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Definition/Summary
Stokes' Theorem (sometimes called the "Generalized Stokes' Theorem") is a theorem pertaining to integration of differential forms in differential geometry that vastly generalizes several theorems in analysis and calculus. Simply stated, it says that the integral of the exterior derivative of a differential form over an orientable manifold is equivalent to the integral of the differential form over the boundary of that manifold.
Equations
Let ##\alpha## be a differential form on an orientable manifold ##M##. Then,
$$\int\limits_M \, d\alpha = \int\limits_{\partial M} \alpha .$$
Extended explanation
Many theorems from calculus and analysis are actually specific cases of Stokes' Theorem. For example, consider the Fundamental Theorem of Calculus, in the form ##\int_{a}^{b}f^\prime(x) \, dx = f(b)-f(a)##. If we write it in the form ##\int\limits_{[a,b]} \, df = \int\limits_{\partial[a,b]}f##, the relation clearly becomes a special case of Stokes' Theorem.
The Divergence Theorem and Green's Theorem are also special cases.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
Stokes' Theorem (sometimes called the "Generalized Stokes' Theorem") is a theorem pertaining to integration of differential forms in differential geometry that vastly generalizes several theorems in analysis and calculus. Simply stated, it says that the integral of the exterior derivative of a differential form over an orientable manifold is equivalent to the integral of the differential form over the boundary of that manifold.
Equations
Let ##\alpha## be a differential form on an orientable manifold ##M##. Then,
$$\int\limits_M \, d\alpha = \int\limits_{\partial M} \alpha .$$
Extended explanation
Many theorems from calculus and analysis are actually specific cases of Stokes' Theorem. For example, consider the Fundamental Theorem of Calculus, in the form ##\int_{a}^{b}f^\prime(x) \, dx = f(b)-f(a)##. If we write it in the form ##\int\limits_{[a,b]} \, df = \int\limits_{\partial[a,b]}f##, the relation clearly becomes a special case of Stokes' Theorem.
The Divergence Theorem and Green's Theorem are also special cases.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!