TANGENTIAL CASE: CLOSED PATH AND NON-ZERO CURL
Here you can use Green's Theorem
to trade your
line integral in on an area integral.
Perhaps suprisingly, this can be very adavantageous. Here's why:
- (1) Differentiation often simplifies things. In particular, if you differentiate a polynomial, you lower its degree. That certainly makes it simpler. Also, in two dimensions, the curl 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, y^2 + xy + 4)
Then curl(F) = y -1.
The curl 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.
For example, if we consider the vector field F from (1),
and the triangle from (2), with the path around the boundary oriented counter clockwise. Green's theorem says that
Hence the line integral equals: