the jacobian matrix is how multivariable systems actually move you don’t deal with one variable anymore you deal with transformations input vector → output vector the jacobian captures how every input dimension affects every output dimension what it is: → a matrix of partial derivatives → each row = one output function → each column = one input variable J(i,j) = ∂f_i / ∂x_j why it matters: → it’s the local linear approximation of a nonlinear system → it tells you how small changes propagate → it converts messy systems into something you can compute in physics: → coordinate transformations → velocity mappings → change of variables in integrals in robotics: → maps joint velocities → end-effector velocity → singularities show up when the jacobian collapses ...