Del in cylindrical and spherical coordinates

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This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.

Notes[edit]

  • This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ):
    • The polar angle is denoted by θ: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
    • The azimuthal angle is denoted by φ: it is the angle between the x-axis and the projection of the radial vector onto the xy-plane.
  • The function atan2(y, x) can be used instead of the mathematical function arctan(y/x) owing to its domain and image. The classical arctan function has an image of (−π/2, +π/2), whereas atan2 is defined to have an image of (−π, π].

Coordinate conversions[edit]

Conversion between Cartesian, cylindrical, and spherical coordinates
From
Cartesian Cylindrical Spherical
To Cartesian N/A
Cylindrical N/A
Spherical N/A

Unit vector conversions[edit]

Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of destination coordinates
Cartesian Cylindrical Spherical
Cartesian N/A
Cylindrical N/A
Spherical N/A
Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of source coordinates
Cartesian Cylindrical Spherical
Cartesian N/A
Cylindrical N/A
Spherical N/A

Del formula[edit]

Table with the del operator in cartesian, cylindrical and spherical coordinates
Operation Cartesian coordinates (x, y, z) Cylindrical coordinates (ρ, φ, z) Spherical coordinates (r, θ, φ), where θ is the polar angle and φ is azimuthal
A vector field A
Gradient f
Divergence ∇ ⋅ A
Curl ∇ × A
Laplace operator 2f ≡ ∆f
Vector Laplacian 2A ≡ ∆A
Material derivativeα[1] (A ⋅ ∇)B
Tensor divergence ∇ ⋅ T
Differential displacement d
Differential normal area dS
Differential volume dV
The source that is used for these formulae uses for the azimuthal angle and for the polar angle, which is common mathematical notation. This page uses for the polar angle and for the azimuthal angle, which is common notation in physics. In order to get the mathematics formulae, switch and in the formulae shown in the table above.

Non-trivial calculation rules[edit]

  1. (Lagrange's formula for del)

Cartesian derivation[edit]

Infinitesimal element in cartesian coordinates

The expressions for and are found in the same way.

Cylindrical derivation[edit]

Infinitesimal element in cylindrical coordinates

Spherical derivation[edit]

Infinitesimal element in spherical coordinates

See also[edit]

References[edit]

  1. ^ Weisstein, Eric W. "Convective Operator". Mathworld. Retrieved 23 March 2011. 

External links[edit]