Lie bracket formula: [X,Y]k = Σ Xi(∂Yk/∂xi) − Σ Yi(∂Xk/∂xi)
This implementation models each vector-field component as a quadratic polynomial with constant, linear, squared, and mixed terms. The calculator evaluates partial derivatives exactly and then computes each bracket component at the chosen point.
For 2D, the calculator uses x and y components. For 3D, it includes x, y, and z. Component magnitudes are also summarized with a Euclidean bracket norm:
- Select 2D or 3D based on your vector fields.
- Enter the evaluation point coordinates.
- Fill coefficients for each component of X and Y.
- Use constant, linear, squared, and mixed-term inputs as needed.
- Press Calculate Lie Bracket to display the result above the form.
- Review component contributions, norm, table output, and Plotly chart.
- Download CSV for data sharing or PDF for reporting.
| Dimension | X | Y | Point | Bracket Result |
|---|---|---|---|---|
| 2D | (y, −x) | (x², y) | (1, 2) | (2, 0) |
| 3D | (y, z, x) | (x, y², z) | (1, 1, 1) | Enter coefficients to evaluate |
The default sample values loaded in the form match the first example row.
1. What does the Lie bracket measure?
It measures how two vector fields fail to commute. If the bracket is zero, the local flows align more smoothly. Nonzero values indicate directional interaction between the fields.
2. Which inputs does this calculator support?
It supports 2D or 3D vector fields whose components are quadratic polynomials. You can enter constants, linear terms, squared terms, and mixed terms like xy, xz, and yz.
3. Why are there many coefficient boxes?
Each component is treated as a polynomial. Separate boxes let you define detailed vector fields without typing symbolic expressions, which keeps the calculation exact and the interface structured.
4. Does the calculator work symbolically?
It performs exact derivative rules on the polynomial coefficients, then evaluates the bracket numerically at your chosen point. So the derivative step is analytic, while the final display is point-based.
5. What does the magnitude output mean?
The magnitude summarizes the overall size of the computed bracket vector at the selected point. It is useful when comparing several field pairs or checking whether the interaction is small or large.
6. Why might my bracket become zero?
A zero result can happen when both directional-derivative terms cancel exactly at the chosen point. It may also mean the fields commute there, even if they do not commute everywhere.
7. What is included in the CSV export?
The CSV file includes dimension, evaluation point, both vector fields, each intermediate directional term, every bracket component, and the final bracket magnitude for documentation.
8. When should I use 3D mode?
Use 3D mode when your vector fields depend on z or include a third component. If your problem only needs x and y, 2D mode keeps the form shorter.