| | This course is a continuation for Calculus I. The course begins with Techniques of integration and it is given as: integration by parts, trigonometric integrals, trigonometric substitutions, partial fractions, rationalizations, half-angle substitution , and improper integrals. Then some applications of definite integrals are given such as: areas between two curves, volumes by washers and cylindrical shells, arc length, and area of a surface. The concept of infinite series is introduced, then tests of convergence are given. Power series, Maclaurin and Taylor expansions are given. Then differentiation and integration of power series are given. Finally, polar coordinates, graphs in polar coordinates, and areas in polar coordinates are given. | first year,Second year | | |
Logic
(a) Axioms and theorems
(b) Negations
(c) Quantifiers
Algebra of sets
(a) Union
(b) Intersection
(c) Symmetric deference
(d) Deference
(e) Complement
Functions
(a) Domain and range
(b) Deferent classes of functions including 1-1 and onto
(c) Graph of a function
Relations on sets
(1) Equivalence relations and equivalence classes
(2) Partial order relation
(3) Total order relation
Cardinality of sets
(a) Finite sets
(b) Countable sets
(c) Uncountable sets
| Second year | | | Real numbers: order, absolute value, bounded subsets, completeness property, Archimedean property; supremum and infimum; sequences: limit, Cauchy sequence, recurrence sequence, increasing, decreasing sequence, lim sup, lim inf of a sequence; functions: limit, right, left limit, continuity at a point, continuity on an interval; uniform continuity (on an interval) relations between continuity and uniform continuity, differentiability: definition, right, left derivative, relation between differentiability and continuity, Rolle’s theorem, mean value theorem, applications on mean value theorem. | Second Year | | | 1. To present Basic properties of complex numbers.
2. To introduce Basic properties of Analytic function.
3. To present Elementary complex functions.
4. To evaluate complex integral.
5. To introduce Laurent series.
6. To present Residue theorem.
7. To use complex integration methods to evaluate Real Improper integrals.
8. To present Basic properties of mappings and conformal mappings.
| Fourth Year | | | Linear Equations
Differentiation
5.Partial Differentiation
6.Integration
7.Matrices | Fourth year | | | Several recent Internet lecture notes.
2. Introduction to Time Series and Forecasting, 2nd edition, by P. J. Brockwell and R. A. Davis, Springer, 2002.
| Third Year | | | 1. أن يطلع الطالب على أدوار الشعوب القديمة في ابتداع الأفكار الرياضية وتطويرها.
2. أن يتعرف الطالب على العباقرة من الأمم الذين أسهموا في تطوير الرياضيات.
3. أن يتعرف الطالب على دور علماء العرب والمسلمين في ابتداع الأفكار الرياضية وتطويرها. ودورهم في نقل وترجمة التراث العلمي القديم ومدى استفادة علماء الغرب منه.
4. أن يتتبع الطالب مراحل ظهور وتطور الأفكار الرياضية حتى الآن.
| Fourth Year | | | Point Estimation
2. Con¯dence Intervals
3. Testing of Statistical Hypotheses and Chi-Square Test.
4. Properties of Estimators.
5. Su±cient Statistic.
6. Completeness and uniqueness
7. Exponential Class of distributions
8. Multi-parameter Case Minimal Su±cient Statistic.
9. Bayesian Estimators
10. Roa-Cramer Bound
11. Asymptotic Distribution of Maximum Likelihood Estimators and Robust Estimation
12. Theory of Statistical Tests, UMP Tests, Likelihood Ratio Tests, The sequential Probability Ratio Tests,
and Minimax | Fourth year | | | Hilbert spaces: the geometry of Hilbert space, the Riesz representation theorem, orthonormal bases, isomorphic Hilbert spaces, operators on Hilbert space: basic properties and examples, adjoints, projections, invariant and reducing subspaces, positive operators and the polar decomposition, self-adjoint operators, normal operators, isometric and unitary operators, the spectrum and the numerical range of an operator, operator inequalities, compact operators, Banach spaces: basic properties and examples, convex sets, subspaces and quotient spaces, linear functionals and the dual spaces, the Hahn-Banach theorem, the uniform boundedness principle, the open mapping theorem, and the closed graph theorem. | MSc | | | 1. Monotone functions on [a, b].
2. Total variation of a function on [a, b].
3. Functions of bounded variation on [a, b].
4. Continous functions of bounded variation.
5. Riemann integral, the definition.
6. Existence of Riemann integral.
7. Basic properties of Rimann integral.
8. Classes of Riemann integrable functions (step functions, Continous functions, monotone functions).
9. Mean value theorems for Riemann integral.
10. Then fundamental theorem of calculus.
11. The Riemann-stieltjes integral, the definition.
12. Basic properties of R-S integral.
13. Integration by parts.
14. Continous functions and the R-S integral.
15. Monotone functions and the R-S integral.
16. Mean value theorems for R-S integral.
17. The fundamental theorem for R-S integral.
18. Linear transformations on Rn and their matrix representation (fast revision).
19. Functions from Rn to Rm (Vector fields)basic setup and examples).
20. The driative of a vector field, the definition.
21. Differentiability implies continuity.
22. Partial derivatives.
23. Matrix representation of the derivative.
24. The gradiant and its properties.
25. The chain rule.
26. The mean value theorem.
27. Higher order derivatives (the second).
28. The inverse function theorem.
29. The function mapping theorem (the statement only).
| Third year | | | Lebesgue measure: outer measure, measurable sets and functions, Egoroff's theorem, Lusin's theorem, convergence in measure, the Lebesgue integral: the integral of a bounded function over a set of finite measure, the integral of a nonnegative function, the general Lebesgue integral, Riemann and Lebesgue integrals, differentiation: differentiation of monotone functions, functions of bounded variation, differentiation of an integral, absolute continuity, Lp classes: the Holder and Minkowski inequalities, completeness of Lp classes, the duals of Lp classes, Banach spaces: linear operators, the Hahn-Banach theorem and other basic results, Hilbert spaces. | MSc | | | Analytic functions: power series, Laurent series, analytic functions as mappings, Mobius transformations, linear fractional transformations, conformal mappings, cross ratio, complex integration: zeros of analytic functions, Cauchy's theorem and formula, the argument principle, the open mapping theorem, the maximum modulus principle, Schwartz lemma, singularities: classification of singularities, residues, residue theorem, evaluation of real definite and improper integrals, normal families: Riemann mapping theorem, Schwartz reflection principle, Schwartz-Christofell formulas, harmonic functions: Dirichlet problem, Poisson’s formula, mean value property. | MSc |
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