Jacobian Determinant Calculator
Calculate the coordinate transformation factor |J| and visualize grid distortion.
Define Transformation T(x,y)
Evaluate at Point (x, y)
Jacobian Matrix J
| … | … |
| … | … |
Determinant |J|
Transformation Visualization
Visual: The gray grid is standard (x,y). The blue lines show how that grid twists into (u,v). The Red Shape is what a small square around your point becomes.
What is the Jacobian?
When you change coordinate systems (like converting from Cartesian to Polar coordinates), you stretch or squish space. The Jacobian Determinant (often just called “the Jacobian”) measures exactly how much the area (or volume) expands or shrinks at any specific point during this transformation.
The Math
For a transformation u(x,y), v(x,y), the Jacobian Matrix is:
[ ∂v/∂x ∂v/∂y ]
The Determinant of this matrix is the scaling factor.
Why it matters
In Multiple Integrals, you can’t just replace dx dy with du dv. You must multiply by the Jacobian:
dx dy = |J| du dv
Interpreting the Value
- |J| = 1: Area is preserved (e.g., pure rotation).
- |J| > 1: Area is expanded (stretching).
- 0 < |J| < 1: Area is compressed (shrinking).
- |J| = 0: The transformation collapses a region into a line or point (singularity).
- Negative J: The transformation involves mirroring or flipping orientation. We usually take the absolute value for area integration.
Frequently Asked Questions (FAQ)
Q: What is the Jacobian for Polar Coordinates?
A: For x = r cos(θ), y = r sin(θ), the Jacobian is simply r. This is why we write r dr dθ instead of just dr dθ in integrals.
Q: Can I use this for 3D?
A: Yes! For 3 variables (u, v, w), it becomes a 3×3 matrix. The determinant then represents the volume scaling factor.
Q: Why is “det” calculated?
A: The matrix contains derivative information (slope). The determinant combines these slopes to tell you the total geometric area scaling, filtering out rotation.