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Green's Theorem Calculator

Verify Green's theorem by comparing the closed line integral with the double integral.

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∮ P dx+Q dy = ∬_D (Q_x − P_y) dA
e.g. x*y
e.g. x^2

📖 Tutorial | Green's Theorem

1. Theorem
Green's theorem: for a closed region D with positively oriented boundary L, ∮_L P dx+Q dy = ∬_D (Q_x−P_y) dA.
2. Symbols
P,Qfield components
Lpositive boundary
Q_x−P_y2D curl
Dregion
3. How it works
  • Integrate around the rectangle counterclockwise;
  • Numerically find Q_x, P_y;
  • Approximate the double integral by the center value;
  • They should match.
4. Steps
  1. Enter P, Q;
  2. Enter rectangle bounds;
  3. Click Calculate.
5. Example
Example: P=xy, Q=x² on [0,1]².
Solution: Q_x=2x, P_y=x; at center difference 0.5, area 1 → double integral ≈0.5, matching the line integral.
6. Pitfalls
Boundary must be positively oriented (CCW);
The double integral uses a center approximation;
D must be simply connected.

❓ FAQ | Green's Theorem

Use of Green's theorem?
Convert hard closed line integrals into double integrals.
Positive orientation?
CCW, region on the left.
Qx−Py?
The 2D curl.
Why approximate?
Center value over the rectangle.
Path-independent?
When Qx=Py (irrotational).