A Morse function is a smooth function from a manifold to the real numbers, satisfying certain conditions at its critical points. Specifically, at each critical point (where the derivative of vanishes), the Hessian matrix (the matrix of second partial derivatives) of is non-degenerate (its determinant is non-zero). This ensures that the critical points are isolated and that we can classify them according to their index—the number of negative eigenvalues of the Hessian.