Vector

Author

Eric W. Weisstein
April 14, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/Vector.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Vector.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

n={Cos[θ]Sin[ϕ],Sin[θ]Sin[ϕ],Cos[ϕ]};

⟨(Subscript[n, i])⟩

Integrate[#Sin[ϕ],{θ,0,2π},{ϕ,0,π}]&/@n

⟨(Subscript[n, i])(Subscript[n, j])⟩

Integrate[Times@@# Sin[ϕ],{θ,0,2π},{ϕ,0,π}]&/@Flatten[Outer[List,n,n],1]/(4Pi)

Flatten[Table[KroneckerDelta[i,j]/3,{i,3},{j,3}]]

⟨(Subscript[n, i])(Subscript[n, j])(Subscript[n, k])⟩

Integrate[Times@@# Sin[ϕ],{θ,0,2π},{ϕ,0,π}]&/@Flatten[Outer[List,n,n,n],2]/(4Pi)

⟨(Subscript[n, i])(Subscript[n, j])(Subscript[n, k])(Subscript[n, l])⟩

Integrate[Times@@# Sin[ϕ],{θ,0,2π},{ϕ,0,π}]&/@Flatten[Outer[List,n,n,n,n],3]/(4Pi)

Table[(#[i,k]#[l,m]+#[i,l]#[k,m]+#[i,m]#[k,l]&@KroneckerDelta)/15,{i,3},{k,3},{l,3},{m,3}]//Flatten

Table[{{i,k,l,m},(#[i,k]#[l,m]+#[i,l]#[k,m]+#[i,m]#[k,l]&@KroneckerDelta)/15},{i,3},{k,3},{l,3},{m,3}]

⟨(((a·(Overscript[n, ̑])))^2)⟩

Integrate[(Array[a,3].n)^2 Sin[ϕ],{θ,0,2π},{ϕ,0,π}]/(4Pi)

⟨(a·(Overscript[n, ̑]))(b·(Overscript[n, ̑]))⟩

Integrate[(Array[a,3].n)(Array[b,3].n) Sin[ϕ],{θ,0,2π},{ϕ,0,π}]/(4Pi)

⟨(a·(Overscript[n, ̑]))(Overscript[n, ̑])⟩

Integrate[(Array[a,3].n)n Sin[ϕ],{θ,0,2π},{ϕ,0,π}]/(4Pi)

⟨(((a×(Overscript[n, ̑])))^2)⟩

Integrate[#.#&@Cross[Array[a,3],n] Sin[ϕ],{θ,0,2π},{ϕ,0,π}]/(4Pi)

⟨(a×(Overscript[n, ̑]))·(b×(Overscript[n, ̑]))⟩

Integrate[Cross[Array[a,3],n].Cross[Array[b,3],n] Sin[ϕ],{θ,0,2π},{ϕ,0,π}]/(4Pi)

⟨(a·(Overscript[n, ̑]))(b·(Overscript[n, ̑]))(c·(Overscript[n, ̑]))(d·(Overscript[n, ̑]))⟩

Integrate[Times@@(Array[#,3].n&/@{a,b,c,d}) Sin[ϕ],{θ,0,2π},{ϕ,0,π}]/(4Pi)//Timing

FullSimplify[Last@%]

Simplify[%5-1/15(((a.b)(c.d)+(a.c)(b.d)+(a.d)(b.c)) /.{a->Array[a,3],b->Array[b,3],c->Array[c,3],d->Array[d,3]})]