Vector Properties

Vector Properties Swapnil Sunil Jain July 18, 2006

Vector Properties

Some Unconventional Syntax:

Axe^x+Aye^y+Aze^z = Axe^x+Aye^y+Aϕe^z
dxdy = dxdy
dxdydz = dxdydz
t1t2𝑑tf(t)g(t) = t1t2f(t)g(t)𝑑t
x = x
dΣ = material surface element
f(r) = f(x,y,z)
𝔼3 = 3


AB i=13AiBi
A×B |𝕖^𝕩𝕖^𝕪𝕖^𝕫AxAyAzBxByBz|
[A,B,C] A(B×C)=(A×B)C=|AxAyAzBxByBzCxCyCz|
Δ 2(Laplace operator or Laplacian)
2 2-1c2t2(D’Alembert or wave operator or D’Alembertian)
2A(A)-×(×A) = (2Ax,2Ay,2Az)=Ax+Ay+Az(Vector Laplacian)
A (a1x1+a2x2++anxn)
F = 0F is incompressible or solenoidal or divergence-free
×F = 0F is irrotational or conservative or curl-free
laplacian(F) div(grad(F))
Duf fu^  (Directional Derivative)
rotuF curl(F)u^  (Rotational Derivative)
helicity(F) curl(F)F
F = limδV01δVδSFn^𝑑S
(×F)n = limδS01δSδCFt^𝑑s
×F = limδV01δVδSn^×F𝑑S
f = limδV01δVδSn^f𝑑S
× ie^ihi×vi

Gradient, Divergence and Curl in Curvilinear Coordinates

For Cartesian coordinates: hx=hy=hz=1
For Cylindrical coordinates: hr=1,hϕ=r,hz=1
For Spherical coordinates: hr=1,hθ=r,hϕ=rsin(θ)
f = i1hifqie^i
×F = 1Ωie^ijkϵijkhi(hkFk)qj
F = i1Ωqi(ΩFihi)
2f = 1Ωiqi(Ωhi2fqi)
where ΩΠhi


u+vu+v  Triangle Inequality
|uv|uv  Cauchy-Schwarz Inequality

Product identities:

A(B×C) = B(C×A)=C(A×B)  Scalar Triple Product
A×(B×C) = B(AC)-C(AB)  Vector Triple Product
A×(B×C)+B×(C×A)+C×(A×B) = 0  Jacobi Identity
(A×B)(C×D) = A[B×(C×D)]=(AC)(BD)-(BC)(AD)Scalar Quadruple Product
(A×B)×(C×D) = (A×BD)C-(A×BC)D=[C,D,A]B-[C,D,B]AVector Quadruple Product
A×[B×(C×D)] = (A×C)(BD)-(BC)(A×D)

Gradient Identities:

(f+g) = f+g
(fg) = fg+gf
(AB) = A×(×B)+B×(×A)+(A)B+(B)A
(A×B) = (A)×B-(B)×A  gradient of vector??
(fA) = (f)A+f(A)  gradient of vector??

Divergence Identities:

(A+B) = A+B
(fA) = f(A)+A(f)
(A×B) = B(×A)-A(×B)
(AB) = (A)B+A(B)=(A)B+(A)B

Curl Identities:

×(A+B) = ×A+×B
×(fA) = f(×A)-A×(f)
×(A×B) = (B)A-(A)B+(B)A-(A)B
×(AB) = (A)B-A×(B)

Laplacian Identities:

2(fg) = g2f+2fg+f2g
2(fA) = =f2A+A2f+2(f)A

Mixed Identities:

(×A) = 0
×f = 0
×(fg) = f×g
fg = f2g+fg
×(×A) = (A)-2Aonly true for rectangular coordinates

DifferentialMathworldPlanetmath Identities:

ddt(fA) = fdAdt+dfdtA
ddt(AB) = AdBdt+BdAdt
ddt(A×B) = A×dBdt+B×dAdt

Integral Identities:

Gauss’/Divergence Thm:

Standard Form:

𝒮F𝑑A = 𝒱(F)𝑑V(𝒮encloses𝒱)


𝒮Fn𝑑S = 𝒱(F)𝑑V(𝒮encloses𝒱)

Stokes’ Thm:

Standard Form:

𝒞F𝑑l = 𝒮(×F)n𝑑S(𝒮boundedby𝒞)


𝒞F𝑑l = 𝒮(×F)𝑑A(𝒮boundedby𝒞)
𝒞Ft𝑑s = 𝒮(×F)n𝑑S(𝒮boundedby𝒞)
𝒞Ft𝑑s = 𝒮(×F)𝑑A(𝒮boundedby𝒞)

Standard Form:

𝒞(f)𝑑l = f(r2)-f(r1)(𝒞goesfromr1tor2)

Integration by Parts for Vectors

𝒱f(A)𝑑V = 𝒮fA𝑑a-𝒱A(f)𝑑V

Integral Form of Maxwell’s Equations:

𝒮Dn^d2A = Qfreeenc
𝒮Bn^d2A = 0
𝒞E𝑑l = -dΦBdt
𝒞H𝑑l = Ifreeenc+dΦDdt
ΦB = 𝒮Bn^d2A
ΦD = 𝒮Dn^d2A

Differential FormMathworldPlanetmath of Maxwell’s Equations:


Complex Differential Form of Maxwell’s Equations:

where McB+iE

EM Equations:

2A-1c22At2 = -μ0J  Vector Poisson’s Equation
2V-1c22Vt2 = -ρϵ  Scalar Poisson’s Equation
A(r) = μ04πJ(r)|r-r|d3r
ϕ(r) = 14πϵ0ρ(r)|r-r|d3r
Title Vector Properties
Canonical name VectorProperties1
Date of creation 2013-03-11 19:25:33
Last modified on 2013-03-11 19:25:33
Owner swapnizzle (13346)
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Numerical id 1
Author swapnizzle (0)
Entry type Definition