Page 382 - Process Modelling and Simulation With Finite Element Methods

A MATLAB/FEMLAB Primer for Vector Calculus 369

A.4.2 Derivatives of vector fields

The vector differential operator V may be applied to a vector field F(x) in two
ways: (1) the scalar product V - F called the divergence, (2) the vector product
VxF, called the curl.
The divergence is given by

aq a~, a4
- -- +-+-
ax ay az

The curl is given by

aF3 dF2 a< aF, aF2 a<
- - -- - -- - -- I
-[ ay aZ ' aZ ax ' ax ay

Eijk is the permutation tensor introduced earlier. Of course one can see readily
that div F is a scalar, while curl F is a vector.
The operator I; - V is often seen in advection terms in heat or mass transport
equations. Clearly, it is not the divergence, since
a a a
F~
F . v = 6 -+ F~ 2+ -
ax ay aZ
which is still an operator, in contrast to (A15), which is a scalar.