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 v^a = |g|^−1/2 (|g|^1/2 v^m)_{, m} .

* Curl: In a 3D flat manifold, the curl of a vector field vⁱ is (∇ × v)_i = ε_ijk ∂_jv_k .
* Twist: In a 4D manifold, the twist of a vector field v^a is

ω_a:= ε_abcd v^b ∇^c v^d ;

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 D_a X^a dv = ∫_∂(M) X^a dS_a .
$ First Green identity: For a spatial region V with boundary S:= ∂V, and all functions f, g on V,

∫_V (f ∇²g + ∇f · ∇g) dv = ∫_S f ∇g · dσ .

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

∫_V (f ∇²g − g ∇²f) dv = ∫_S (f ∇g − g ∇f) · 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].
> Online resources: see Wikipedia page on the divergence theorem.

Other Topics > s.a. lie derivative; tensor fields [derivatives].
@ Fractional: Meerschaert et al PhyA(06) [and advection-dispersion equation]; Tarasov AP(08)-a0907 [and Maxwell's equations].

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