mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Hadamard's lemma, named after

Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry, and partial differential equations. Biography The son of a tea ...

, is essentially a first-order form of

Taylor's theorem In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the k-th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation a ...

, in which we can express a smooth, real-valued function exactly in a convenient manner.

Statement

Proof

Consequences and applications

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References