Optimal Quadrature Formulas for Curvilinear Integrals of the First Kind

We consider the problem on optimal quadrature formulas for curvilinear integrals ofthe first kind that are exact for constant functions. This problem is reduced to the minimizationproblem for a quadratic form in many variables whose matrix is symmetric and positive definite.We prove that the objective quadratic function attains its minimum at a single point ofthe corresponding multi-dimensional space. Hence, for a prescribed set of nodes, there existsa unique optimal quadrature formula over a closed smooth contour, i.e., a formula with the leastpossible norm of the error functional in the conjugate space. We show that the tuple of weights ofthe optimal quadrature formula is a solution of a special nondegenerate system of linear algebraicequations.