Hadamard's lemma

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In mathematics, Hadamard's lemma, named after Jacques Hadamard, is essentially a first-order form of Taylor's theorem, in which we can express a smooth, real-valued function exactly in a convenient manner.


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 gi is a smooth function on U, a = (a1,...,an), and x = (x1,...,xn).


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.


  • Nestruev, Jet (2002). Smooth manifolds and observables. Berlin: Springer. ISBN 0-387-95543-7.