Jacobian Calculator
Find the Jacobian matrix, determinant, and area scaling for a 2-variable linear map.
Table of Contents
How to Use
- Enter the partial derivatives a11, a12, a21, a22
- Set the point (x, y) where you want to evaluate
- Calculate to see the Jacobian matrix and determinant
- Review the area scale and mapped point
What the Jacobian tells you
The determinant of the Jacobian captures local scaling and orientation: positive values preserve orientation, negative values flip it, and zero collapses area to a line or point.
- det > 0 → orientation preserved
- det < 0 → orientation reversed
- det = 0 → mapping collapses area
Tips for linear maps
- Use the determinant magnitude as the local area scale
- If the determinant is near zero, the map is ill-conditioned at that point
- For nonlinear maps, evaluate the Jacobian at several points to study local behavior
Frequently Asked Questions
- Why is the determinant important?
- It measures local scaling: |det J| is the factor by which areas are stretched or compressed near the point. The sign indicates whether orientation is preserved or flipped.
- What if the determinant is zero?
- The mapping collapses area, so it is not locally invertible at that point. Adjust the transformation or evaluate at another point.
- Can I model nonlinear functions?
- This tool focuses on linear maps. For nonlinear functions, compute the analytical partial derivatives first, then plug their values in at the point of interest.