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)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(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
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
The Stokes theorem immediately implies that the integration of an exact differential form with compact support vanishes. But in fact also the converse is true: If the integration of a differential form with compact support of top degree vanishes, then it is exact (prop. below) .
This statement underlies for instance
higher Lie integration by the “path method”
the interpretation of the BV-BRST complex as computing expectation values (see there).
Let $X$ be an oriented connected smooth manifold of finite dimension. Let $n=\dim X$ and write $\Omega^n_{cp}(X)$ for the vector space of differential n-forms with compact support and
for the linear map to the real numbers given by integration of differential forms.
Then the kernel of this map is precisely the exact differential forms
hence the image of the de Rham differential $d \colon \Omega^{n-1}_{cp} \to \Omega^n_{cp}(X)$.
(e.g. Lafointaine 15, section 7.3, theorem 7.5)
At least when $X = \mathbb{R}^n$ is a Cartesian space, then the statement of prop. also holds in smoothly indexed sets of smooth differential forms.
(e.g. Lafointaine 15, section 7.3, lemma 7.3)
Lie integration (proof of this prop.)