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MATH 350/351 (Honors Analysis I and II)

DESCRIPTION	MATH 350-351 is a year long course giving a rigorous treatment of calculus in one and several variables. THIS COURSE IS NO LONGER OFFERED: it has been replaced by the less advanced sequence MATH 340/341. Very advanced entering students who have finished the content of MATH 340/341 should discuss appropriate coursework with a math department advisor or faculty member.
PREREQUISITES	Permission of the department.
TOPICS	Real and complex number systems Completeness (least-upper-bound properly or Dedekind cuts) Real numbers Countability Complex numbers Real n-dimensional space Norms, inner products, metrices Topology of Real n-dimensional spaces Open sets, closed sets, boundaries, closures Accumulation points, sequences, and subsequences Compactness, Bolzano-Weierstrass Theorem, Heine-Borel Theorem Connectedness Numerical Sequences and Series Convergence Cauchy sequences Lim sup and lim inf. Convergence tests Continuity Definition, relation to topological concepts Intermediate-value theorem Uniform continuity Maxima and minima Differentiation of functions of one and several variables The derivative as a linear transformation Continuity of the derivative Chain rule Mean-value theorem L'Hôpital's Rule Taylor's theorem Necessary conditions for maxima and minima Introduction to calculus of variations (optional) Integration Riemann integral (Riemann-Stieltjes integral is optional) Multiple integrals, change of variables Introduction to Lebesque integration (optional) Vector Analysis Vector algebra Curves and surfaces Line integrals Surface integrals The Divergence theorem and Stokes' theorem Sequences and series of functions Pointwise convergence and uniform convergence Weierstrass M-test Integration and differentiation of series The space of continuous functions Arzela-Ascoli theorem Weierstrass approximation theorem (Stone-Weierstrass theorem is optional) Other modes of convergence Fourier series Implicit functions and related topics Contraction - mapping principle Local inverse - and Implicit Function Theorem Existence theorem for ordinary differential equations Constrained extrema, Lagrange multipliers Global Implicit function theorems for variational problems Analytical degree theory, solvability of nonlinear vectorial equations Global Implicit Function theorems (optional)
TEXT	Advanced Calculus, 3rd. ed. by R.C. Buck or Elementary Classical Analysis, 2nd. ed., J.E. Marsden & M. J. Hoffman or Principles of Mathematical Analysis, 3rd. ed., W. Rudin