Green Theorem Calculator
Work out green theorem instantly with clear inputs, formula shown and shareable results.
About Green Theorem Calculator
Green's theorem converts a closed line integral into a double integral over the enclosed region: ∮(P dx + Q dy) = ∬(∂Q/∂x − ∂P/∂y) dA. When the integrand is constant it is simply that value times the area.
Formula
Green's theorem
∮C (P dx + Q dy) = ∬R (∂Q/∂x − ∂P/∂y) dA
Worked example
With the default values below, Green Theorem Calculator returns Line integral around the boundary of 24.00000000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- ∂Q/∂x
- 3
- ∂P/∂y
- 1
- Area of the enclosed regionminimum 0
- 12
Results
- Line integral around the boundary
- 24.00000000
- ∂Q/∂x − ∂P/∂y
- 2.00000000
- Enclosed area
- 12.000000
- Scalar curl per unit area
- 2.00000000
Step by step
- 1Values used
- 2Green's theorem
- 3Line integral around the boundary= 24.00000000
- 4∂Q/∂x − ∂P/∂y= 2.00000000
- 5Enclosed area= 12.000000
- 6Scalar curl per unit area= 2.00000000
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
With ∂Q/∂x = 3, ∂P/∂y = 1 and area 12?
The circulation is (3 − 1) × 12 = 24.
How does it give area?
Choosing P = −y/2 and Q = x/2 makes the integrand 1, so the line integral equals the area — the principle behind the planimeter.