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)
Let $(X,g)$ be a Riemannian manifold and $f \in C^\infty(X)$ a function.
The gradient of $f$ is the vector field
where $d_{dR} : C^\infty(X) \to \Omega^1(X)$ is the de Rham differential.
This is the unique vector field $\nabla f$ such that
or equivalently, if the manifold is oriented, this is the unique vector field such that
where $vol_g$ is the volume form and $\star_g$ is the Hodge star operator induced by $g$. (The result is independent of orientation, which can be made explicit by interpreting both $vol$ and $\star$ as valued in pseudoforms.)
Alternatively, the gradient of a scalar field $A$ in some point $x\in M$ is calculated (or alternatively defined) by the integral formula
where $D$ runs over the domains with smooth boundary $\partial D$ containing point $x$ and $\vec{n}$ is the unit vector of outer normal to the surface $S$. The formula does not depend on the shape of boundaries taken in limiting process, so one can typically take a coordinate chart and balls with decreasing radius in this particular coordinate chart.
If $(M,g)$ is the Cartesian space $\mathbb{R}^n$ endowed with the standard Euclidean metric, then
This is the classical gradient from vector analysis?.
In many classical applications of the gradient in vector analysis?, the Riemannian structure is actually irrelevant, and the gradient can be replaced with the differential 1-form.