Potential Function Calculator

Calculator Inputs

This tool builds a scalar potential φ(x, y) from a conservative two-variable field and evaluates its value, gradient, sample table, and graph.

A coefficient for Ax

B coefficient for shared xy structure

C coefficient for x² term in P

D coefficient for xy coupling in P

E coefficient for y² influence in P

F constant term in P

G coefficient for Gy term

H coefficient for y² term in Q

I constant term in Q

Potential constant K

Evaluate at x

Evaluate at y

Graph x minimum

Graph x maximum

Fixed y line for graph

Plot sample points

Field used by this calculator

P(x, y) = Ax + By + Cx² + Dxy + Ey² + F
Q(x, y) = Bx + (D/2)x² + 2Exy + Gy + Hy² + I

Plotly Graph

The graph below shows φ(x, yline), P(x, yline), and Q(x, yline) across the chosen x-range.

Example Data Table

Point	x	y	φ(x, y)	P(x, y)	Q(x, y)
1	0.000000	0.000000	0.000000	3.000000	1.500000
2	1.000000	1.000000	9.116667	8.100000	7.000000
3	2.000000	1.500000	25.470833	15.000000	13.400000
4	3.000000	2.000000	53.700000	24.300000	22.100000

You can export this table and the summary values using the CSV or PDF buttons.

Formula Used

1) Potential definition
A scalar potential function φ(x, y) satisfies ∇φ = (∂φ/∂x, ∂φ/∂y) = (P, Q).

2) Conservative condition
A two-variable field is conservative when ∂P/∂y = ∂Q/∂x over the domain.

3) Potential recovered by integration
φ(x, y) = (A/2)x² + Bxy + (C/3)x³ + (D/2)x²y + Exy² + Fx + (G/2)y² + (H/3)y³ + Iy + K

4) Gradient at the chosen point
∂φ/∂x = P(x, y)
∂φ/∂y = Q(x, y)

5) Interpretation
Larger positive φ values indicate higher scalar potential under the chosen coefficients. The gradient shows the local change direction and rate.

How to Use This Calculator

Enter the field coefficients A through I and the optional constant K.
Choose the point (x, y) where you want the potential and gradient.
Set the graph range and the fixed y-line for the Plotly chart.
Press Calculate Potential Function.
Read the result block placed above the form, directly below the header.
Review the potential equation, gradient components, example table, and graph.
Use CSV for spreadsheet-friendly output and PDF for printable reporting.

FAQs

1) What does this calculator compute?

It computes a scalar potential function, evaluates it at a chosen point, returns gradient components, checks conservative structure, plots sampled values, and creates an example table.

2) Why is the field written in a special form?

The paired component formulas are arranged so the conservative condition is satisfied automatically. That makes the recovered potential exact for the chosen coefficient set.

3) What is the meaning of the constant K?

K shifts the entire potential function upward or downward without changing the gradient. It changes absolute potential values but not directional change rates.

4) What does the gradient represent?

The gradient gives the steepest local increase of the scalar potential. Its x- and y-components match the field values P and Q at the evaluation point.

5) Why does the conservative check show zero?

The calculator uses a compatible field design where ∂P/∂y and ∂Q/∂x are equal. A zero difference confirms consistency at the chosen point.

6) What does the Plotly graph show?

It shows how the potential and both gradient components vary along x while y stays fixed. This helps compare scalar growth and directional rates together.

7) When would I use this in mathematics?

It is useful in vector calculus, conservative field analysis, gradient-based interpretation, path-independent work studies, and classroom demonstrations involving scalar potential construction.

8) Can I export the results for reports?

Yes. The CSV button exports summary values and example rows, while the PDF button creates a neat report-style snapshot suitable for printing or sharing.