GREEN’ S THEOREM - QUICK VERSION

GREEN’ S THEOREM - VERY QUICK

Definition: Let C be a curve in thexy-plane (sayx =

x(t) andy = y(t) with a ≤ t ≤ b). Then thelineintegral, denoted ∫

Cf(x, y) ds,

is the area of the region directly above this curve and belowthe surfacez = f(x, y).

Definition: Let C be a curve in thexy-plane (sayx =

x(t) andy = y(t) with a ≤ t ≤ b). Then thelineintegral, denoted ∫

Cf(x, y) ds,

is the area of the region directly above this curve and belowthe surfacez = f(x, y).

C: x = x(t), y = y(t) with a ≤ t ≤ b

C: x = x(t), y = y(t) with a ≤ t ≤ b

Important: A variation (really a sum of two line integrals)is ∫

CM(x, y) dx + N(x, y) dy

C: x = x(t), y = y(t) with a ≤ t ≤ b

Important: A variation (really a sum of two line integrals)is ∫

CM(x, y) dx + N(x, y) dy

=

∫ b

a

(M(x, y)

dx

dt+ N(x, y)

dy

dt

)dt.

Green’s Theorem: Let C be a piecewise smooth, simpleclosed curve having a counterclockwise orientation thatforms the boundary of a regionS in the xy-plane.

Green’s Theorem: Let C be a piecewise smooth, simpleclosed curve having a counterclockwise orientation thatforms the boundary of a regionS in the xy-plane. IfM(x, y) and N(x, y) have continuous partial deriva-tives onS and its boundaryC, then∮

CM(x, y) dx + N(x, y) dy

=

∫∫S

(∂N

∂x−

∂M

∂y

)dA.