In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel. Let the Taylor series be a power series with real coefficients with radius of convergence Suppose that the series converges. Then is continuous from the left at that is, The same theorem holds for complex power series provided that entirely within a single Stolz sector, that is, a region of the open unit disk where for some fixed finite .
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. Initially proved as Riemann's inequality by , the theorem reached its definitive form for Riemann surfaces after work of Riemann's short-lived student .
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, Green’s theorem and Stokes' theorem are the cases of a surface in or and the divergence theorem is the case of a volume in Hence, the theorem is sometimes referred to as the Fundamental Theorem of Multivariate Calculus.
This article lists some of the most useful coordinate charts in some of the most useful examples of Riemannian manifolds. The notion of a coordinate chart is fundamental to various notions of a manifold which are used in mathematics. In order of increasing level of structure: topological manifold smooth manifold Riemannian manifold and semi-Riemannian manifold The key feature of the last two examples is a defined metric tensor which can be used to integrate along a curve, such as a geodesic curve.
In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves. For a compact, connected, orientable surface , the Euler characteristic is where g is the genus (the number of handles), since the Betti numbers are .