Rahul Sharma

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.

Contents

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:

f ( x ) = f ( a ) + i = 1 n ( x i a i ) g i ( x ) ,

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

Proof

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

h ( t ) = f ( a + t ( x a ) ) .

Then since

h ( t ) = i = 1 n f x i ( a + t ( x a ) ) ( x i a i ) ,

we have

h ( 1 ) h ( 0 ) = 0 1 h ( t ) d t = 0 1 i = 1 n f x i ( a + t ( x a ) ) ( x i a i ) d t = i = 1 n ( x i a i ) 0 1 f x i ( a + t ( x a ) ) d t .

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

g i ( x ) = 0 1 f x i ( a + t ( x a ) ) d t ,

we have proven the theorem.

References

Hadamard's lemma Wikipedia


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