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} .

* __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**) − ∇^{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].

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 20 feb 2021