Calculator inputs
This page uses a single stacked layout, while the input controls switch between three, two, and one columns across screen sizes.
Formula used
For a scalar field f and a driver vector field X, the Lie derivative is the directional derivative of f along X.
LXf = X · ∇f = X1 ∂f/∂x + X2 ∂f/∂y + X3 ∂f/∂z
For a vector field Y, the Lie derivative along X equals the Lie bracket of X and Y.
LXY = [X,Y] = JYX - JXY
This calculator uses a quadratic model for f and affine models for X and Y, making the derivatives exact for the entered coefficients.
f(x,y,z) = c0 + cx x + cy y + cz z + cxx x² + cyy y² + czz z² + cxy xy + cxz xz + cyz yz
Xi(x,y,z) = ai1x + ai2y + ai3z + ai4
Yi(x,y,z) = bi1x + bi2y + bi3z + bi4
How to use this calculator
- Enter the evaluation point coordinates.
- Provide scalar field coefficients for the quadratic scalar model.
- Enter the affine coefficients for the driver field X.
- Enter the affine coefficients for the reference field Y.
- Press the calculate button to place the result section below the header and above the form.
- Review the scalar derivative, bracket vector, magnitude, table, and graph.
- Use the CSV button for spreadsheet work and the PDF button for reports.
- Try the example loader to verify the structure quickly.
Example data table
| Item | Sample value | Notes |
|---|---|---|
| Point p | (1, 2, 0.5) | Evaluation location used by the sample loader. |
| Scalar field f | 2x² + xy + z | This yields f(p) = 4.5 for the sample point. |
| Driver field X | (y + 1, -x + 2z, z + 1) | At p, this becomes X(p) = (3, 0, 1.5). |
| Reference field Y | (x - z, 2y + 1, x + y) | At p, this becomes Y(p) = (0.5, 5, 3). |
| Expected outputs | LXf(p) = 19.5, [X,Y](p) = (-3.5, -5.5, 0) | The bracket magnitude is approximately 6.5192. |
FAQs
1. What does the scalar Lie derivative measure?
It measures how a scalar field changes when you move in the direction of the driver vector field. A positive value means local increase, a negative value means local decrease, and zero indicates local stationarity at the chosen point.
2. What does the vector Lie derivative represent here?
Here it is the Lie bracket [X,Y]. It compares the infinitesimal action of two vector fields and shows whether their local flows commute. A zero bracket means the fields commute locally at the chosen point.
3. Why are the fields coefficient-based instead of symbolic text boxes?
Coefficient inputs keep the calculations exact, stable, and easy to validate in one standalone file. They also avoid unreliable expression parsing while still covering many useful quadratic and affine examples from coursework and applied geometry.
4. Can I use decimals and negative coefficients?
Yes. All numeric fields accept decimal and negative values. That lets you model signed transport directions, shifted affine fields, and mixed quadratic terms without changing the calculator structure.
5. What does the Plotly graph show?
The graph samples points along a normalized path through the selected location, aligned with X(p). It then plots the sampled scalar Lie derivative and the sampled bracket magnitude to reveal how local behavior changes nearby.
6. When should I inspect bracket magnitude?
Inspect it when you want a simple strength measure for local non-commutativity. Larger magnitudes indicate stronger mismatch between the infinitesimal actions of X and Y at the selected point.
7. Does this support tensors or differential forms?
This version focuses on scalar fields and vector fields. The same geometric ideas extend to tensors and forms, but those require a broader input model and more component bookkeeping than this single-page calculator uses.
8. How should I verify my input set?
Use the example loader first. Confirm the returned values match the example table, then replace the coefficients with your own case. This makes sign errors and misplaced coefficients much easier to detect.