Jacobian Matrix Formula
The Jacobian matrix contains all first-order partial derivatives of a vector-valued function.
Used in coordinate transformations, optimization, and robotics.
The Formula
J = [∂f/∂x] = [[∂f&sub1;/∂x&sub1;, ..., ∂f&sub1;/∂x_n], [...], [∂f_m/∂x&sub1;, ..., ∂f_m/∂x_n]]
The Jacobian matrix J of a function f: R^n → R^m contains all first-order partial derivatives. Row i contains the partial derivatives of f_i (the i-th output component) with respect to all inputs. The Jacobian is the best linear approximation of f near a point — it generalizes the derivative to multi-dimensional functions.
Variables
| Symbol | Meaning |
|---|---|
| J | Jacobian matrix (m × n) |
| f: R^n → R^m | Vector-valued function with m outputs and n inputs |
| Jij | Element in row i, column j = ∂fi/∂xj |
| det(J) | Jacobian determinant — scaling factor for coordinate change (requires n = m) |
Example 1 — Polar Coordinates
Find the Jacobian of the polar-to-Cartesian transformation: x = r cos(θ), y = r sin(θ).
J = [[∂x/∂r, ∂x/∂θ], [∂y/∂r, ∂y/∂θ]]
= [[cos(θ), −r sin(θ)], [sin(θ), r cos(θ)]]
det(J) = cos(θ) × r cos(θ) − (−r sin(θ)) × sin(θ)
= r cos²(θ) + r sin²(θ) = r
det(J) = r — this is why dx dy = r dr dθ in polar coordinate integration. The r factor accounts for the stretching of area elements.
Example 2 — Newton's Method in 2D
Solve the system f(x,y) = (x² + y² − 1, x − y) = 0 using Newton's method from (x,y) = (1, 0).
J = [[∂f&sub1;/∂x, ∂f&sub1;/∂y], [∂f&sub2;/∂x, ∂f&sub2;/∂y]] = [[2x, 2y], [1, −1]]
At (1, 0): J = [[2, 0], [1, −1]], f = (0, 1)
Newton step: (x,y) = (1,0) − J&sup-1; f(1,0)
Solution converges to (√2/2, √2/2) ≈ (0.707, 0.707), the point on the unit circle where x = y
When to Use It
The Jacobian matrix is used when:
- Changing variables in multiple integrals — the Jacobian determinant gives the volume scaling factor
- Solving systems of nonlinear equations using Newton-Raphson iteration
- Robotics and kinematics — the Jacobian relates joint velocities to end-effector velocities
- Analyzing stability of dynamical systems near fixed points
- Implementing backpropagation in neural networks (chain rule through layers)
- Linearizing nonlinear control systems around an operating point