Gregory J. Quadratic Form Theory and Differential Equations

Academic Press, 1980. - 248 Pages.Historically, quadratic form theory has been treated as a rich but misunderstood uncle. It appears briefly, almost as an afterthought, when needed to solve a variety of problems. A partial list of such problems includes the Hessian matrix in n-dimensional calculus; the second variational (Jacobi or accessory) problem in the calculus of variations and optimal control theory; Rayleigh- Ritz methods for finding eigenvalues of real symmetric matrices; the Aronszajn-Weinstein methods for solving problems of vibrating rods, membranes, and plates; oscillation, conjugate point, and Sturm comparison criteria in differential equations; Sturm- Liouville boundary value problems; spline approximation ideas for numerical approximations; Gershgorin-type ideas (and the Euler- Lagrange
equations) for banded symmetric matrices; Schrodinger equations; and limit-point-limit-circle ideas of singular differential equations in mathematical physics.