6.6 Integration around closed curves, and green's theorem  (Page 4/5)

The proof of Green's Theorem is tough, and we break it into several steps.

Suppose $S$ is the rectangle $[a,b]\times [c,d].$ Then Green's Theorem is true.

We think of the closed curve $C_S$ bounding the rectangle as the union of four straight lines, $C_1,C_2,C_3$ and $C_4,$ and we parameterize them as follows: Let $\phi :[a,b]\to C_1$ be defined by $\phi \left(t\right)=(t,c);$ let $\phi :[b,b+d-c]\to C_2$ be defined by $\phi \left(t\right)=(b,t-b+c);$ let $\phi :[b+d-c,b+d-c+b-a]\to C_3$ be defined by $\phi \left(t\right)=(b+d-c+b-t,d);$ and let $\phi :[b+d-c+b-a,b+d-c+b-a+d-c]\to C_4$ be defined by $\phi \left(t\right)=(a,b+d-c+b-a+d-t).$ One can check directly to see that this $\phi$ parameterizes the boundary of the rectangle $S=[a,b]\times [c,d].$

As usual, we write $\phi \left(t\right)=\left(x\right(t),y(t\left)\right).$ Now, we just compute, use the Fundamental Theorem of Calculus in the middle, and use part (d) of [link] at the end.

proving the lemma.

Suppose $S$ is a right triangle whose vertices are of the form $(a,c),(b,c)$ and $(b,d).$ Then Green's Theorem is true.

We parameterize the boundary $C_S$ of this triangle as follows:For $t\in [a,b],$ set $\phi \left(t\right)=(t,c);$ for $t\in [b,b+d-c],$ set $\phi \left(t\right)=(b,t+c-b);$ and for $t\in [b+d-c,b+d-c+b-a],$ set $\phi \left(t\right)=(b+d-c+b-t,b+d-c+d-t).$ Again, one can check that this $\phi$ parameterizes the boundary of the triangle $S.$

Write $\phi \left(t\right)=\left(x\right(t),y(t\left)\right).$ Again, using the Fundamental Theorem and [link] , we have

which proves [link] .

Suppose $S_1,...,S_n$ is a partition of the geometric set $S,$ and that the boundary $C_{S_k}$ has finite length for all $1\le k\le n.$ If Green's Theorem holds for each geometric set $S_k,$ then it holds for $S.$

From [link] we have

and from [link] we have

Since Green's Theorem holds for each $k,$ we have that

and therefore

as desired.