Green's Theorem Calculator
Verify Green's theorem by computing both sides: line integral and double integral.
Vector Field F = (P, Q)
F = P(x,y)i + Q(x,y)j
Region
Green's Theorem
oint_C P dx + Q dy = integral integral_D (dQ/dx - dP/dy) dA
Theorem Verification
Verified
Difference: 0.000000
ointLine Integral
-0.000000
iintArea Integral
-0.000000
ARegion Area
3.141593
DeltaDifference
0.000000
Calculation Details
Left side (line integral):
oint_C P dx + Q dy = -0.000000
Right side (area integral):
iint_D (dQ/dx - dP/dy) dA = -0.000000
Applications
- Computing areas using line integrals
- Work done by force fields
- Circulation of vector fields
- Relating boundary to interior properties
About Green's Theorem
Statement
Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It's a special case of Stokes' theorem.
Conditions
- C must be a simple, closed curve
- C must be traversed counterclockwise
- P and Q must have continuous partials
- D must be simply connected