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* v*^{i} is (∇ × **v**)_{i}
= *ε*_{ijk} ∂_{j}*v _{k} .*

*

*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].

**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**) – ∇^{2}**A** .

@ __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 }d*v* = ∫_{∂(M)} *X*^{a }d*S*_{a} .

$ __First Green identity__: For
a spatial region *V* with boundary *S*:= ∂*V*,
and all functions *f*, *g* on *V*,

∫_{V} (*f* ∇^{2}*g* +
∇*f* · ∇*g*)
d*v* = ∫_{S} *f* ∇*g* ·
d*σ* .

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

∫_{V} (*f* ∇^{2}*g* – *g* ∇^{2}*f*)
d*v* = ∫_{S}
(*f* ∇*g* – *g* ∇*f*) · d*σ* .

@ __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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