Calculator Inputs
Enter the required mixed partial derivatives at a sample point. This tool checks whether the equalities expected from a conservative field hold within a chosen tolerance.
Example Data Table
These sample records show how the calculator interprets mixed partial matches under different assumptions.
| Sample | Dimension | Domain | ∂P/∂y | ∂Q/∂x | ∂P/∂z | ∂R/∂x | ∂Q/∂z | ∂R/∂y | Tolerance | Interpretation |
|---|---|---|---|---|---|---|---|---|---|---|
| Point A | 3D | Simply Connected | 4 | 4 | 1.5 | 1.5 | -2 | -2 | 0.0001 | Likely conservative |
| Point B | 3D | Simply Connected | 3 | 3.2 | 2 | 2 | 1 | 1.4 | 0.01 | Not conservative at point |
| Point C | 2D | Not Simply Connected | 6 | 6 | — | — | — | — | 0.0001 | Region-dependent conclusion |
Formula Used
For a two-dimensional field F = (P, Q), a standard local test is:
For a three-dimensional field F = (P, Q, R), the field is locally curl-free when:
∂P/∂y = ∂Q/∂x
∂P/∂z = ∂R/∂x
∂Q/∂z = ∂R/∂y
The calculator measures each mismatch with:
If every required difference is less than or equal to the chosen tolerance, the derivative test passes numerically. A final global conclusion also depends on the region. Passing equalities on a simply connected region supports conservativeness. Passing equalities on a punctured or unknown region can remain inconclusive.
How to Use This Calculator
- Select whether your vector field is 2D or 3D.
- Choose the domain assumption that best matches the problem statement.
- Enter a sample point where the required derivatives are evaluated.
- Type the mixed partial derivative values from your manual work or symbolic tool.
- Set a small tolerance for numerical comparison.
- Press Test Field to show the result above the form.
- Review the table, chart, maximum mismatch, and curl magnitude estimate.
- Export the output with the CSV or PDF buttons if needed.
Important Notes
- This tool performs a numeric consistency check at one sample point.
- A single passing point does not replace a full symbolic proof over the whole region.
- Zero curl together with a simply connected domain is the key practical criterion used here.
- For line integrals, a valid potential function gives path independence.
Frequently Asked Questions
1) What does this calculator test?
It checks whether the mixed partial equalities expected from a conservative vector field match within a chosen tolerance. It also considers whether the region is simply connected before giving the strongest conclusion.
2) Why is the domain assumption important?
A curl-free test can fail to guarantee a global potential on regions with holes. Even when derivatives match, a non-simply connected domain may still produce path-dependent behavior around singularities.
3) Can this prove a field is conservative everywhere?
Not by itself. This version evaluates numeric derivative data at a sample point. For a full proof, the equalities must hold on the relevant region, together with the correct domain assumptions.
4) What tolerance should I use?
Use a small positive value based on your data quality. Exact symbolic work can use a very tiny tolerance. Approximate numerical derivatives usually need a slightly larger margin to avoid false failures.
5) What does curl magnitude estimate mean here?
It summarizes the size of the derivative mismatches. In 2D it reduces to one scalar comparison. In 3D it combines the three curl-related components into one magnitude for quick interpretation.
6) Why can a field pass the derivative test but stay inconclusive?
Because the derivative equalities are only part of the story. On a region with holes, the field may still lack a single-valued potential function, so the calculator reports a cautious conclusion.
7) Can I use this for homework checking?
Yes. It is useful for validating mixed partial calculations, checking curl conditions, and organizing numeric evidence. Still, you should present a complete mathematical argument in your final written solution.
8) Why does the result appear above the form?
That layout keeps the decision, statistics, chart, and export actions visible immediately after submission. It makes repeated testing faster because the form remains available just below the output.