Introduction to Green's theorem circulation checks
Green's theorem gives two equivalent descriptions of circulation in a planar vector field. One description follows a closed boundary and accumulates the tangential contribution of the field. The other measures scalar curl throughout the enclosed area. This calculator evaluates both descriptions for a rectangle, placing the numerical line integral and double integral side by side.
For a vector field written as F = (P,Q), the circulation form is the closed integral of P dx + Q dy. The corresponding interior quantity is ∂Q/∂x − ∂P/∂y. If P and Q have continuous first partial derivatives on an open region containing the rectangle, Green's theorem says these values are equal. A small displayed difference can arise from numerical differentiation, midpoint sampling, Simpson integration, and rounding.
This is a verification tool rather than a symbolic proof engine. It is useful when checking an exercise, exploring how curl affects circulation, or catching a reversed edge before completing a longer calculation. The calculator specifically handles axis-aligned rectangular regions; it does not parameterize circles, triangles, holes, or arbitrary curves.
What this Green's theorem calculator compares
This Green's theorem calculator computes the positively oriented boundary in four pieces. It travels from left to right along the bottom edge, upward along the right edge, right to left along the top edge, and downward along the left edge. That counterclockwise route matters: reversing the route would reverse the sign of the line integral.
The interior calculation samples the scalar curl at the midpoint of each cell in a 20 × 20 grid. The boundary calculation uses Simpson's rule with 200 subintervals on each edge. Smooth polynomial and trigonometric examples generally agree closely, while rapidly varying expressions may require finer sampling than this educational calculator provides.
How to use the Green's theorem calculator
Enter the two field components using x and y as variables. Write multiplication explicitly: use 2*x rather than 2x. Powers may be entered with a caret, as in x^2. Common functions include sin, cos, tan, exp, log, sqrt, and abs. Parentheses are recommended whenever an expression has several operations.
- Enter P(x,y), the coefficient multiplying dx in the circulation integral.
- Enter Q(x,y), the coefficient multiplying dy.
- Set the left and right bounds with x start and x end.
- Set the bottom and top bounds with y start and y end.
- Select Compute Integrals and compare the two reported values.
The starting x bound must be smaller than the ending x bound, and the starting y bound must be smaller than the ending y bound. This requirement defines a nonempty rectangle and lets the calculator apply the standard counterclockwise orientation consistently. Negative coordinates and decimal bounds are allowed.
Inputs for P(x,y), Q(x,y), and the rectangle
P(x,y) is the horizontal component of the vector field and contributes on horizontal boundary segments because dy is zero there. Q(x,y) is the vertical component and contributes on vertical segments because dx is zero there. Both expressions may still depend on both variables, even though one coordinate remains fixed while evaluating any particular edge.
The values x start and x end define the rectangle's width. The values y start and y end define its height. Coordinates carry whatever length unit belongs to the original problem. If P and Q represent velocity, for example, integrating a component against a distance produces circulation units of velocity times length. The curl has field-per-length units, and integrating it over area produces the same circulation units. The calculator does not attach a unit label because the correct unit depends on the field you enter.
A constant added to P or Q can affect individual edge integrals, but it cancels around a closed rectangular path. By contrast, changing how P depends on y or how Q depends on x changes the scalar curl and therefore changes the net circulation. This distinction is one reason Green's theorem is more informative than adding unrelated input values.
Formulas for rectangular circulation and interior curl
The Green's theorem identity used by the calculator is:
The rectangle and its boundary are represented by:
On the bottom edge, y is fixed at c and x runs from a to b. On the right edge, x is fixed at b and y runs from c to d. The top and left edges run in the reverse coordinate directions to preserve counterclockwise orientation. Adding those four signed contributions produces the boundary circulation.
The interior side is approximated by multiplying the sampled curl in each cell by the cell area and adding the results. For a constant or linear curl, midpoint sampling is exact apart from floating-point rounding. For a strongly curved function, the result is an approximation whose quality depends on how much the function changes within each cell.
Worked example: the field P = −y and Q = x
Consider P(x,y) = −y and Q(x,y) = x on the rectangle 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1. The scalar curl is ∂Q/∂x − ∂P/∂y = 1 − (−1) = 2. The rectangle has area 2, so the interior integral is exactly 2 × 2 = 4.
The boundary calculation reaches the same value. Along the bottom, P is zero, so that contribution is 0. Along the right side, Q is 2 and y rises from 0 to 1, giving 2. Along the top, P is −1 while x travels from 2 back to 0, giving (−1)(0 − 2) = 2. Along the left side, Q is zero, so the final contribution is 0. The total is 0 + 2 + 2 + 0 = 4.
To reproduce the example, enter -y, x, and the bounds 0, 2, 0, and 1. Because the curl is constant and the edge expressions are linear, both numerical methods should display 4.000000. This is a meaningful check because every number comes from the line or area integral; unlike quantities are never added merely to create a score.
Comparison table: changing the y-dependence of P
A useful sensitivity check changes a genuine parameter in the field while holding the same rectangle. Let P = −k*y, Q = x, and D = [0,2] × [0,1]. The scalar curl is 1 + k, so the exact circulation is the area 2 multiplied by 1 + k. The values below are therefore actual Green's theorem outputs, not totals assembled from unrelated inputs.
| Field parameter k |
P(x,y) |
Scalar curl |
Line and area integral |
Interpretation |
| 0.5 |
−0.5*y |
1.5 |
3 |
A weaker y-gradient in P produces less positive circulation. |
| 1 |
−y |
2 |
4 |
This is the worked example's constant-curl field. |
| 1.5 |
−1.5*y |
2.5 |
5 |
A stronger negative y-gradient in P increases positive curl. |
This comparison illustrates the correct kind of sensitivity analysis: vary one mathematically defined parameter, derive its effect on curl, and compare valid integral results. The relationship is linear here because both the field and the theorem's integral are linear.
How to interpret the Green's theorem result
The first reported number is the sum of the four counterclockwise boundary integrals. The second is the sampled double integral of curl. Agreement to several decimal places supports the setup, but it does not establish the theorem's smoothness assumptions. If the signs are opposite, inspect the direction of any hand calculation and confirm that P accompanies dx while Q accompanies dy.
A modest mismatch can indicate rapid variation that the fixed grid does not resolve. A large mismatch more often points to an invalid expression, a discontinuity, a singularity inside the rectangle, or a misunderstanding of the field. Try a simpler nearby rectangle and inspect each partial derivative analytically when possible. Changing both field components and all bounds at once makes diagnosis unnecessarily difficult.
Limitations of this rectangular Green's theorem check
This calculator assumes that expressions evaluate to finite real numbers throughout the rectangle and on its boundary. It is not suitable for a field with a pole, branch discontinuity, undefined logarithm, or other singular behavior in the sampled region. Green's theorem itself requires appropriate continuity, even if a coarse numerical sample happens to miss a problematic point.
- The region must be one axis-aligned rectangle with x start less than x end and y start less than y end.
- The boundary orientation is fixed as counterclockwise.
- The area method uses a 20 × 20 midpoint grid, while each edge uses 200-step Simpson integration.
- Numerical derivatives can amplify floating-point error for highly oscillatory or extremely scaled expressions.
- The tool reports circulation form, not the flux form of Green's theorem.
Use the result as a transparent numerical check alongside algebra, not as a replacement for identifying the domain and verifying the hypotheses. The strongest workflow is to compute the curl by hand, predict the sign and approximate magnitude, run the calculator, and then explain why the two integrals should agree.