Mathematics for theoretical physics (1209.5665v2)

Abstract: This book intends to give the main definitions and theorems in mathematics which could be useful for workers in theoretical physics. It gives an extensive and precise coverage of the subjects which are addressed, in a consistent and intelligible manner.The first part addresses the Foundations (mathematical logic, set theory, categories), the second Algebra (algebraic strucutes, groups, vector spaces tensors, matrices, Clifford algebra). The third Analysis (general topology, measure theory, Banach Spaces, Spectral theory). The fourth Differential Geometry (derivatives, manifolds, tensorial bundle, pseudo-riemannian manifolds, symplectic manifolds). The fifth Lie Algebras, Lie Groups.and representation theory. The sixth Fiber bundles and jets. The last one Functional Analysis (differential operators, distributions, ODE, PDE, variational calculus). Several signficant new results are presented (distributions over vector bundles, functional derivative, spin bundle and manifolds with boundary).