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
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
A vector field is a section of a vector bundle.
More specifically, a vector field is a tangent vector field which is a section of a tangent bundle. Hence this is a function which picks a tangent vector at each point of a manifold, such that this assignment is suitably differentiable.
Equivalently this is a rank $(1,0)$-tensor field on $X$
Vector fields may be identified with derivations on the algebra of smooth functions. See the article derivations of smooth functions are vector fields.
For instance section 1.3 and 1.4 of
Last revised on June 21, 2024 at 09:49:46. See the history of this page for a list of all contributions to it.