Potential Function Solver Calculator

Build potential functions from structured field coefficients. Validate gradient compatibility before integrating terms carefully first. Review outputs, tables, and downloads for multivariable problem solving.

Enter Vector Field Coefficients

This solver reconstructs scalar potentials for polynomial vector fields whose component terms are constant, linear, bilinear, and squared terms.

Reset

P(x, y, z)

x-direction gradient component.

Q(x, y, z)

y-direction gradient component.

R(x, y, z)

z-direction gradient component.

Interpretation note: The input defines a vector field F. The calculator checks whether F can be written as ∇φ and, if valid, rebuilds φ.

Example Data Table

Case Dimension Vector Field Potential Function Outcome
Example A 2D ⟨2xy + y2, x2 + 2xy⟩ x2y + xy2 + C Conservative
Example B 3D ⟨2xy + z2, x2 + 3, 2xz + 2⟩ x2y + xz2 + 3y + 2z + C Conservative
Example C 3D ⟨y, x + z, y⟩ No single scalar potential Non-conservative

Formula Used

The solver assumes a coefficient-based polynomial vector field F = ⟨P, Q, R⟩ in two or three variables. A scalar potential φ exists when the mixed partial derivatives match.

Conservative field conditions

For 2D fields, the requirement is:

∂P/∂y = ∂Q/∂x

For 3D fields, all of these must hold:

  • ∂P/∂y = ∂Q/∂x
  • ∂P/∂z = ∂R/∂x
  • ∂Q/∂z = ∂R/∂y

Potential reconstruction

When the checks pass, the solver integrates the x-component and assigns the remaining coefficients through compatibility conditions.

Typical recovered terms include:

φ = ax + by + cz + dxy + exz + fyz + gx² + hy² + iz² + … + C

How to Use This Calculator

  1. Choose whether your field is two-dimensional or three-dimensional.
  2. Enter the coefficients for each component P, Q, and R.
  3. Provide an evaluation point to test the field and potential numerically.
  4. Set a constant C if you want a specific potential value.
  5. Click Solve Potential Function to run the consistency checks.
  6. Review the reconstructed potential, point values, and compatibility table.
  7. Use the CSV or PDF button to export the result summary.

Frequently Asked Questions

1. What does this calculator solve?

It checks whether a polynomial vector field is conservative and, when possible, rebuilds a scalar potential whose gradient matches the entered field.

2. What kinds of fields are supported?

This version supports coefficient-based polynomial fields containing constants, first-degree terms, bilinear terms, and squared terms in each component.

3. Why can the result say no potential exists?

A scalar potential requires matching mixed partial derivatives. If any compatibility difference is nonzero, the field is not conservative in this model.

4. What does the constant C mean?

Potential functions are defined up to an additive constant. Changing C shifts potential values without changing the gradient field.

5. Does a zero curl guarantee a potential here?

Within this structured polynomial setup and ordinary simply connected settings, the matching partial checks are the practical test used by the solver.

6. Can I use fractions or decimals?

Yes. Every coefficient and evaluation coordinate accepts decimal values, so rational approximations and measured data can be entered directly.

7. What is shown at the evaluation point?

The calculator evaluates the vector field at your chosen point. If a potential exists, it also computes the corresponding scalar potential value there.

8. Why export the result?

Exports help you document solved examples, share compatibility checks, and keep a record of the reconstructed potential for later review.

Related Calculators

divergence theorem calculatorhessian matrix calculatorstream function calculatortransport equation solveradvection equation solvervibration equation solver

Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.