proof of Green’s theorem
Consider the region R bounded by the closed curve P in a simply connected space. P can be given by a vector valued function →F(x,y)=(f(x,y),g(x,y)).
The region R can then be described by
| ∫∫R(∂g∂x-∂f∂y)𝑑A=∫∫R∂g∂x𝑑A-∫∫R∂f∂y𝑑A |
The double integrals above can be evaluated separately. Let’s look at
| ∫∫R∂g∂x𝑑A=∫ba∫B(y)A(y)∂g∂x𝑑x𝑑y |
Evaluating the above double integral, we get
| ∫ba(g(A(y),y)-g(B(y),y))𝑑y=∫bag(A(y),y)𝑑y-∫bag(B(y),y)𝑑y |
According to the fundamental theorem of line integrals, the above equation is actually equivalent
to the evaluation of the line integral of the function →F1(x,y)=(0,g(x,y)) over a path P=P1+P2, where P1=(A(y),y) and P2=(B(y),y).
| ∫bag(A(y),y)𝑑y-∫bag(B(y),y)𝑑y=∫P1→F1⋅𝑑→t+∫P2→F1⋅𝑑→t=∮P→F1⋅𝑑→t |
Thus we have
| ∫∫R∂g∂x𝑑A=∮P→F1⋅𝑑→t |
By a similar argument, we can show that
| ∫∫R∂f∂y𝑑A=-∮P→F2⋅𝑑→t |
where →F2=(f(x,y),0). Putting all of the above together, we can see that
| ∫∫R(∂g∂x-∂f∂y)𝑑A=∮P→F1⋅𝑑→t+∮P→F2⋅𝑑→t=∮P(→F1+→F2)⋅𝑑→t=∮P(f(x,y),g(x,y))⋅𝑑→t |
which is Green’s theorem.
| Title | proof of Green’s theorem |
|---|---|
| Canonical name | ProofOfGreensTheorem |
| Date of creation | 2013-03-22 12:28:47 |
| Last modified on | 2013-03-22 12:28:47 |
| Owner | mathcam (2727) |
| Last modified by | mathcam (2727) |
| Numerical id | 11 |
| Author | mathcam (2727) |
| Entry type | Proof |
| Classification | msc 26B12 |