 given a critical point x = (x1, ..., xn), i.e., a point where the first partial derivatives ∂f/∂xi{\displaystyle \partial f/\partial x_{i}} of ƒ vanish, the function has a local minimum if the Hessian matrix is positive definite. Quadratic programming can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see above).[65]Another matrix frequently used in geometrical situations is the Jacobi matrix of a differentiable map f: Rn → Rm. If f1, ..., fm denote the components of f, then the Jacobi matrix is defined as [66]J(f)=[∂fi∂xj]1≤i≤m,1≤j≤n.{\displaystyle J(f)=\left[{\frac {\partial f_{i}}{\partial x_{j}}}\right]_{1\leq i\leq m,1\leq j\leq n}.}If n > m, and if the rank of the Jacobi matrix attains its maximal value m, f is locally invertible at that point, by the implicit function theorem.[67]Partial differential equations can be classified by con