The convective operator associated with a vector field 
is 
, where 
is the gradient operator.
Applied in a three-dimensional orthogonal coordinate system to a vector field 
, the convective operator becomes
![[(A·del )B]_j=sum_(k=1)^3[(A_k)/(h_k)(partialB_j)/(partialq_k)+(B_k)/(h_kh_j)(A_j(partialh_j)/(partialq_k)-A_k(partialh_k)/(partialq_j))],](/images/equations/ConvectiveOperator/NumberedEquation1.svg) |
(1)
|
where 
and 
are physical components relative to the local orthonormal
basis, and the scale factors 
are related to the metric tensors
by 
(Binney and Tremaine 2008, pp. 777-778). In Cartesian
coordinates,
 |
(2)
|
In cylindrical coordinates, with 
the azimuth,
 |
(3)
|
In spherical coordinates, with 
the azimuth and 
the polar angle,
 |
(4)
|
