I would like to verify the identity $ \oint \vec F \cdot (\hat i dx + \hat j dy) + \oint \vec F \cdot (\hat i dx + \hat j dy) + \oint \vec F \cdot (\hat i dx + \hat j dy) = \oint \vec F \cdot (\hat i dx + \hat j dy + \hat k dz) $ If it is incorrect then what would be the correct identity. Green's theorem is special case of Stokes's theorem. How do we arrive at Stokes's theorem using Green's theorem?
How to arrive at Stokes's theorem from Green's theorem?
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0Perhaps this would be more productive if you explicitly stated the version of Stokes's Theorem that you are talking about. Some statements are **very** general, and nowhere near Green's Theorem. Some are more specific, though, and we might actually be able to do what you wish. – 2012-06-19
1 Answers
Time and space does not permit a complete answer, but here is an outline of one way to do it.
First, note that Green's theorem in the plane (applied to $f\partial g/\partial u$ and $f\partial g/\partial v$) leads to $\newcommand{\pd}[2]{\frac{\partial#1}{\partial#2}} \iint\limits_D\Big(\pd fu\pd gv-\pd fv\pd gu\Big)\,du\,dv =\oint\limits_J f\,dg$ where $D$ is a “sufficiently nice” region in the plane and $J$ is its boundary curve.
Next, assume a three-dimensional surface $S$ is parametrized by $\mathbf{r}\colon D\to\mathbb{R}^3$, and that $\mathbf{F}$ is a vector field. Now you can prove the identity $ (\operatorname{curl}\mathbf{F})\cdot \Big(\pd{\mathbf{r}}{u}\times\pd{\mathbf{r}}{v}\Big) =\pd{\mathbf{F}}{u}\cdot\pd{\mathbf{r}}{v} -\pd{\mathbf{F}}{v}\cdot\pd{\mathbf{r}}{u}$ and discover that each component function of $\mathbf{F}$ in this equation gives rise to a term of the form of the integrand on the left in the first equation. I.e., you let $f$ be each of the components of $\mathbf{F}\circ\mathbf{r}$ in turn, with $g$ being the corresponding comonent of $\mathbf{r}$, and add the three resulting equations together. You now have Stokes's theorem as written out using the given parametrization of $S$.
“Some assembly required.”
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0The above answer is extracted from [a note](http://www.math.ntnu.no/~hanche/kurs/mat2/2003v/stokes.pdf) I wrote about this for a class in 2003; unfortunately, the note is in Norwegian, but perhaps you can glean some of the missing details from it anyhow. – 2012-06-19