Jacobian Determinant Calculator
The Jacobian determinant describes how a two-variable transformation stretches, shrinks, or flips area near a point. It is the determinant of the Jacobian matrix, whose entries are the four first-order partial derivatives of the transformation. You enter those four partial derivatives, evaluated at your point of interest, and this calculator returns the determinant along with the absolute area-scaling factor and an orientation flag. The Jacobian determinant is central to changing variables in multiple integrals and to the inverse function theorem.
Jacobian determinant formula
J = | du/dx du/dy |
| dv/dx dv/dy |
det(J) = (du/dx)*(dv/dy) - (du/dy)*(dv/dx)
Area factor = | det(J) |
The determinant follows the standard 2x2 cross-product rule: the product of the main diagonal minus the product of the anti-diagonal. Its absolute value gives the local area scaling and its sign gives the orientation.
Notes on the Jacobian
- The four inputs are partial derivatives already evaluated at your chosen point.
- The absolute value of the determinant is the local area-scaling factor.
- A negative determinant means the transformation reverses orientation locally.
- A zero determinant means the map is not locally invertible there.
- In a change-of-variables integral, you multiply by the absolute Jacobian determinant.
Jacobian determinant: frequently asked questions
What is the Jacobian determinant?
The Jacobian determinant is the determinant of the Jacobian matrix, the matrix of first-order partial derivatives of a vector-valued transformation. For a 2D transformation it equals the local factor by which the map stretches or shrinks area.
What is the formula for a 2x2 Jacobian determinant?
For partial derivatives arranged as [[du/dx, du/dy], [dv/dx, dv/dy]], the Jacobian determinant is (du/dx)*(dv/dy) minus (du/dy)*(dv/dx). It is the same cross-product rule as any 2x2 determinant.
Why does the Jacobian matter in changing variables?
When you change variables in a double integral, the absolute value of the Jacobian determinant is the factor that corrects for how the new coordinates distort area. It is the multivariable analogue of the derivative in single-variable substitution.
What does a negative Jacobian determinant mean?
A negative Jacobian determinant means the transformation reverses orientation, flipping the handedness of the coordinate system locally. For area scaling you take the absolute value, but the sign itself carries orientation information.
What does a zero Jacobian determinant indicate?
A zero Jacobian determinant means the transformation is locally degenerate at that point: it collapses area to zero and is not invertible there. The inverse function theorem requires a nonzero Jacobian for a local inverse to exist.
Official sources
- NIST Digital Library of Mathematical Functions: Vectors and vector-valued functions.
- NIST Digital Library of Mathematical Functions: Determinants and linear operators.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.