This page is about the lemma on Taylor polynomials of smooth functions. For Hadamard's formula in Lie theory see there.
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
The Hadamard lemma says that the Taylor series of a smooth function on the real line around the origin has a remainder at order $n$ which is the product of $x^{n+1}$ (for $x$ the canonical coordinate function) with another smooth function.
Simple as this may sound, it has profound consequences, as it means that smooth functions behave more like polynomials than the classical definition might suggest. For instance the Hadamard lemma directly implies that
derivations of smooth functions are vector fields: For $X$ a smooth manifold and $C^\infty(X)$ its $\mathbb{R}$-algebra of smooth functions (under pointwise multiplication), then there is a natural bijection between the smooth tangent vector fields on $X$ and the purely algebraic derivations of the algebra $C^\infty(X)$;
the smooth infinitesimally thickened point $\mathbb{D}$ is the same as in algebraic geometry: The quotient of the algebra of smooth functions $C^\infty(\mathbb{R}^1)$ on the real line by the ideal generated by the square $x^2$ of the canonical coordinate function $x$ is the ring of dual numbers:
together this implies that a tangent vector in a smooth manifold $X$ is equivalently a morphism of the form
of formal duals of $\mathbb{R}$-alghebras, from the infinitesimally thickened point $\mathbb{D}$.
This means that differential geometry has more in common with algebraic geometry than is manifest from the traditional definitions. In synthetic differential geometry one makes use of these facts to find a useful unified perspective. For exposition of this point see at geometry of physics -- supergeometry.
More generally one may ask for other types of function algebras which satisfy the conclusion of the Hadamard lemma. These turn out to be the algebras over those algebraic theories which are called Fermat theories. These are hence a crucial ingredient for well-adapted models of synthetic differential geometry.
(Hadamard lemma)
For every smooth function $f \in C^\infty(\mathbb{R})$ on the real line there is a smooth function $g$ such that
This function $g$ is also called a Hadamard quotient.
It follows that $g(0) = f'(0)$ is the derivative of $f$ at 0. By applying this repeatedly the lemma says that $f$ has a partial Taylor series expansion whose remainder $h$ is a smooth function:
More generally, for smooth functions on any Cartesian space $\mathbb{R}^n$ the lemma says that there are for each $f \in C^\infty(X)$ $n$ smooth functions $g_i$ such that
So at the origin these smooth functions compute the partial derivatives of $f$
Holding $x$ fixed, put $h(t) = f(t x)$. Then
where the second equality uses the chain rule. The lemma follows by putting
The Hadamard lemma implies in particular that
derivations of smooth functions are vector fields (see there for the proof);
the Taylor series of a smooth function is an asymptotic series (this example).
The Hadamard lemma is due to Jacques Hadamard.
Review includes
The Hadamard lemma is what makes the standard convenient models for synthetic differential geometry tick. Its role in this respect can be seen from proposition 1.2 on in