Potential Function Calculator

Field dimension

Potential constant

Decimal precision

First component P or M

Second component Q or N

Third component R

Evaluation x

Evaluation y

Evaluation z

Start x

Start y

Start z

End x

End y

End z

Example Data Table

Mode	P	Q	R	Expected potential
2D	2xy	x^2 + 2*y	0	x^2*y + y^2 + C
3D	2xy + z	x^2	x	x^2y + xz + C
2D	y	x	0	x*y + C
3D	2*x	2*y	2*z	x^2 + y^2 + z^2 + C

Formula Used

For a two variable field V = <P,Q>, a potential function F satisfies F_x = P and F_y = Q. A polynomial field is conservative when Q_x - P_y = 0.

For a three variable field V = <P,Q,R>, the calculator checks curl V = <R_y - Q_z, P_z - R_x, Q_x - P_y>. When the curl is zero and all rebuilt derivatives match, F is accepted.

The endpoint work estimate is W = F(end) - F(start). This applies only when the field is conservative.

How To Use This Calculator

Select two dimensional or three dimensional mode.
Enter polynomial vector components using x, y, and z.
Use multiplication signs for the clearest parsing.
Enter one evaluation point and two endpoint sets.
Press calculate to show results above the form.
Use CSV or PDF options to save the output.

Understanding Potential Functions

A potential function turns a vector field into one scalar expression. It is useful when a field is conservative. In that case, every component of the field is a partial derivative of the same scalar function. This calculator focuses on polynomial fields in two or three variables. It reads the entered components, tests the needed derivative matches, and builds a possible potential function when the field passes the checks.

Why The Test Matters

A conservative field has path independent work. This means the work from one point to another depends only on the endpoints. The curve shape does not change the final value. That property is important in mechanics, electricity, optimization, and coordinate geometry. When the field is not conservative, a single potential function cannot represent all components at once.

What The Calculator Does

The tool first parses each component as a polynomial in x, y, and z. Then it integrates the first component with respect to its matching variable. After that, it compares the next component with the derivative already produced. Any missing part is integrated and added. The same process repeats for the third component when three dimensional mode is selected. Finally, the calculator differentiates the finished potential again. If every derivative matches the original field, the result is accepted.

Reading The Result

The potential function is shown with an optional constant. The curl or mixed partial check is also displayed. A zero check supports the conservative conclusion. The value at one point can help verify a specific location. The endpoint work estimate uses the potential difference between the final and starting points.

Input Tips

Use terms such as 2*x^2*y, -3*y, z, or 5. Multiplication signs are recommended. Decimal coefficients are accepted. Keep powers as non-negative integers. Avoid trigonometric, logarithmic, or rational expressions, because this page is designed for polynomial-style entries. For clear results, enter zero in unused components and choose two dimensional mode when z is not needed.

Exporting Results

Use the CSV option for spreadsheet records. Use the PDF button for a quick report. The example table gives safe sample inputs. Compare it with your own field before submitting complex entries. This supports repeatable notes during careful problem solving.

FAQs

What is a potential function?

It is a scalar function whose gradient equals the vector field. Each vector component becomes one partial derivative of that scalar function.

What inputs does this calculator support?

It supports polynomial terms using x, y, and z. Use products, powers, decimals, fractions, addition, and subtraction.

Can I use sine or cosine terms?

No. This version is focused on polynomial-style entries. Rewrite the field as supported terms, or use a symbolic algebra system.

How does it test a conservative field?

It compares mixed partials through curl checks. It also rebuilds the potential and verifies that its derivatives match the entered components.

Why is endpoint work included?

For conservative fields, work depends only on endpoints. The calculator estimates work by subtracting starting potential from ending potential.

Should I enter R in two dimensional mode?

You may leave R as zero. The calculator ignores the third component when two dimensional mode is selected.

What does a failed derivative match mean?

It means the entered field cannot be represented by one supported potential function. Recheck the field components and signs.

Can I save my result?

Yes. Use the CSV button for spreadsheet output. Use the PDF button after calculating to create a simple report.