Stokes Theorem
Stokes' theorem
In differential geometry, Stokes' theorem (or Stokes's theorem, also called the generalized Stokes' theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. The general formulation reads: If is an (n − 1)-form with compact support on , and denotes the boundary of with its induced orientation, and denotes the exterior differential operator, then.
The modern Stokes' theorem is a generalization of a classical result first discovered by Lord Kelvin, who communicated it to George Stokes in July 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name. The classical Kelvin–Stokes theorem which relates the surface integral of the curl of a vector field over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary, is a special case of the general Stokes theorem (with n = 2) once we identify a vector field with a 1 form (see more below). Likewise, the classical Gauss or divergence theorem is a special case of the general Stokes theorem once we identify a vectorfield with a n - 1 form. Similar ideas apply for Green's theorem and the Gradient theorem.
Stokes' theorem is a vast generalization of this theorem in the following sense.
By the choice of F, . In the parlance of differential forms, this is saying that f(x) dx is the exterior derivative of the 0-form, i.e. function, F: in other words, that dF = f dx. The general Stokes theorem applies to higher differential forms instead of F.
A closed interval [a, b] is a simple example of a one-dimensional manifold with boundary. Its boundary is the set consisting of the two points a and b. Integrating f over the interval may be generalized to integrating forms on a higher-dimensional manifold. Two technical conditions are needed: the manifold has to be orientable, and the form has to be compactly supported
In differential geometry, Stokes' theorem (or Stokes's theorem, also called the generalized Stokes' theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. The general formulation reads: If is an (n − 1)-form with compact support on , and denotes the boundary of with its induced orientation, and denotes the exterior differential operator, then.
The modern Stokes' theorem is a generalization of a classical result first discovered by Lord Kelvin, who communicated it to George Stokes in July 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name. The classical Kelvin–Stokes theorem which relates the surface integral of the curl of a vector field over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary, is a special case of the general Stokes theorem (with n = 2) once we identify a vector field with a 1 form (see more below). Likewise, the classical Gauss or divergence theorem is a special case of the general Stokes theorem once we identify a vectorfield with a n - 1 form. Similar ideas apply for Green's theorem and the Gradient theorem.
Stokes' theorem is a vast generalization of this theorem in the following sense.
By the choice of F, . In the parlance of differential forms, this is saying that f(x) dx is the exterior derivative of the 0-form, i.e. function, F: in other words, that dF = f dx. The general Stokes theorem applies to higher differential forms instead of F.
A closed interval [a, b] is a simple example of a one-dimensional manifold with boundary. Its boundary is the set consisting of the two points a and b. Integrating f over the interval may be generalized to integrating forms on a higher-dimensional manifold. Two technical conditions are needed: the manifold has to be orientable, and the form has to be compactly supported