Calculator Inputs
Enter expressions using x, y, and z.
Use explicit multiplication like 2*x and powers like x^2.
Plotly Graph
The graph samples the computed exterior derivative magnitude across nearby x-values while keeping the remaining coordinates fixed.
Example Data Table
| Case | Dimension | Input Form | Point | Expected Exterior Derivative |
|---|---|---|---|---|
| Scalar field | 2D | f = x^2*y + sin(y) | (1, 0.5) | d f ≈ 1.000000 dx + 1.877583 dy |
| One-form | 3D | ω = y dx + z dy + x dz | (1, 1, 1) | dω = -1 dx∧dy + 1 dx∧dz - 1 dy∧dz |
| Two-form | 3D | ω = z dx∧dy + 0 dx∧dz + x dy∧dz | (1, 1, 1) | dω = 2 dx∧dy∧dz |
Formula Used
For a 0-form f
The exterior derivative becomes the differential: d f = ∂f/∂x dx + ∂f/∂y dy + ∂f/∂z dz. In 2D, the z-term is omitted.
For a 1-form ω = A_x dx + A_y dy + A_z dz
The result is a 2-form: dω = (∂A_y/∂x − ∂A_x/∂y) dx∧dy + (∂A_z/∂x − ∂A_x/∂z) dx∧dz + (∂A_z/∂y − ∂A_y/∂z) dy∧dz.
For a 2-form ω = B_xy dx∧dy + B_xz dx∧dz + B_yz dy∧dz
In 3D, the result is a 3-form: dω = (∂B_yz/∂x − ∂B_xz/∂y + ∂B_xy/∂z) dx∧dy∧dz. In 2D, the derivative of a 2-form is zero.
Numerical Differentiation
This calculator uses the central-difference approximation: ∂f/∂x ≈ [f(x+h) − f(x−h)] / (2h). The same pattern is used for y and z derivatives.
How to Use This Calculator
- Choose the dimension, then select whether you are differentiating a 0-form, 1-form, or 2-form.
- Enter the coordinate point where the derivative should be approximated.
- Provide a small positive step size
hfor numerical differentiation. - Type each coefficient expression using explicit multiplication, such as
3*x*y. - Submit the form. The result will appear above the form, followed by coefficient tables and a graph.
- Use the export buttons to save a CSV or PDF copy of the computed results.
Frequently Asked Questions
1) What does this calculator compute?
It numerically approximates the exterior derivative of scalar fields, one-forms, and two-forms at a chosen point in two or three dimensions.
2) Does it perform symbolic differentiation?
No. It uses central differences, so the result is numerical. This makes it practical for quick evaluation, visualization, and export without a symbolic algebra engine.
3) Which variables can I use?
Use x, y, and z. Supported functions include sin, cos, tan, asin, acos, atan, sqrt, abs, log, exp, pow, and pi.
4) Why must multiplication be explicit?
The parser expects expressions like 2*x or x*y. Writing 2x may fail because implicit multiplication is not interpreted safely.
5) What step size should I choose?
A small value like 0.001 is a good starting point. Very large values reduce accuracy, while extremely tiny values can amplify floating-point noise.
6) Why is the derivative of a 2-form zero in 2D?
Because a 2-form is already top-degree in two dimensions. Applying d would produce a 3-form, and no nonzero 3-forms exist there.
7) What does the graph show?
It plots the magnitude of the computed exterior derivative while x changes near the chosen point. The other coordinates stay fixed.
8) What do the CSV and PDF files contain?
They include the displayed result coefficients, the derivative details, and summary labels from the current calculation for reporting or study notes.