Example data table
| Mode | Vector field | Exact? | Potential function | Example work A→B |
|---|---|---|---|---|
| 2D | F(x, y) = ⟨4 + 2x + 3y, -1 + 3x + 5y⟩ | Yes | φ = 4x - y + x2 + 3xy + 2.5y2 | A(0,0), B(1,2) gives 19 |
| 3D | F(x, y, z) = ⟨1 + 2x + 4y + z, 3 + 4x + y + 5z, 2 + x + 5y + 6z⟩ | Yes | φ = x + 3y + 2z + x2 + 4xy + xz + 0.5y2 + 5yz + 3z2 | A(0,0,0), B(1,1,1) gives 20.5 |
| 2D repaired | F(x, y) = ⟨2 + x + 6y, 3 + 4x + y⟩ | No | Repair uses average mixed term 5 | Shows approximate potential for sensitivity checks |
Formula used
2D exact affine field
For F(x, y) = ⟨a0 + a1x + a2y, b0 + b1x + b2y⟩, exactness requires a2 = b1.
φ(x, y) = C + a0x + b0y + 0.5a1x2 + a2xy + 0.5b2y2
3D exact affine field
For F = ⟨P, Q, R⟩, exactness requires ∂P/∂y = ∂Q/∂x, ∂P/∂z = ∂R/∂x, and ∂Q/∂z = ∂R/∂y.
φ = C + a0x + b0y + c0z + 0.5a1x2 + a2xy + a3xz + 0.5b2y2 + b3yz + 0.5c3z2
Potential difference and work: For conservative fields, path-independent work from point A to point B equals φ(B) − φ(A).
Repair option: When mixed derivatives disagree, the calculator can average each conflicting pair to build a nearest symmetric affine approximation for exploratory analysis.
How to use this calculator
- Select 2D or 3D affine field mode.
- Enter coefficients for each component of the vector field.
- Choose display precision, tolerance, and the integration constant.
- Provide an evaluation point and start and end coordinates.
- Submit to test exactness and generate the potential function.
- Review potential values, work, curl diagnostics, and the Hessian matrix.
- Enable repair when you want a symmetric approximation for study.
- Use the export buttons to download result summaries as CSV or PDF.
Recent calculation history
| Time | Mode | Status | φ at point | Δφ | Work | Potential |
|---|---|---|---|---|---|---|
| No calculations saved yet. | ||||||
FAQs
1. What kind of vector field does this finder support?
It supports 2D and 3D affine vector fields, where each component has a constant term and first-degree coefficients in x, y, and z.
2. Why does the calculator test exactness first?
A scalar potential exists only when the field is conservative on the domain. The exactness checks compare cross-derivatives or curl conditions before building φ.
3. What does the integration constant C change?
It shifts every potential value by the same amount. Differences such as φ(B) − φ(A) and conservative work remain unchanged.
4. When should I use the repair option?
Use it when measured or estimated coefficients nearly satisfy exactness. It gives a symmetric affine approximation for sensitivity checks, not an exact potential for the original field.
5. Why can work be unavailable?
If the field is not conservative and repair is off, path-independent work from a potential function is undefined. The calculator then reports diagnostics only.
6. What does the Hessian matrix represent here?
The Hessian stores second partial derivatives of the potential. For affine gradient fields, it is constant and mirrors the symmetric coefficient structure.
7. Can this tool handle nonlinear trigonometric fields?
No. This page is designed for affine fields only. Nonlinear symbolic integration requires a broader expression parser and a different solving engine.
8. What does the symmetry gap mean?
It is the largest mismatch among paired cross-derivatives. Smaller gaps indicate a field that is closer to conservative behavior under the chosen tolerance.