Shortcut model

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:

$f(x)=f(a)+\sum _{i=1}^n(x_i-a_i)g_i(x),$

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

$h^{\prime }(t)=\sum _{i=1}^n\frac{\partial f}{\partial x_i}(a+t(x-a))(x_i-a_i),$

we have

$h(1)-h(0)={\int }_0^1{h}^{\prime }(t)dt={\int }_0^1\sum _{i=1}^n\frac{\partial f}{\partial x_i}(a+t(x-a))(x_i-a_i)dt=\sum _{i=1}^n(x_i-a_i){\int }_0^1\frac{\partial f}{\partial x_i}(a+t(x-a))dt.$

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

$g_i(x)={\int }_0^1\frac{\partial f}{\partial x_i}(a+t(x-a))dt,$

we have proven the theorem.

References

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