In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Hadamard's lemma, named after Jacques Hadamard

Jacques Hadamard

Jacques Salomon Hadamard FRS was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.-Biography:...

, is essentially a first-order form of Taylor's theorem

Taylor's theorem

In calculus, Taylor's theorem gives an approximation of a k times differentiable function around a given point by a k-th order Taylor-polynomial. For analytic functions the Taylor polynomials at a given point are finite order truncations of its Taylor's series, which completely determines the...

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

Statement

Let ƒ be a smooth, real-valued function defined on an open, star-convex neighborhood U of a point a in n-dimensional Euclidean space. Then ƒ(x) can be expressed, for all x in U, in the form:


where each g_i is a smooth function on U, a = (a₁,...,a_n), and x = (x₁,...,x_n).

Proof

Let x be in U. Let h be the map from [0,1] to the real numbers defined by


Then since


we have


But additionally, h(1) − h(0) = f(x) − f(a), so if we let


we have proven the theorem.