This page is about the lemma on Taylor polynomials of smooth functions. For Hadamard's formula in Lie theory see there.
synthetic differential geometry
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from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
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The Hadamard lemma says that the Taylor series of a smooth function on the real line around the origin has a remainder at order which is the product of (for 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 a smooth manifold and its -algebra of smooth functions (under pointwise multiplication), then there is a natural bijection between the smooth tangent vector fields on and the purely algebraic derivations of the algebra ;
the smooth infinitesimally thickened point is the same as in algebraic geometry: The quotient of the algebra of smooth functions on the real line by the ideal generated by the square of the canonical coordinate function is the ring of dual numbers:
together this implies that a tangent vector in a smooth manifold is equivalently a morphism of the form
of formal duals of -alghebras, from the infinitesimally thickened point .
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 on the real line there is a smooth function such that
This function is also called a Hadamard quotient.
It follows that
is the derivative of at 0. By applying this repeatedly the lemma says that has a partial Taylor series expansion whose remainder is a smooth function:
More generally, for smooth functions on any Cartesian space the lemma says that there are for each smooth functions such that
So at the origin these smooth functions compute the partial derivatives of
Holding fixed, put . 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 notion of a Fermat theory makes Hadamard’s lemma into an axiom. See there for more information.
The Hadamard lemma is due to Jacques Hadamard.
Review:
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
Mathema-philosophical discussion of the ingredients of the Hadamard lemma as analogous to an adjoint cylinder resembling a duality of opposites:
Last revised on September 6, 2024 at 18:05:31. See the history of this page for a list of all contributions to it.