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  "body": "\n## Summary\n\nGlobal analysis is the study of differential equations, variational problems and analysis on [[smooth manifolds|F45VQ47GPV]] as whole objects.\nIt looks at properties that depend on the entire manifold and not only on one chart.\nIts tools include [[differential operators on vector bundles|CGBX1N9V8H]], [[spaces of smooth maps|BW12J14X0Z]], [[critical point theory|9ZQC7ARWTT]] and [[index theory|675MAXBQN6]].\nIt links analysis with the topology of manifolds, for instance through Morse theory and the Atiyah–Singer index theorem.\n\n## Explanations\n\n### L1 human\n\nClassical calculus studies functions near one point or on a piece of flat space.\nGlobal analysis asks what happens on a curved space, such as a sphere or a torus, when the whole space is taken into account.\nA typical question is how many solutions an equation has on the whole space, or how many critical points a function must have.\nFor example, every smooth function on a sphere has at least two critical points, a maximum and a minimum, because the sphere is compact.\nThe shape of the space thus forces facts about the analysis on it.\n",
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