manifold
(noun)
a topological space that looks locally like the "ordinary" Euclidean space and is Hausdorff
Examples of manifold in the following topics:
-
Surfaces in Space
- A surface is a two-dimensional, topological manifold.
-
Stokes' Theorem
- 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.
- Stokes' theorem says that the integral of a differential form $\omega$ over the boundary of some orientable manifold $\Omega$ is equal to the integral of its exterior derivative $d\omega$ over the whole of $\Omega$, i.e.:
-
Parametric Equations
- The notion of parametric equation has been generalized to surfaces of higher dimension with a number of parameters equal to the dimension of the manifold (dimension one and one parameter for curves, dimension two and two parameters for surfaces, etc.)
-
Functions of Several Variables
- In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes' theorem, which applies to the integration of differential forms over manifolds.