This course begins the formal study of limits, differentiability of algebraic and transcendental functions, applications of differentiation, and an introduction to integration. Study of integration by standard forms and special methods, indeterminate forms and improper integrals, transcendental functions, analytic geometry, and polar coordinates. Study of systems of linear equations, vector spaces, matrices, determinants, linear dependence, bases, dimension, linear transformations, eigenvectors, applications. Study of ordinary differential equations. Elements of modern probability theory and its application in statistical inference.
Axiomatic structure is emphasized.
An introduction to several geometries developed after 1800. Study of non-Euclidean geometries, particularly hyperbolic geometry, projective geometry, geometric topology and geometry of inversion. Axiomatic structure and invariant properties are emphasized. An axiomatic study of important algebraic systems, including groups, rings, fields, and vector spaces. Introduction to numerical algorithms essential in scientific computation.