This paper combines some ideas to obtain the multitime variants of Hamilton-Jacobi theory. Section I introduces some Riemannian first order jet bundles. Section 2 describes the multitime Hamilton-Jacobi PDE system connected to the path independent curvilinear integral action. Section 3 analyzes the multitime Hamilton-Jacobl divergence PDE connected to multiple integral action. Section 4 studies the harmonic map Lagrangian. Section 5 underlines the novelty of our results.