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Del in cylindrical and spherical coordinates

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

Contents

NotesEdit

  • 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 conversionsEdit

Conversion between Cartesian, cylindrical, and spherical coordinates[1]
From
Cartesian Cylindrical Spherical
To Cartesian      
Cylindrical      
Spherical      

Unit vector conversionsEdit

Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of destination coordinates[1]
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 formulaEdit

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 and φ is the azimuthal angleα
A vector field A      
Gradient f[1]      
Divergence ∇ ⋅ A[1]      
Curl ∇ × A[1]      
Laplace operator 2f ≡ ∆f[1]      
Vector Laplacian 2A ≡ ∆A  
Material derivativeα[2] (A ⋅ ∇)B    
Tensor divergence ∇ ⋅ T
Differential displacement d[1]      
Differential normal area dS      
Differential volume dV[1]      
This page uses   for the polar angle and   for the azimuthal angle, which is common notation in physics. The source that is used for these formulae uses   for the azimuthal angle and   for the polar angle, which is common mathematical notation. In order to get the mathematics formulae, switch   and   in the formulae shown in the table above.

Non-trivial calculation rulesEdit

  1.  
  2.  
  3.  
  4.   (Lagrange's formula for del)
  5.  

Cartesian derivationEdit

 

 


 

The expressions for   and   are found in the same way.

Cylindrical derivationEdit

 

 

 

 

 

 

Spherical derivationEdit

 

 

 

 

 

 

Unit vector conversion formulaEdit

The unit vector of a coordinate parameter u is defined in such a way that a small positive change in u causes the position vector   to change in   direction.

Therefore,   where s is the arc length parameter.

For two sets of coordinate systems   and  , according to chain rule,  

Now, let all of   but one and then divide both sides by the corresponding differential of that coordinate parameter, we find:

 

ReferencesEdit

  1. ^ a b c d e f g h Griffiths, David J. (2012). Introduction to Electrodynamics. Pearson. ISBN 978-0-321-85656-2.
  2. ^ Weisstein, Eric W. "Convective Operator". Mathworld. Retrieved 23 March 2011.

External linksEdit