrac {\partial f_3} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t} + \dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t} \Biggr)\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ - \Biggl( \, \) | \(\displaystyle \dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_1} {\partial y} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_1} {\partial z} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \dfrac {\partial f_2} {\partial x} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_2} {\partial z} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t}\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ + \, \) | \(\displaystyle \dfrac {\partial f_3} {\partial x} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_3} {\partial y} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t} \Biggr) \Biggr) \rd s \rd t\) | rearranging into groups,
in order to clarify use of the Chain Rule for Real-Valued Functions in the next step |
|||||||||
| \(\displaystyle \) | \(=\) | \(\, \displaystyle \ \ \iint_R \Biggl( \, \) | \(\displaystyle \paren {\dfrac {\partial f_1} {\partial s} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_2} {\partial s} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_3} {\partial s} \dfrac{\partial z}{\partial t} }\) | ||||||||||
| \(\displaystyle \) | \(\) | \(\, \displaystyle \ \ \ \ \ \ \ \ - \, \) | \(\displaystyle \paren {\dfrac {\partial f_1} {\partial t} \dfrac{\partial x}{\partial t} + \dfrac {\partial f_2} {\partial t} \dfrac{\partial y}{\partial t} + \dfrac {\partial f_3} {\partial t} \dfrac{\partial z}{\partial t} } \Biggr) \rd s \rd t\) | Chain Rule for Real-Valued Functions | |||||||||
| \(\displaystyle \) | \(=\) | \(\, \displaystyle \ \ \iint_R \Biggl( \, \) | \(\displaystyle \dfrac {\partial \mathbf F} {\partial s} \cdot \dfrac{\partial \mathbf r}{\partial t} - \dfrac {\partial \mathbf F} {\partial t} \cdot \dfrac{\partial \mathbf r}{\partial s} \Biggr) \rd s \rd t\) | Definition:Dot Product | |||||||||
| \(\displaystyle \) | \(=\) | \(\, \displaystyle \ \ \iint_R \Biggl( \, \) | \(\displaystyle \frac{\partial G_2}{\partial s} - \frac{\partial G_1}{\partial t} \Biggr) \rd s \rd t\) |
By Green's Theorem, this can be written as:
- $\displaystyle \int_{\partial R} \mathbf G \cdot \rd \mathbf s$
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
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Stokes's theorem (G.G. Stokes, 1854)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: Stokes's Theorem
