Vector
Calculus| Vector
Calculus |
In General > s.a. integration
on manifolds; vector fields.
* Useful formulas: Gradients,
divergences and curls of products satisfy, for all functions f, g and all vectors
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 92.
Green's Identities
$ First 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 ·
d
.
$ Second 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)
· d .
@ Generalization: Goldberg & Newman JMP(69); Mazur PLA(84)
[for non-linear -models].
Other Topics > s.a. lie
derivative; tensor
fields [derivatives].
@ Generalizations: Meerschaert et al PhyA(06) [fractional, and advection–dispersion
equation]
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
23 aug 2007