Credit: (4-2)5 

Content:  Review of multidimensional Calculus, Derivatives of multivariable functions, continuity of multivariable functions. Fundamental Lemma for differentiability, Chain rule and Taylor’s Theorem for multivariable functions. Jacobian. Inverse and Implicit function Theorems. Topology of R2 and R3. Riemann - Stieltjes Integral, integrability, integrability of continuous functions, sequences of integrable functions. Bounded convergence and Riesz Representation Theorems. Theorems of Integral Calculus: Integration in Cartesian spaces. Improper and infinite integrals. Series of functions. 

Goals: This is a course designed for science students to complete and complement the sequences of math courses, Math 151, Math 152, Math 260. The aim of this sequence is to equip the student with the necessary mathematical background that he would be needing through his undergraduate studies. 
 

Course Outline: 

        (Weeks 2) Review of Multidimensional Calculus, Limits of multivariable functions, continuity of multivariable functions. Derivatives of multivariable
        functions: The total derivative. Directional derivative. Fundamental Lemma for differentiability. Chain rule and Taylor’s Theorem for multivariable functions.
        Jacobian. Inverse and Implicit function theorems. 

        (Week 1) Topology of R2 and R3. Open, connected and compact sets. Sequences of functions: Convergence of a sequence of functions, uniform
        convergence. Cauchy criterion for uniform convergence, uniform continuity. 

        (Week 2) Riemann-Stieltjes Integral: Definition, Cauchy criterion for integrability, integration by parts, integrability of continuous functions, sequences of
        integrable functions. Bounded convergence and Riesz Representation Theorems. 

        (Week 2) Theorems of Integral Calculus: First and second Mean Value Theorems, fundamental and change of variable theorem. Integrals depending on a
        parameter. Leibniz’s formula, interchange the order of integration. 

        (Week 2) Integration in Cartesian spaces: Content of a set, integrability Theorems, Mean Value Theorem, reduction to iterated integrals, transformation of
        integrals and the Jacobian theorem. 

        (Week 2) Improper and infinite integrals: Improper integrals of unbounded functions, Cauchy principal value, definition of infinite integrals, Cauchy
        criterion, tests for convergence absolute convergence, uniform convergence. Infinite integrals of sequences, Dominated Convergence Theorem. 

        (Week 2) Series of functions: Absolute and uniform convergence, continuity of the limit, term-by-term differentiation and integration of series. Tests for
        uniform convergence. 

Suggested textbooks: R.G.Bartle; The elements of real Analysis, Wiley, 1964 ; A.I.Khuri, Advanced Calculus with Applications in Statistics.
 

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