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 Order Differential Equations
Second Order Linear Equations
Higher Order Linear Equations
Series Solutions of Second Order Linear Equations
The Laplace Transform
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.
1- Functions of several variables
2- The three linear operators:
3- Different types of integrals:
4- Six main theorems
5- Calculus of variation: Functional of one variable
Vector Integral Calculus, Integral Theorems
Fourier Series, Integrals, and Transforms
Partial Differential Equations
1- Classification of PDEs.
2- Some physical models (Heat, Wave and Laplece equations).
3- Second order Linear PDEs. With constant coefficients.
4- Separation of variables.
5- Orthogonal functions.
6- Sturm-Liouville BVP.
7- Fourier series and Fourier transform.
8- Boundary value problems involving rectangular and circular regions.
9- Special functions (Bessel and Legendre).
10- Boundary value problems involing cylindrical and spherical regions.
This course begins with the idea of solving a system of linear equations. The concept of matrices is introduced and the algebra of matrices is discussed. The notion of the determinant of a matrix is also introduced. Vector spaces, subspaces, linear independence, spanning sets and bases are introduced. Then the concept of inner product spaces is discussed. The eigenvalues and eigenvectors and linear transformations are also introduced. Basic facts and properties of matrices are used through this course.
1. The completeness property of R.
2. The Archimedean principle in R.
3. Limit of a sequence.
4. Convergent sequences.
5. Monotone and bounded sequences.
6. Cauchy sequences.
7. Subsequences and limit points. Liminf, and limsup.
8. Bolzano-Weierstrass Theorem.
9. Open sets, closed sets, bounded sets and compact sets in R.
10. Limits of real valued functions.
11. Definition of limits by neighborhoods.
12. Definition of limits by sequences.
13. Continuous functions on R.
14. Sequence definition and neighborhood definition of continuity.
15. Boundedness of continous functions on compact intervals.
16. The extreme value theorem.
17. The intermediate value theorem.
18. Uniformly continuous functions.
19. The sequential criterion for uniform continuity.
20. The derivative of functions.
21. Roles Theorem.
22. Mean value theorem.
23. Generalized Mean value theorem.
24. Taylor Theorem with remainder.
25. L’Hospital,s rule.
Upon completion of this course, the student should be able to
1. Write mathematical proofs and reason abstractly in exploring properties of groups and rings.
2. Define, construct examples of, and explore properties of groups, including symmetry groups, permutation groups and cyclic groups.
3. Determine subgroups and factor groups of finite groups.
4. Determine, use and apply homomorphism's between groups.
5. Understand definitions, examples, and theorems pertaining to groups.
. 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,