Calculator Inputs
Enter a quadratic vector field and a rectangular region. The calculator checks Green's theorem by comparing the double integral with the boundary integral.
Example Data Table
This example uses P(x, y) = -y and Q(x, y) = x on the rectangle from x = -1 to 1 and y = -1 to 1.
| Example | P(x, y) | Q(x, y) | Region | Mode | Integrand | Exact Value |
|---|---|---|---|---|---|---|
| Unit rotation field | -y | x | [-1, 1] × [-1, 1] | Circulation | ∂Q/∂x - ∂P/∂y = 2 | 8 |
| Unit rotation field | -y | x | [-1, 1] × [-1, 1] | Flux | ∂P/∂x + ∂Q/∂y = 0 | 0 |
Formula Used
Circulation form: ∮C P dx + Q dy = ∬R (∂Q/∂x − ∂P/∂y) dA
Flux form: ∮C P dy − Q dx = ∬R (∂P/∂x + ∂Q/∂y) dA
This calculator assumes a counterclockwise boundary and a rectangular region. It supports quadratic coefficient models for P and Q, then evaluates both sides exactly.
How to Use This Calculator
- Select either circulation form or outward flux form.
- Enter the rectangular bounds for x and y.
- Fill in the six coefficients for P(x, y).
- Fill in the six coefficients for Q(x, y).
- Choose the plot grid size for the heatmap.
- Press Calculate Now to show results above the form.
- Review the area integral, boundary integral, and difference.
- Download a CSV summary or PDF report if needed.
FAQs
1. What does this calculator evaluate?
It evaluates Green's theorem for rectangular regions. You can compute circulation or outward flux, then compare the boundary integral with the matching double integral.
2. Which vector fields are supported?
This version supports coefficient-based quadratic fields. Each component includes constant, x, y, x², xy, and y² terms.
3. Why is the region restricted to rectangles?
Rectangles allow exact evaluation with simple inputs. They also make the heatmap and edge-by-edge boundary interpretation much easier to inspect.
4. What is the difference between circulation and flux?
Circulation measures tangential flow around the boundary. Flux measures outward flow across the boundary. Green's theorem connects both to different double-integral integrands.
5. Why might the difference be zero?
A zero difference means both sides match perfectly. That confirms the theorem for the entered field and rectangle, within the calculator's exact model.
6. What does the Plotly graph show?
The graph shows the selected integrand over the rectangle. It helps you see where curl or divergence is positive, negative, or nearly constant.
7. Can I use decimal coefficients?
Yes. All numeric fields accept decimals, negative values, and zero. That makes the tool useful for homework, examples, and quick checks.
8. What should I export?
Use CSV for spreadsheets or records. Use PDF for reports, assignments, or sharing a quick summary with the formulas and results.