 Del in cylindrical and spherical coordinates

This is a list of some vector calculus formulae of general use in working with various curvilinear coordinate systems.
Contents
Note
 This page uses standard physics notation. For spherical coordinates, θ is the angle between the z axis and the radius vector connecting the origin to the point in question. ϕ is the angle between the projection of the radius vector onto the xy plane and the x axis. Some sources reverse the definitions of θ and ϕ, so the meaning should be inferred from the context.
 The function atan2(y, x) can be used instead of the mathematical function arctan(y/x) due to its domain and image. The classical arctan(y/x) has an image of (π/2, +π/2), whereas atan2(y, x) is defined to have an image of (π, π]. (The expressions for the Del in spherical coordinates may need to be corrected)
Table with the del operator in cylindrical, spherical and parabolic cylindrical coordinates Operation Cartesian coordinates (x,y,z) Cylindrical coordinates (ρ,φ,z) Spherical coordinates (r,θ,φ) Parabolic cylindrical coordinates (σ,τ,z) Definition
of
coordinatesDefinition
of
unit
vectorsA vector field Gradient Divergence Curl Laplace operator Vector Laplacian Material derivative ^{[1]}
Differential displacement Differential normal area Differential volume Nontrivial calculation rules:  (Laplacian)
 (using Lagrange's formula for the cross product)
References
 ^ Weisstein, Eric W.. "Convective Operator". Mathworld. http://mathworld.wolfram.com/ConvectiveOperator.html. Retrieved 23 March 2011.
See also
 Del
 Orthogonal coordinates
 Curvilinear coordinates
 Vector fields in cylindrical and spherical coordinates
External links
 Maxima Computer Algebra system scripts to generate some of these operators in cylindrical and spherical coordinates.
Categories: Vector calculus
 Coordinate systems
Wikimedia Foundation. 2010.