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.