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 gab ∂b f if there is a metric gab 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
= εijk ∂jvk .
* Twist: In a 4D manifold, the twist of a vector field va is
ωa:= εabcd vb ∇c 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].
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 (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 σ-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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