global analysis

{{Short description|Field of mathematical analysis}}

In mathematics, global analysis, also called analysis on manifolds, is the study of the global and topological properties of differential equations on manifolds and vector bundles.{{cite journal|last=Smale|first=S.|title=What is Global Analysis|journal=American Mathematical Monthly|date=January 1969|volume=76|issue=1|pages=4–9|doi=10.2307/2316777}}{{cite book|last=Richard S. Palais|title=Foundations of Global Non-Linear Analysis|url=http://vmm.math.uci.edu/PalaisPapers/FoundationsOfGlobalNonlinearAnalysis.pdf|date=1968|publisher=W.A. Benjamin, Inc.}} Global analysis uses techniques in infinite-dimensional manifold theory and topological spaces of mappings to classify behaviors of differential equations, particularly nonlinear differential equations.{{cite book|last=Andreas Kriegl and Peter W. Michor|title=The Convenient Setting of Global Analysis|url=https://www.mat.univie.ac.at/~michor/apbookh-ams.pdf|date=1991|publisher=American Mathematical Society|isbn=0-8218-0780-3|pages=1–7}} These spaces can include singularities and hence catastrophe theory is a part of global analysis. Optimization problems, such as finding geodesics on Riemannian manifolds, can be solved using differential equations, so that the calculus of variations overlaps with global analysis. Global analysis finds application in physics in the study of dynamical systems{{cite book|last=Marsden|first=Jerrold E.|title=Applications of global analysis in mathematical physics|date=1974|publisher=Publish or Perish, Inc.|location=Berkeley, CA.|isbn=0-914098-11-X|page=Chapter 2|url=https://archive.org/details/applicationsofgl00mars|url-access=registration}} and topological quantum field theory.