Fundamentals of Advanced Mathematics V3, Differential Calculus, Tensor Calculus, Differential Geometry, Global Analysis

*Fundamentals of Advanced Mathematics V3* begins with the study of differential and analytic infinite-dimensional manifolds. Then, fibered bundles are considered, in particular than tangent bundle and the cotangent bundle. The tensor calculus on manifolds is the developed. Then comes the differential and integral calculus on manifolds (general Stokes formula, integral curves and manifolds). The analysis over Lie groups is then treated. The Haar measure is introduced, the convolution of functions and distributions as well, and the harmonic analysis over a Lie group. The theory of connections is treated (linear connections, principal connexions, and Cartan connections). Then the calculus of variations is developed both in the Lagrangian and the Hamiltonian formulations; last, the passage to optimal control is made. This volume is the prerequisite to the analytic and geometric study of nonlinear systems.

Differential and analytic manifolds, vector bundles, tensors, Lie derivatives, applications to algebraic topology, differential equations on manifolds, analysis over Lie groups, connections, calculus of variations, optimal control