Jacobian Calculator
Type 2 or 3 functions and their variables (x, y, z or r, θ, φ) to get the Jacobian matrix with every partial derivative as a formula, the simplified Jacobian determinant with steps, and the values at a point.
The Jacobian matrix holds every first partial derivative, J[i][j] = ∂fᵢ/∂xⱼ, and its determinant gives the area or volume scale factor of a change of variables: r for polar coordinates and ρ² sin φ for spherical coordinates.
Use the Jacobian Calculator
Syntax
Enter functions and click Compute Jacobian
Jacobian matrix
Jacobian determinant
At
|J| =
Steps
Common Jacobian determinants
Change-of-variables factors used in double and triple integrals.
| Coordinates | Transformation | |J| | Differential |
|---|---|---|---|
| Polar | x = r cos θ, y = r sin θ | r | dx dy = r dr dθ |
| Cylindrical | x = r cos θ, y = r sin θ, z = z | r | dV = r dr dθ dz |
| Spherical (ρ, φ, θ) | x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ | ρ² sin φ | dV = ρ² sin φ dρ dφ dθ |
| Linear map | u = ax + by, v = cx + dy | ad − bc | du dv = |ad − bc| dx dy |
| Scaling (ellipse) | x = a u, y = b v | ab | dx dy = ab du dv |
Quick answer
For functions f₁ … fₘ of variables x₁ … xₙ, the Jacobian is the m × n matrix whose row i is the gradient of fᵢ: J[i][j] = ∂fᵢ/∂xⱼ. When m = n the Jacobian determinant |J| measures how the map stretches area (2D) or volume (3D) near a point; it is the factor in dx dy = |J| du dv when changing variables in an integral, and |J| ≠ 0 means the map is locally invertible there. For a 2 × 2 matrix, |J| = (∂f/∂x)(∂g/∂y) − (∂f/∂y)(∂g/∂x); for 3 × 3, expand along the first row. Polar coordinates x = r cos θ, y = r sin θ give |J| = r, cylindrical gives r, and spherical x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ with variables (ρ, φ, θ) gives ρ² sin φ.
Examples
| Functions | Variables | Jacobian determinant |
|---|---|---|
| 2x + 3y, 4x + 5y | x, y | −2 |
| x² − y², 2xy | x, y | 4x² + 4y² |
| r cos θ, r sin θ | r, θ | r |
| r cos θ, r sin θ, z | r, θ, z | r |
| ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ | ρ, φ, θ | ρ² sin φ |
Enter your functions above or load an example to see the matrix, the determinant and each step.
Summary
This Jacobian calculator works symbolically, like a computer algebra system. Enter 2 or 3 functions and the variables they depend on, in Cartesian or polar, cylindrical and spherical form, and it returns the Jacobian matrix with each partial derivative as a simplified formula, the determinant expanded and simplified (including sin² + cos² = 1), and each step. Give a point to evaluate the matrix and determinant numerically and check local invertibility. Everything runs in your browser.
How it works
- Choose 2 × 2 or 3 × 3, or load an example such as polar or spherical coordinates.
- Type the variables (x, y, z or r, θ, φ — write theta, phi and rho or paste the Greek letters) and each function: x^2 y, r cos(θ), e^(x y), sqrt(x^2 + y^2).
- The calculator differentiates every function with respect to every variable symbolically, using the power, product, quotient and chain rules.
- It builds the Jacobian matrix, expands the determinant and simplifies it, including sin² + cos² = 1.
- Enter a point to evaluate the matrix and determinant numerically and see whether the map is locally invertible there.
Use cases
- Checking Jacobian matrices and determinants for multivariable calculus homework.
- Finding the change-of-variables factor for double and triple integrals (polar, cylindrical, spherical).
- Deriving custom coordinate transformations such as u = x² − y², v = 2xy.
- Testing whether a map is locally invertible at a point (non-zero determinant).
- Linearising nonlinear systems in dynamics, control and Newton's method.
- Checking hand-computed partial derivatives one entry at a time.