# B.2: Vector Operators

- Page ID
- 6378

[m0139_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 (\(\hat{~}\)) over the symbol. The vector operand \({\bf A}\) 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:

f &= + + &

Gradient in cylindrical coordinates:

f &= + + &

Gradient in spherical coordinates:

f &= + + &

**Divergence**

Divergence in Cartesian coordinates:

&= + + &

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:

^2 f &= + + &

Laplacian in cylindrical coordinates:

^2 f &= ( ) + + &

Laplacian in spherical coordinates:

^2 f &= (r^2 ) &

& +() &

& + &