## Divergence and Curl of Vector Fields

**Divergence of a Vector Field**

The divergence of a vector field
**F**=<P(x,y,z),Q(x,y,z),R(x,y,z)>,
denoted by div **F**, is the scalar function defined by the
dot product

Here is an example. Let

The divergence is given by:

**Curl of a Vector Field**

The curl of a vector field **F**=<P(x,y,z),Q(x,y,z),R(x,y,z)>,
denoted curl**F**, is the vector field defined by the
cross product

An alternative notation is

The above formula for the curl is difficult to remember. An
alternative formula for the curl is

det means the determinant of the 3x3 matrix. Recall that the
determinant consists of a bunch of terms which are products
of terms from each row. The product of the terms on the
diagonal is

As you can see, this term is part of the x-component of the curl.

Consider the following example: **F**=<xyz,ysin z, ycos x>.

curl F = <cos x - ycos z, xy + ysin z, -xz>.

**[Vector Calculus Home]
[Math
254 Home] [Math 255 Home]
[Notation]
[References]**
**Copyright **© **1996 Department
of Mathematics, Oregon State
University**

If you have questions or comments, don't hestitate to
contact us.