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MTH631 : Real Analysis II

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Course Info

Course Category

Mathematics

Course Level

Undergraduate

Credit Hours

3

Pre-requisites

N/A

Instructor

Dr. Salman Amin Malik
PhD
University of La Rochelle, France (2012)

Course Contents

Sequence of functions, pointwise convergence, Uniform convergence, Theorem 4.4.4 Necessary and Sufficient conditions for pointwise and uniform convergence, Uniform convergence implies pointwise convergence, Uniform Convergence some conclusions,  Integrability of the uniform limit, Uniform convergence of derivatives of a sequence of functions, Infinite Series of functions, Cauchy's Uniform Convergence Criterion, Dominated series of real numbers for a series of functions, Weierstrass's Test, Dirichlet's test for uniform convergence, Series of product of two functions, Continutiy of uniformly convergent series of functions, Interchange of Summation and Integration, Integration of sequences of integrable functions, Interchange of summation and integration, Differentiation of a sequence of functions, Interchange of summation and differentiation of infinite series, Properties of functions defined by power series, kth order derivative of a power seires, uniquness of the power series, definate integral of a function represented by power series, Arithematic operations with power series, product of two functions represented by power series, the reciprocal of pwer series and example, Abel's Theorem, Equicontinuous functions on a set, Uniformly covergent sequence of functions is equicontinuous, The Stone-Weierstrass Theorem, Fourier Series,  Periodic functions, Trignometric Polynomials, The space E and inner product Lemma, Orthonormal set of functions, complete set of functions, Fourier coefficients, Even and odd functions, Convergence of Fourier Series, Dirichlet Theorem,  Best Approximation Theorem, The Euler Gama functions Theorem, Convex function, The beta function, Functions of several variables, the structure of R^n, Inner product and Schwarz's inequality in R^n, Line segments in R^n, Neighbourhoods and open sets in R^n, Cauhcy's Convergence Criterion, Principle of Nested Sets Theorem, Heine-Borel Theorem in R^n Theorem, Connected sets and Regions in R^n, Polygonally connected set in R^n, Limit of real valued functions of n variables in R^n, Algebra of limits, infinte limits and limits at infinity of function with n variables, Vector-Valued Functions, Composite vector valued Functions and limits of vactor valued functions, Bounded functions, Intermedate value Theorem in R^n, Uniform continuity, Directional Derivative, Differentiable Functions of Several Variables, The differential in one variable, The differential in functions of several  variables, Maxima and minima for functions of n variables, Differentiability of vector valued functions, Higher derivatives of Composite functions, Higher Differentials, Vector valued functions using matrices, Linear transformations, A New Notation for the Differential, The Norm of a Matrix, Square matrix, Continuous Transformations Theorem , Differentiable Transformations Theorem, Local invertibility of linear trasformation , The implicit function theorem, Jacobians , Locally integrable functions , Absolute intgerability, Conditional convergence of improper integrals, The Dirichlet's Test Theorem, Riemann sum in R^n, Upper and Lower Integrals, Sets with zero content, Intgerals over more general subsets of R^n, Differentiable surfaces, Itegrated intgerals, Fubini's Theorem, Functions of bounded variations, Additive property of total variation.