Given a relation between two variables expressed by an equation of the form f (x,y) = k, we often want to “solve for y.” That is, for each given x in some interval, we expect to find one and only one value y = φ(x) that satisfies the relation. The function j is thus implicit in the relation; geometrically, the locus of the equation f (x,y) =k is a curve in the (x,y)-plane that serves as the graph of the function y = φ(x). The problem of implicit functions—and the aim of this chapter—is to determine the function φ from the relation f, or at least to determine that φ existswhen its exact form cannot be found. There are analogues of this problem in all dimensions;that is, x and y can be vectors, and the relation f (x,y) = k can expand intoa set of equations. However, we begin our analysis with a single equation, becausethe various impediments to finding the implicit function already occur there.
KeywordsPartial Derivative Tangent Plane Implicit Function Theorem Regular Point Maximal Rank
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