Hadamard's lemma

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.

Hadamard's lemma[1] — Let

be a smooth, real-valued function defined on an open, star-convex neighborhood

-dimensional Euclidean space.

is a smooth function on

Let

{\displaystyle h(t)=f(a+t(x-a))\qquad {\text{ for all }}t\in [0,1].}

{\displaystyle h'(t)=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}(a+t(x-a))\left(x_{i}-a_{i}\right),}

which implies

{\displaystyle {\begin{aligned}h(1)-h(0)&=\int _{0}^{1}h'(t)\,dt\\&=\int _{0}^{1}\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}(a+t(x-a))\left(x_{i}-a_{i}\right)\,dt\\&=\sum _{i=1}^{n}\left(x_{i}-a_{i}\right)\int _{0}^{1}{\frac {\partial f}{\partial x_{i}}}(a+t(x-a))\,dt.\end{aligned}}}

But additionally,

so by letting

{\displaystyle g_{i}(x)=\int _{0}^{1}{\frac {\partial f}{\partial x_{i}}}(a+t(x-a))\,dt,}

the theorem has been proven.

Corollary[1] — If

is a smooth function on

Explicitly, this conclusion means that the function

that sends

{\displaystyle {\begin{cases}f(x)/x&{\text{ if }}x\neq 0\\\lim _{t\to 0}f(t)/t&{\text{ if }}x=0\\\end{cases}}}

is a well-defined smooth function on

By Hadamard's lemma, there exists some

implies

Corollary[1] — If

are distinct points and

is a smooth function that satisfies

then there exist smooth functions

By applying an invertible affine linear change in coordinates, it may be assumed without loss of generality that

By Hadamard's lemma, there exist

let

terms above has the desired properties.