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# B.2: Vector Operators

[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 ) &
&   +() &
&   + &