Hadamard's lemma - Alchetron, The Free Social Encyclopedia

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:

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

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

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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Contents

Statement

Proof

References