School of Science - Department of Mathematics

0301313Real Analysis II

• Course Description :Series of real numbers: the definition and the algebraic properties. Convergence: the definition and the basic properties. Absolute and conditional convergence. Tests of absolute convergence (Ratio, nth root and comparison tests). Rearrangements of series. Abel test. Dirichlet test. Sequences of functions, the definition and examples. Pointwise convergence. Uniform convergence. Uniform convergence and continuity on [a,b]. Uniform convergence and integrability on [a,b]. Uniform convergence of sequences of derivatives. Dini's Theorem. Uniform convergence and interchange limit theorems. Series of functions: definition and basic properties. Pointwise and uniform convergence of series of functions. Weierstrass M-test. Uniformly convergent series of continuous functions. Uniformly convergent series of integrable functions. Interchange of summation and integration. The space C[a,b], the definition, metric and algebraic properties. The Weierstrass approximation theorem. Linear transformations on Rⁿ and their matrix representation (fast revision). Functions from Rⁿ to Rm (basic setup and examples). The derivative of vector valued functions of several variables, The definition. directional derivatives. Differentiability implies continuity. Partial derivatives. Matrix representation of the derivative. The gradiant and its properties. The chain rule. The mean value theorem. Higher order derivatives. Inverse and implicit mapping theorems (statements).
• Pre Request : 0301213
• Credit Hour :3
• Department : Mathematics
• Program :
• Course Level :Bachelor
• Course Outline :

  0301313 Syllabus.pdf