Calculus on Euclidean space

rdf:langString Calculus on Euclidean space

rdf:langString Lemma

rdf:langString Theorem

rdf:langString Corollary

rdf:langString Inverse function theorem

rdf:langString Mean value inequality

rdf:langString The Gauss–Bonnet theorem

rdf:langString In mathematics, calculus on Euclidean space is a generalization of calculus of functions in one or several variables to calculus of functions on Euclidean space as well as a finite-dimensional real vector space. This calculus is also known as advanced calculus, especially in the United States. It is similar to multivariable calculus but is somehow more sophisticated in that it uses linear algebra (or some functional analysis) more extensively and covers some concepts from differential geometry such as differential forms and Stokes' formula in terms of differential forms. This extensive use of linear algebra also allows a natural generalization of multivariable calculus to calculus on Banach spaces or topological vector spaces. Calculus on Euclidean space is also a local model of calculus on manifolds, a theory of functions on manifolds.

rdf:langString For a bounded region in of dimension whose boundary is a union of finitely many -subsets, if is oriented, then : for any differential -form on the boundary of .

rdf:langString If is a simply connected open subset of , then each closed form on is exact.

rdf:langString Let be a differentiable function from an open subset of such that has rank at every point in . For a differentiable function , if attains either a maximum or minimum at a point in , then there exists real numbers such that :. In other words, is a stationary point of .

rdf:langString Each -manifold can be embedded into .

rdf:langString If is a continuous function on a closed rectangle , then :

rdf:langString Let be as above and a map for some open subset of . If is differentiable at and differentiable at , then the composition is differentiable at with the derivative :

rdf:langString A bounded function on a closed rectangle is integrable if and only if the set has measure zero.

rdf:langString If are continuously differentiable, then is continuously differentiable.

rdf:langString For each bounded surface in , we have: : where is the Euler characteristic of and the curvature.

rdf:langString Let be a map between open subsets in . If is continuously differentiable and is bijective, there exists neighborhoods of and the inverse that is continuously differentiable .

rdf:langString If are locally integrable functions on an open subset such that : for every . Then almost everywhere. If, in addition, are continuous, then .

rdf:langString Given the map as above and points in such that the line segment between lies in , if is continuous on and is differentiable on the interior, then, for any vector , : where