“Hadamard's lemma”的意思、由来-开放百科全书

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Hadamard's lemma

释义

Statement
Proof
References

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.

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,{{nbsp}}…,{{nbsp}}a_n), and x = (x₁,{{nbsp}}…,{{nbsp}}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.

References

{{cite book |author=Nestruev, Jet |title=Smooth manifolds and observables |publisher=Springer |location=Berlin |year=2002 |pages= |isbn=0-387-95543-7 |oclc= |doi=}}

2 : Real analysis|Theorems in analysis

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