Calculator Input
Enter the dimension, source variables, transformed variables, evaluation point, mapped point, and the partial derivatives that make up the Jacobian matrix.
Example data table
| Case | Dimension | Point | Jacobian matrix | Determinant | Interpretation |
|---|---|---|---|---|---|
| Planar shear-stretch | 2 × 2 | (x, y) = (1, 0) | [[2, 1], [-1, 3]] | 7 | Area expands sevenfold and orientation is preserved. |
| Planar singular map | 2 × 2 | (x, y) = (2, 2) | [[1, 2], [2, 4]] | 0 | Local collapse occurs, so no inverse exists there. |
| Spatial mapping | 3 × 3 | (x, y, z) = (1, 1, 1) | [[2, 1, 0], [0, 3, -1], [1, 0, 2]] | 13 | Volume expands by thirteen, with invertible local behavior. |
Formula used
General Jacobian matrix
For a transformation F(x₁, x₂, …, xₙ) = (f₁, f₂, …, fₙ), the Jacobian is the matrix
J = [∂fᵢ / ∂xⱼ].
Two-variable determinant
For (u, v) = F(x, y),
det(J) = (∂u/∂x)(∂v/∂y) - (∂u/∂y)(∂v/∂x).
Three-variable determinant
For (u, v, w) = F(x, y, z), the determinant is the determinant of the
3 × 3 partial-derivative matrix. A nonzero value means the local inverse exists.
Differential scaling rule
The calculator uses dU = |det(J)| · dX. In 2D, this scales area elements. In 3D, it scales volume elements.
Local linear approximation
Small source changes satisfy ΔF ≈ J · ΔX. Therefore, the columns of the Jacobian show how source basis directions deform locally.
How to use this calculator
1) Choose the Jacobian size
Select 2 × 2 for two variables or 3 × 3 for three variables.
2) Enter variable labels
Type source names such as x, y, z and target names such as u, v, w to match your transformation.
3) Provide the evaluation point
Enter the source coordinates where the Jacobian should be evaluated. This is the local point for your transformation analysis.
4) Add mapped values
Enter the transformed coordinates at that same point. These values help position the plotted transformed cell or basis tips.
5) Fill in the partial derivatives
Enter every derivative that belongs in the Jacobian matrix. Each row is a target component, and each column is a source variable.
6) Review the outputs
Press Compute Jacobian. The page shows the determinant, inverse matrix, rank, norms, scaling effect, and Plotly graph above the form.
Frequently asked questions
1) What does the Jacobian determinant tell me?
It measures local scaling. In two variables, it scales tiny areas. In three variables, it scales tiny volumes. Its sign also indicates whether orientation is preserved or reversed near the chosen point.
2) Why can the determinant be zero?
A zero determinant means the mapping collapses local dimensions. Nearby points may flatten onto a line or plane, so the transformation loses invertibility at that location.
3) What does the inverse Jacobian represent?
It estimates how small changes in transformed coordinates map back to the original coordinates. It exists only when the Jacobian determinant is nonzero.
4) Why are partial derivatives entered manually?
This page is designed for reliable numeric evaluation at a point. Manual entries let you use derivatives from hand calculations, symbolic tools, or external algebra systems.
5) What is the difference between row norms and column norms?
Row norms summarize how sensitive each transformed component is. Column norms summarize how strongly each source direction influences the mapped coordinates.
6) What does the condition number mean here?
It estimates local sensitivity to perturbations. A larger condition number suggests the transformation is numerically less stable and more sensitive to small input changes.
7) Can this calculator handle coordinate changes?
Yes. It is useful for Cartesian, polar, cylindrical, spherical, or custom mappings, provided you know the partial derivatives at the evaluation point.
8) What does the Plotly graph show?
It visualizes local deformation. For two variables, it shows a transformed cell. For three variables, it shows transformed basis directions anchored at the mapped point.