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)
Given a linear differential operator $P$, then a generalized solution or weak solution of the corresponding homogenous differential equation is a distribution $u$ (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 $b \in \mathcal{D}$ we have
where $P^\ast$ is the formally adjoint differential operator.
This is such that in the case that $u = u_f$ happens to be a non-singular distribution given by an ordinary smooth function $f$, then $u_f$ is a generalized solution precisely if $f$ 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 $u$ are the fundamental solutions or Green's functions of $P$.