Differential Geometry
Exterior Derivative
The exterior derivative d raises the degree of a differential form by one and satisfies d^2 = 0
Stokes' Theorem (general)
The integral of a differential form over the boundary equals the integral of its exterior derivative over the interior
Green's Theorem
A line integral around a closed curve equals the double integral of the curl over the enclosed region
Divergence Theorem (Gauss)
The flux of a vector field through a closed surface equals the integral of its divergence over the enclosed volume
Poincaré Lemma
On a contractible domain, every closed differential form is exact
Frobenius Theorem (Integrability)
A smooth distribution is integrable to a foliation if and only if it is involutive under the Lie bracket
Gauss–Bonnet Theorem
The total Gaussian curvature of a closed surface equals 2π times its Euler characteristic
Riemannian Metric
A smooth positive-definite symmetric 2-tensor that measures lengths and angles on a manifold