 Vector Calculus

Main Vector Derivatives
* Gradient: Given a (differentiable) function f on a manifold M, its gradient is the 1-form ∂a f, which can be made into a vector $$g^{ab} \partial^~_b\,f$$ if there is a metric $$g^~_{ab}$$ available.
* Divergence: Given a volume element ω on M, (div X) ω = $$\cal L$$X ω; A convenient formula is

a va = |g|−1/2 (|g|1/2 vm), m .

* Curl: In a 3D flat manifold, the curl of a vector field vi is (∇ × v)i = εijkjvk .
* Twist: In a 4D manifold, the twist of a vector field va is

ωa:= εabcd vbc vd ;

In an n-dimensional manifold the twist, defined as above, will be an (n−3)-form, but one may be able to define a 1-form if there are preferred vector fields to contract it with.
@ References: Romano & Price AJP(12)jun [shear and vector gradient in the undergraduate curriculum]; Kim et al a1911 [graphical notation].

Differential Identities > s.a. vector fields.
* Useful formulas: Gradients, divergences and curls of products satisfy, for all functions f, g and all vector fields A,

∇(fg) = (∇f) g + f (∇g)

∇ · (f A) = (∇f) · A + f (∇ · A)

∇ × (f A) = (∇f) × A + f (∇ × A)

∇ × (∇ × A) = ∇(∇ · A) − ∇2A .

@ References: Tonti 75; Schey 05.

Integral Identities > s.a. integration on manifolds.
$Divergence (Gauss) theorem: For any manifold M with boundary ∂M, and any vector field X on M, M Da Xa dv = ∂(M) Xa dSa .$ First Green identity: For a spatial region V with boundary S:= ∂V, and all functions f, g on V,

V (f2g + ∇f · ∇g) dv = S fg · dσ .

\$ Second Green identity: (Green's theorem) For a spatial region V with boundary S:= ∂V, and all functions f, g on V,

V (f2gg2f) dv = S (fggf) · dσ .

@ General references: Pfeffer 12 [divergence theorem using Lebesgue integration, and sets of finite perimeter].
@ Generalized divergence theorem: Milton PRS(13) [sharp inequalities that generalize the divergence theorem]; > s.a. finsler spaces.
@ Other generalizations: Goldberg & Newman JMP(69); Mazur PLA(84) [Green identity for non-linear $$\sigma$$-models]; Dray & Hellaby JMP(94) [Gauss theorem for any signature]; Tarasov CNSNS(15)-a1503 [non-integer dimension and fractal media].