Green's theorem and application

Question 1

Necessary condition

Answer 1

Let F: Ω-> |R^n be a conservative vector field whose components are smooth. If F = (F1, …., Fn) then for all i, j = 1, …, n, we have ∂Fi/∂xj = ∂Fj/∂xi.

Question 2

Star-shaped set

Answer 2

A set Ω C |R^n is called star shaped if there exists a point xo e Ω so that for all x e Ω the segment xo and x lie entirely in Ω.

Question 3

Poincare’s lemma

Answer 3

If Ω C |R^n is a star-shaped set and F = (F1, …, Fn) : Ω -> |R^n is a smooth vector field with the property that
for all i, j e {1, …, n}, we have ∂Fi/∂xj = ∂Fj/∂xi. Then F is conservative and more precisely, we can define the potential as
f(x) = ∫ (over 0 to 1) F ((1-t)xo + tx) . (x-xo) .dt.

Question 4

Simple closed curve

Answer 4

A curve parametrised by r: [a.b] -> |R^2 is called a simple closed curve if r us injective on [a,b) and r(a) = r(b).
More explicitly, a simple closed curve is one that does not intersect itself.

Question 5

Green’s theorem

Answer 5

Let C be a piecewise smooth simple closed curve in the plane with parametrisation r : [a,b] -> |R^2, that traverses C counterclockwise and F = (F1, F2) a smooth vector field defined in the region D bounded by C. Then
∫ (over C) F.dr = ∫ (over C) F1 .dx + F2 .dy
= ∫∫(over D) (∂F2/∂x - ∂F1/∂y) dxdy

Question 6

Converse of the necessary condition

Answer 6

If Ω is the region in |R^2 that is bounded by a simple plane curve C and F = (P, Q) : Ω -> |R^2 is a vector field so that ∂Q/∂x = ∂P/∂y, then F is conservative.

Question 7

Area of Ω

Answer 7

If C is a simple closed planar curve and Ω is the region bounded by C then the area of Ω is
Area Ω = ∫ (over C) x.dy = - ∫(over C) y.dx = 1/2 ∫ (over C) -y.dx +x.dy