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Physics LibreTexts

12.2: Vector Operators

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This section contains a summary of vector operators expressed in each of the three major coordinate systems:

  • Cartesian (x,y,z)
  • cylindrical (ρ,ϕ,z)
  • spherical (r,θ,ϕ)

Associated basis vectors are identified using a caret (ˆ ) over the symbol. The vector operand A is expressed in terms of components in the basis directions as follows:

  • Cartesian: A=ˆxAx+ˆyAy+ˆzAz
  • cylindrical: A=ˆρAρ+ˆϕAϕ+ˆzAz
  • spherical: A=ˆrAr+ˆθAθ+ˆϕAϕ

Gradient

Gradient in Cartesian coordinates:

f=ˆxfx+ˆyfy+ˆzfz

Gradient in cylindrical coordinates:

f=ˆρfρ+ˆϕ1ρfϕ+ˆzfz

Gradient in spherical coordinates:

f=ˆrfr+ˆθ1rfθ+ˆϕ1rsinθfϕ

Divergence

Divergence in Cartesian coordinates:

A=Axx+Ayy+Azz

Divergence in cylindrical coordinates:

A=1ρρ(ρAρ)+1ρAϕϕ+Azz

Divergence in spherical coordinates:

A=  1r2r(r2Ar)  +1rsinθθ(Aθsinθ)  +1rsinθAϕϕ

Curl

Curl in Cartesian coordinates:

×A=  ˆx(AzyAyz)  +ˆy(AxzAzx)  +ˆz(AyxAxy)

Curl in cylindrical coordinates:

×A=  ˆρ(1ρAzϕAϕz)  +ˆϕ(AρzAzρ)  +ˆz1ρ[ρ(ρAϕ)Aρϕ]

Curl in spherical coordinates:

×A=  ˆr1rsinθ[θ(Aϕsinθ)Aθϕ]  +ˆθ1r[1sinθArϕr(rAϕ)]  +ˆϕ1r[r(rAθ)Arθ]

Laplacian

Laplacian in Cartesian coordinates:

2f=2fx2+2fy2+2fz2

Laplacian in cylindrical coordinates:

2f=1ρρ(ρfρ)+1ρ22fϕ2+2fz2

Laplacian in spherical coordinates:

2f=  1r2r(r2fr)  +1r2sinθθ(fθsinθ)  +1r2sin2θ2fϕ2


This page titled 12.2: Vector Operators is shared under a CC BY-SA license and was authored, remixed, and/or curated by Steven W. Ellingson (Virginia Tech Libraries' Open Education Initiative) .

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