Section Summary: 17.5

Curl and Divergence

1. Definitions

   • The del operator:

     

     We've seen before, of course: when it acts on a scalar function f, it returns the gradient of f.

   • Curl of vector field F:

     

     If curl at a point, then is said to be irrotational at that point.

     Making use of , then, curl F = , where is the cross product (sorry, my ``X'' symbol isn't working!).

   • Divergence:

     

     If div , then is said to be incompressible.

2. Theorems

   • If f is a function of three variables that has continuous second-order partial derivatives, then

     

     (note that that's a vector ). This says that if F is conservative, then curl .

   • If F is a vector field defined on all of whose component functions have continuous partial derivatives and curl , then F is a conservative vector field.

   • If is a vector field on and P, Q, and R have continuous second-order partial derivatives, then

     

   • Green's theorem:

     

   • Green's theorem (normal components):

     

3. Properties/Tricks/Hints/Etc.

   If curl , then F is not conservative.

   In the context of fluid flow,

   1. The divergence measures the tendency of a fluid to diverge from the point (x,y,z), whereas
   2. the curl measures the rotational tendency of the fluid (about the axis given by the direction of the curl) at the point (x,y,z).

4. Summary

   In this section we encounter two important extensions of the gradient operator (also known as ``del''): del operates on a scalar function to produce the gradient. In addition,

   • produces a vector field called the curl of F; and
   • produces a scalar field called the divergence of F.

   We discover two equivalent vector-formulation of Green's theorem which allows us to use the result in three-space, and understand it in the context of fluid flow.

   The curl also provides us with a way of determining whether a vector field is conservative (that is, a gradient field).