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
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
When a multifunction is differentiated with respect to any one of its arguments alone, holding the others fixed, then we are engaged in partial differentiation.
Very generally, let be a family of differentiable spaces (in some sense), let be another such space, and let be a differentiable map to from a subspace of the cartesian product . Let be a relevant differential or derivative operator, and let be the composite
of the inclusion map of and the th product projection (the th coordinate). Then under good conditions, we have
for a unique family of linear operators, the partial derivatives of with respect to this decomposition of . The term , which may be denoted , is similarly a partial differential of .
More precisely, we choose a category of differentiable spaces and differentiable maps between them, on which there is an endofunctor that takes each space to a notion of tangent bundle , which is assumed to be a vector bundle over , and takes a map to . (Note that this isn’t the case for generalised smooth spaces, but we could take microlinear spaces, as well as more familiar examples such as differentiable manifolds.) Then , is a linear operator between stalks (for a point in ), and the sum takes place in the vector space .
We can extend this if we work in a cartesian closed category of generalised smooth spaces. As in the above, let be a family of smooth spaces and another smooth space. For simplicity, let be a smooth map (aka morphism in the category) defined on the whole product (so we take in the above). For we can use the cartesian closed structure to define a morphism
Thus given a morphism we get a parametrised family of morphisms which we could write (using parameters) as . As taking the derivative is a smooth functor, we can partially differentiate the morphisms by applying differentiation to the morphisms , thus yielding as a morphism . In full, is the image of under the chain of morphisms:
This is the partial derivative of along .
When the coordinates are given individual names , one usually writes for (where replaces ); but is less ambiguous. Similarly, one can write for the partial differential , which is when replaces . (If is thought of as an infinitesimal change in , then is an infinitesimal change subject to the condition that are fixed.) Then
which explains the notation and why ‘’ looks like ‘’. (The reason for the latter equality is that is the Kronecker delta .)
The Kock-Lawvere axiom for the axiomatization of differentiation in synthetic differential geometry was introduced in
Last revised on November 4, 2017 at 23:54:35. See the history of this page for a list of all contributions to it.