Del in cylindrical and spherical coordinates

Note

• This page uses standard physics notation. For spherical coordinates, is the angle between the z axis and the radius vector connecting the origin to the point in question. is the angle between the projection of the radius vector onto the x-y plane and the x axis. Some sources reverse the definitions of and , so the meaning should be inferred from the context.
• The function atan2
  Atan2
  In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real arguments and not both equal to zero, is the angle in radians between the positive -axis of a plane and the point given by the coordinates on it...
  
  (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

Del

In vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus...

 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 Parabolic cylindrical coordinates In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the...  (σ,τ,z)
Definition of coordinates
Definition of unit vectors
A vector field Vector field In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
Gradient Gradient In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
Divergence Divergence In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
Curl
Laplace operator Laplace operator In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...
Vector Laplacian Vector Laplacian In mathematics and physics, the vector Laplace operator, denoted by \scriptstyle \nabla^2, named after Pierre-Simon Laplace, is a differential operator defined over a vector field. The vector Laplacian is similar to the scalar Laplacian...
Material derivative
Differential displacement
Differential normal area
Differential volume
Non-trivial calculation rules: (Laplacian) (using Lagrange's formula for the cross product Cross product In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them... )

See also

• Del
  Del
  In vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus...
  
  
• Orthogonal coordinates
  Orthogonal coordinates
  In mathematics, orthogonal coordinates are defined as a set of d coordinates q = in which the coordinate surfaces all meet at right angles . A coordinate surface for a particular coordinate qk is the curve, surface, or hypersurface on which qk is a constant...
  
  
• Curvilinear coordinates
  Curvilinear coordinates
  Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible at each point. This means that one can convert a point given...
  
  
• Vector fields in cylindrical and spherical coordinates
  Vector fields in cylindrical and spherical coordinates
  * This page uses standard physics notation. For spherical coordinates, \theta is the angle between the z axis and the radius vector connecting the origin to the point in question. \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis...
  
  

External links

• Maxima Computer Algebra system scripts to generate some of these operators in cylindrical and spherical coordinates.