the Jacobian matrix
/ ya-KOH-bee-an /
When a function takes in several numbers and spits out several numbers, asking for its single 'slope' no longer makes sense — there are many directions to differentiate in. The Jacobian matrix is the honest replacement for the derivative in this multivariable setting. It is a grid that records how each output responds to a small change in each input, and it is exactly the linear map that best approximates the function near a point.
Concretely, for a planar system x' = f(x,y), y' = g(x,y), the Jacobian is the 2-by-2 grid J = [∂f/∂x, ∂f/∂y; ∂g/∂x, ∂g/∂y], where each entry is a partial derivative — how fast one component changes as you tweak one variable, holding the other fixed. Evaluate every entry at a particular point (usually an equilibrium) to get a matrix of plain numbers. That number-matrix is the linear system u' = J u that the original equations look like nearby.
The Jacobian is the bridge between nonlinear systems and the linear theory you already know. Its eigenvalues at an equilibrium decide the local type — node, saddle, spiral, center — and its determinant and trace feed straight into the trace-determinant classification. The same matrix appears all over mathematics: as the stretch factor in changing variables in multiple integrals, and as the step in Newton's method for several unknowns.
For f = x - x y, g = -y + x y, the Jacobian is [1 - y, -x; y, -1 + x]. At the equilibrium (1,1) it becomes [0, -1; 1, 0], whose eigenvalues are plus and minus i — the linearization predicts a center.
Each entry is a partial derivative; evaluating at an equilibrium converts the matrix of functions into a matrix of numbers ready for eigenvalue analysis.
Do not confuse the Jacobian matrix with the Jacobian determinant — engineers often call the determinant alone 'the Jacobian'. For stability you need the whole matrix (and its eigenvalues), not just its determinant.