Other ways of writing Green’s theorem
In the introductory reading on Green’s theorem, we wrote Green’s theorem
as
∫
CF ⋅ ds = D(curl F) ⋅ kdA | | |
or
∫
CF ⋅ ds = D dA, | | |
where
F(
x,y) = (
F1(
x,y)
,F2(
x,y)) is a two-dimensional vector field,
D is a
region in plane, and
C is its positively oriented boundary. I used this
notation to tie Green’s theorem in with the concept of the
curl of a vector
field.
We often denote C by ∂D to make it explicit that the curve C is the (positively
oriented) boundary of D. This notation is also more natural when the region D
has more than one boundary component. Then, Green’s theorem can look like, for
example,
∫
∂DF ⋅ ds = D dA, | | |
Note that the notation
∂D simply means the boundary of
D. It has nothing to do
with a partial derivative.
However, people frequently write Green’s theorem differently. First, they like to
change the formula by writing the line integral at the left in terms of
components:
∫
∂DF1dx + F2dy = D dA. | | |
Then, they like to let the vector field be
F(
x,y) = (
P(
x,y)
,Q(
x,y)), so that
Green’s theorem becomes
∫
∂DPdx + Qdy = D dA. | | |
Sometimes, F = (P,Q) won’t be referred to as a vector field. Instead, one can
discuss the above version of Green’s theorem as applied to the two scalar valued
functions Q : D → R and P : D → R.