Mathematical Formulas - Vector Operators

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

  • Cartesian (,,)
  • cylindrical (,,)
  • spherical (,,)

Associated basis vectors are identified using a caret (

) over the symbol. The vector operand

is expressed in terms of components in the basis directions as follows:

  • Cartesian:
  • cylindrical:
  • spherical:


Gradient in Cartesian coordinates:


Gradient in cylindrical coordinates:


Gradient in spherical coordinates:



Divergence in Cartesian coordinates:


Divergence in cylindrical coordinates:

Divergence in spherical coordinates:


Curl in Cartesian coordinates:

Curl in cylindrical coordinates:

Curl in spherical coordinates:


Laplacian in Cartesian coordinates:


Laplacian in cylindrical coordinates:


Laplacian in spherical coordinates:


Ellingson, Steven W. (2018) Electromagnetics, Vol. 1. Blacksburg, VA: VT Publishing. CC BY-SA 4.0

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