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)
In Riemannian geometry, the curl or rotation of a vector field $X$ over an oriented $3$-dimensional Riemannian manifold $(M,g)$ is the vector field $curl(X)$ (or $rot(X)$) defined by
where $\star_g$ is the Hodge star operator of $(M,g)$,
Alternatively, the curl/rotation of a vector field $\vec\mathcal{A}$ in some point $x\in M$ is calculated (or alternatively defined) by the integral formula
where $D$ runs over the smooth (pseudo)-oriented surfaces (smooth submanifolds of dimension $2$) containing the point $x$ and with smooth boundary $\partial D$, $\vec{n}$ is the unit vector of outer normal to the surface $S$, and $\vec{t}$ is the unit vector tangent to the curve $\partial S$. 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.
(We use the orientation of $M$ in the Hodge dual, or alternatively in determining the direction of $\vec{n}$ from the orientation of $S$ or the direction of $\vec{t}$ fom the pseudo-orientation of $S$.)
More generally, if $(M,g)$ is a Riemannian manifold whose cotangent spaces (equivalently, tangent spaces) are smoothly equipped with a binary cross product $⨉\colon \Omega^2(M;R) \to \Omega^1(M;R)$, then the curl of any vector field $X$ is
However, this is not as general as it may appear:
There are also cross products of other arity? in other dimensions; using essentially the same formula, we can take the curl of a $k$-vector field? if we have a smooth $(k+1)$-ary cross product.
If $(M,g)$ is $\mathbb{R}^3$ endowed with the canonical Euclidean metric, then the curl of a vector field $(X^1,X^2,X^3) = X^1\partial_1 + X^2\partial_2 + X^3\partial_3$ is
This is the classical curl from vector analysis?.
In many classical applications of the curl in vector analysis?, the Riemannian structure is actually irrelevant, and the gradient can be replaced with the deRham differential $d_{dR}$. That is, $X$ is treated as the $1$-form $g(X)$, its curl is treated as the $2$-form $d_{dR} g(X)$, and once these identifications are made there is no need to involve $g$ at all.