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)
Given a linear differential operator , then a generalized solution or weak solution of the corresponding homogenous differential equation is a distribution (hence a “generalized function” via this prop) satisfying
where the differential operator on the left now acts via derivatives of distributions. This means that for all we have
where is the formally adjoint differential operator.
This is such that in the case that happens to be a non-singular distribution given by an ordinary smooth function , then is a generalized solution precisely if is an ordinary solution:
Similarly there are generalized solutions for the inhomogeneous equation, and in fact now the inhomogeneity may itself be a distribution. In particular for delta distribution-inhomogeneity
the generalized solutions are the fundamental solutions or Green's functions of .
Last revised on November 23, 2017 at 14:27:03. See the history of this page for a list of all contributions to it.