This book, written by the eminent mathematician Henri Cartan, contains a summary of his lectures on the course "Mathematics II" at the University of Paris. They include differential calculus, the theory of differential equations in Banach spaces, the theory of differential forms and the theory of multidimensional integrals built on its basis, as well as initial information on the calculus of variations in differential geometry. The presentation is elementary, although it is conducted at the modern scientific level.