Calculator Input
This tool evaluates ∮(P dx + Q dy) for linear planar fields. It compares numerical circulation with Green theorem for quick physics validation.
Example Data Table
| Example | Field P(x,y) | Field Q(x,y) | Path | Area | Curl z | Expected Integral |
|---|---|---|---|---|---|---|
| Rigid rotation field | -y | x | Circle, radius 2 | 4π | 2 | 8π ≈ 25.132741 |
| Linear circulation field | 2x + 3y | 6x - y | Ellipse, rx 3, ry 2 | 6π | 3 | 18π ≈ 56.548668 |
| Shifted affine field | 4x - y + 1 | 2x + 3y | Rectangle, width 5, height 2 | 10 | 3 | 30.000000 |
Formula Used
The calculator evaluates the closed line integral ∮(P dx + Q dy) around a selected loop.
The field uses a linear planar model: P(x,y) = ax + by + c and Q(x,y) = dx + ey + f.
Numerical evaluation uses midpoint summation over many short path segments: ∮(P dx + Q dy) ≈ Σ [P(xm,ym)Δx + Q(xm,ym)Δy].
Green theorem provides the comparison value: ∮(P dx + Q dy) = ∬(∂Q/∂x − ∂P/∂y) dA.
For this linear field, the scalar curl is constant: ∂Q/∂x − ∂P/∂y = d − b. Therefore, the theorem estimate becomes: (d − b) × Area, with sign set by loop orientation.
How to Use This Calculator
- Choose a closed path shape: circle, ellipse, or rectangle.
- Set the traversal direction as clockwise or counterclockwise.
- Enter the six coefficients for P(x,y) and Q(x,y).
- Enter the center coordinates and shape dimensions.
- Increase the segment count for tighter numerical accuracy.
- Press calculate to show the result above the form.
- Review the numerical integral, theorem estimate, and error values.
- Use the CSV and PDF buttons to save the report.
Frequently Asked Questions
1) What does this calculator measure?
It measures the closed line integral of a planar vector field around a loop. In physics, this often represents circulation, work around a path, or rotational behavior inside the enclosed region.
2) Why does the tool compare with Green theorem?
Green theorem offers a second calculation route. It converts the boundary integral into an area integral. Comparing both values helps you verify the numerical result and judge whether your segment count is sufficient.
3) Which fields are supported here?
This version supports linear planar fields: P(x,y)=ax+by+c and Q(x,y)=dx+ey+f. That structure is powerful enough for many teaching, engineering, and introductory physics circulation examples.
4) What changes when I reverse the loop orientation?
Reversing orientation changes the sign of the closed integral. Counterclockwise traversal is treated as positive. Clockwise traversal returns the same magnitude when conditions match, but with opposite sign.
5) Why should I increase the segment count?
More segments create shorter path pieces. That improves midpoint approximation accuracy and usually reduces error against the theorem estimate. Smooth curves benefit the most when you raise the segment count.
6) Can I use this for electromagnetism or fluid flow examples?
Yes. Closed loop integrals appear in circulation, induced effects, and field rotation studies. This calculator is especially useful for simplified two-dimensional teaching models in those topics.
7) Why can the integral be zero?
The value can be zero when positive and negative contributions cancel, or when the enclosed curl is zero. A conservative or locally balanced field often produces that behavior on closed loops.
8) What do the CSV and PDF files contain?
They contain the main result summary. That includes shape, orientation, field equations, enclosed area, curl, numerical integral, theorem estimate, and accuracy indicators for quick reporting or record keeping.