NORMAL CASE: CLOSED PATH AND NON-ZERO DIVERGENCE

to trade your line integral in on an area integral.

Perhaps suprisingly, this can be very adavantageous. Heres why:

• (1) Differentiation often simplifies things. In particular, if you differentiate a polynomial, you lower its degree. That certainly makes it simpler. Also, the divergence is just an ordinary function; not a vector field. This also makes it simpler to deal with.

  Here is an exapmle:

  F(x,y) = (x^2 + y, x^2 + xy + 4)

  Then div(F) = 3x.

  The divergence is simpler, isn't it?

• (2) Often a complicated path bounds a simple region.

  Here is an example:

  Consider the triangle

  0 < x < y

  x > 0

  2y < x + 1

  The limits of integration for an area integral over it would be:

  x < y < (x+1)/2

  0 < x < 1

  But if one wanted to parameterize the boundary, one would need three separate parameterizations -- one for each side. This is probably worse.