Calculator Inputs
Enter a constant vector field F = <Fx, Fy, Fz>, a point r = <x, y, z>, and a path for graphing.
Plotly Graph
The graph tracks Ax, Ay, Az, and |A| along the selected path r(t) = r0 + td.
Saved Calculation Table
This table keeps the most recent results from your current session.
| # | Field F | Point r | k | Vector Potential A | |A| | curl(A) |
|---|---|---|---|---|---|---|
| No saved calculations yet. Submit the form to populate this table. | ||||||
Example Data Table
This worked example uses F = <2, -1, 4>, r = <1, 3, -2>, and k = 1.5.
| Fx | Fy | Fz | x | y | z | k | Ax | Ay | Az | |A| |
|---|---|---|---|---|---|---|---|---|---|---|
| 2 | -1 | 4 | 1 | 3 | -2 | 1.5 | -7.5000 | 6.0000 | 5.2500 | 10.9459 |
Formula Used
This calculator uses a canonical vector potential for a constant vector field. It is useful for vector calculus study, curl verification, and coordinate-based analysis.
r = <x, y, z>
A = (k/2)(F × r)
Ax = (k/2)(Fyz - Fzy)
Ay = (k/2)(Fzx - Fxz)
Az = (k/2)(Fxy - Fyx)
|A| = √(Ax2 + Ay2 + Az2)
curl(A) = kF
Important note: vector potentials are not unique. This page returns one convenient canonical choice for constant fields.
The graph evaluates the same equation along the line r(t) = r0 + td, so you can inspect how each component changes with path parameter t.
How to Use This Calculator
- Enter the constant field components Fx, Fy, and Fz.
- Enter the evaluation point coordinates x, y, and z.
- Set the scaling factor k. Use 1 for the standard canonical form.
- Choose a direction vector for the graph path.
- Set path start, path end, and number of graph steps.
- Pick your preferred decimal precision.
- Press the calculate button to show the result above the form.
- Review the metrics, graph, session table, and export files if needed.
Frequently Asked Questions
1) What does this calculator actually compute?
It computes one canonical vector potential A for a constant vector field F at a selected point r. It also reports magnitude, curl verification, path samples, and a graph.
2) Why does the formula use a cross product?
For constant fields, A = (1/2)(F × r) is a compact vector-calculus construction whose curl equals F. The scale factor k lets you study curl(A) = kF instead.
3) Is the vector potential unique?
No. Many vector potentials can produce the same curl. This page returns a clean canonical form, which is helpful for calculation, plotting, and classroom verification.
4) Can I use this for non-constant fields?
Not directly. This implementation assumes the target field components are constant. Non-constant fields usually require symbolic integration, gauge choices, or numerical methods beyond this page.
5) What does the graph represent?
The graph evaluates vector potential components along the line r(t) = r0 + td. It helps you see how A changes with position and direction.
6) Why is curl(A) shown in the results?
The curl check confirms the construction. For the chosen canonical formula, curl(A) should equal kF. That makes the output easy to verify numerically.
7) What does A · F tell me here?
It provides a quick orthogonality check. Because A comes from a cross product with F, the dot product often helps illustrate geometric relationships in the computed result.
8) What do the CSV and PDF exports contain?
The CSV export contains path sample values for t, point coordinates, vector potential components, and magnitudes. The PDF export captures the current result summary for documentation.