Electromagnetics I
Ch 10: Appendices
Loc 10.5

Electromagnetics I
Ch 10
Loc 10.5

Mathematical Formulas - Vector Operators

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

Cartesian ( $x$ , $y$ , $z$ )
cylindrical ( $\rho$ , $\phi$ , $z$ )
spherical ( $r$ , $\theta$ , $\phi$ )

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: ${\bf A} = \hat{\bf x}A_x + \hat{\bf y}A_y + \hat{\bf z}A_z$
cylindrical: ${\bf A} = \hat{\bf \rho}A_{\rho} + \hat{\bf \phi}A_{\phi} + \hat{\bf z}A_z$
spherical: ${\bf A} = \hat{\bf r}A_r + \hat{\bf \theta}A_{\theta} + \hat{\bf \phi}A_{\phi}$

Gradient

Gradient in Cartesian coordinates:

(10.5.1)

Gradient in cylindrical coordinates:

(10.5.2)

Gradient in spherical coordinates:

(10.5.3)

Divergence

Divergence in Cartesian coordinates:

(10.5.4)

Divergence in cylindrical coordinates:

Divergence in spherical coordinates:

Curl

Curl in Cartesian coordinates:

Curl in cylindrical coordinates:

Curl in spherical coordinates:

Laplacian

Laplacian in Cartesian coordinates:

(10.5.5)

Laplacian in cylindrical coordinates:

(10.5.6)

Laplacian in spherical coordinates:

Ellingson, Steven W. (2018) Electromagnetics, Vol. 1. Blacksburg, VA: VT Publishing. https://doi.org/10.21061/electromagnetics-vol-1 CC BY-SA 4.0

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