Jacobian Determinant Calculator

Inputs

Supported: + − * / ^, parentheses, and functions like sin(), cos(), sqrt(), ln().

Dimension

Use 2D for planar mappings, 3D for spatial mappings.

Precision (decimals)

Higher precision shows more digits in derivatives and determinant.

Step size (h)

Typical range: 1e-6 to 1e-4 depending on scale.

u(x, y, z)

Use variables x, y, and optionally z.

v(x, y, z)

Constants: pi, e. Functions: exp(), ln().

w(x, y, z) (3D only)

Ignored in 2D mode.

x value

y value

z value (3D only)

Ignored in 2D mode.

Reset

Tip: For polar mapping u=r cos(θ), v=r sin(θ), use x=r and y=θ.

Example data table

Mode	Functions	Point	Expected determinant
2D	u = x^2 + y^2, v = x*y	(x, y) = (1, 2)	≈ 0 (near, due to chosen mapping point)
2D	u = x, v = y	(x, y) = (3, 4)	1 (identity mapping)
3D	u = x, v = y, w = z	(1, 2, 3)	1 (identity mapping)
3D	u = x^2, v = y^2, w = z^2	(1, 2, 3)	8·x·y·z = 48 at (1,2,3)

Notes: This calculator uses numerical derivatives, so results can vary slightly with step size.

For a 2D transformation (u, v) = (u(x, y), v(x, y)), the Jacobian matrix is:

J = [[∂u/∂x, ∂u/∂y],
[∂v/∂x, ∂v/∂y]]

The Jacobian determinant is:

det(J) = (∂u/∂x)(∂v/∂y) − (∂u/∂y)(∂v/∂x)

For 3D, the Jacobian is 3×3 and the determinant is computed using the standard cofactor expansion.

Partial derivatives are approximated with the central difference: ∂f/∂x ≈ (f(x+h) − f(x−h)) / (2h).