Second Stage


Ordinary Differential Equations

Units : 6
Lecturer : Asst. Prof. Dr. Shayma Adil Murad

Description :

Course overview:  

This course includes the study of first order differential equations, higher order linear differential equations, Laplace transforms, boundary value and initial value problems, qualitative analysis of solutions, and applications of differential equations. Very detailed information is often available. After successful completion of this course, students will be able to develop key concepts in Special functions and demonstrate the usefulness of ordinary differential equations for modeling physical and other phenomena. Complementary mathematical approaches for their solution will be presented, including analytical methods, graphical analysis and numerical techniques. The basic content of the course includes (First order equations,  Mathematical models, Linear equations of second order,  The Laplace transform, Linear systems of arbitrary order and matrices, Nonlinear systems and phase plane analysis).

Course objective:

  The student will be able to:

1-Classify ordinary differential equations according  to order and linearity, as well as distinguish between initial value problems and boundary value problems.

2- Find general solutions to first-order, second-order, and higher-order homogeneous and non-homogeneous differential equations by manual and technology-based methods.

3- Identify and apply initial and boundary values to find particular solutions to differential equations by manual and technology-based methods, and analyze and interpret the results.

4- Select and apply appropriate methods to solve differential equations; these methods will include, but are not limited to, undetermined coefficients, variation of parameters, the operator D, eigenvalues and eigenvectors, Laplace and inverse Laplace transforms.

5- Select and apply Existence and uniqueness theorems of differential equation.  

6- Power Series Method, Ordinary points and singular points, Roots if indicial equation (equal, unequal and non integer), Solutions a round ordinary points, Frobenius series solution.

7- Phase Portraits of Linear Systems.

To achieve this goal, students will completion of homework assignments, and exams.

Advanced Calculus

Description :

Course overview:

This course studies real sequences and infinite series without assumption of previous knowledge. The notions of upper and lower limits are introduced and used sparingly as a simplifying device; with their aid, the theory is given in almost complete form. The usual tests are given: in particular, the root test. With its aid, the treatment of power series is greatly simplified. Uniform convergence is presented with great care and applied to power series. Final sections point out the parallel with improper integrals; in particular, power series are shown to correspond to the Laplace transform. Also we opens with a review of determinants, and linear equations, and then develops matrix algebra, including Gaussian elimination, and n-dimensional geometry, with stress on linear We discuss partial derivatives and develops them with the aid of vectors (gradient, for example) and matrices; partial derivatives are applied to geometry and to maximum-minimum problems. The course introduces divergence and curl and the basic identities; orthogonal coordinates are treated concisely. One of the chapters is on integration, reviews definite and indefinite integrals, using numerical methods to show how the latter can be constructed; multiple integrals are treated carefully, with emphasis on the rule for change of variables; Leibnitz's Rule for differentiating under the integral sign is proved. Improper integrals are also covered; the discussion of these is completed , where they are related to infinite series. The concept of line and surface integrals is introduced. Although the notions are first presented without vectors, it very soon becomes clear how natural the vector approach is for this subject. Line integrals are used to provide an exceptionally complete treatment of transformation of variables in a double integral. Many physical applications. including potential theory, are given.

Course objective:

The goals of the Advanced Calculus course are to give an introduction to the subject of Real Analysis, and to continue the rigorous development of the theorems of Calculus to the areas of derivatives, integrals, and series of functions. Students should gain a more in depth understanding of these areas of Calculus and should increase their proficiency in proving various results. I hope to convey the usefulness of series of functions, the importance of the uniform norm, and the limitations of the Riemann integral. A focus is on learning to prove mathematical statements rigorously.

The Topics:

1.Infinite Sequences and Series.
1.1: Sequences. definition infinite sequence, convergence and divergence, definition diverges to infinity, calculating limits of sequences, using hopital's rule, bounded non decreasing sequences, decreasing sequence, bounded, upper bound, least upper bound
1.2: Infinite Series, Infinite Series, Geometric Series, Divergent Series, The nth-Term Test for Divergence, Combining Series, The Integral Test, The p –Series, The Comparison Test, The Limit Comparison Test, The Ratio and Root Tests.
1.3: Alternating Series. Absolute and Conditional Convergence. The Absolute Convergence Test.
2.4: Power Series. Power Series and Convergence. The Radius of Convergence of a Power Series. How to Test a Power Series for Convergence
2.5: Taylor and Maclaurin Series.

2. Polar Coordinates
2.1 : Definition of Polar Coordinates.
2.2 : Polar Equations and Graphs.
2.3 : Relating Polar and Cartesian Coordinates.
2.4 : Graphing in Polar Coordinates. Symmetry. Symmetry Tests for Polar Graphs. Finding Points Where Polar Graphs Intersect.
2.5 : Areas in Polar Coordinates.
2.6 : Length of a Polar Curve.

3. Vectors and the Geometry of Space.
3.1: Vectors. Vector Addition and Multiplication of a Vector by a Scalar. Properties of Vector Operations. Unit Vectors.
3.2: The Dot Product. Angle Between Vectors. Perpendicular (Orthogonal) Vectors. Properties of the Dot Product
3.3: The Cross Product. The Cross Product of Two Vectors in Space. Parallel Vectors. Properties of the Cross Product.
3.4: Lines and Planes in Space. Vector Equation for a Line. Parametric Equations for a Line. The Distance from a Point to a Line in Space. An Equation for a Plane in Space.Equation for a Plane.Lines of Intersection. The Distance from a Point to a Plane. Angles Between Planes

4. Partial Derivatives.
4.1: Functions of Several Variables. Domains and Ranges.
4.2: Limits and Continuity in Higher Dimensions. Limits. Limit of a Function of Two Variables. Properties of Limits of Functions of Two Variables. Continuity. Continuous Function of Two Variables.
4.3: Partial Derivatives. Partial Derivatives of a Function of Two Variables. Functions of More Than Two Variables. Partial Derivatives and Continuity. The Chain Rule.
4.4: Directional Derivatives and Gradient Vectors. Directional Derivatives in the Plane. Gradient Vector.
4.5: Tangent Planes and Differentials.
4.6: Extreme Values and Saddle Points. Lagrange Multipliers.

5. Multiple Integrals.
5.1: Double Integrals. Double Integrals over Rectangles. Double Integrals as Volumes. Fubini’s Theorem for Calculating Double Integrals. Properties of Double Integrals. Areas of Bounded Regions in the Plane. Double Integrals in Polar Form.
5.2: Integrals in Polar Coordinates. Area in Polar Coordinates. Changing Cartesian Integrals into Polar Integrals.
5.3:Triple Integrals in Rectangular Coordinates. Triple Integrals. Volume of a Region in Space. Properties of Triple Integrals.
5.4: Triple Integrals in Cylindrical and Spherical Coordinates. Integration in Cylindrical Coordinates. Equations Relating Rectangular (x,y,z) and Cylindrical Coordinates. Spherical Coordinates and Integration. Equations Relating Spherical Coordinates to Cartesian and Cylindrical Coordinates.
5.5: Substitutions in Multiple Integrals.

6. Integration in Vector Fields.
6.1: Line Integrals, Vector Fields, Gradient Fields
6.2: Green's Theorem in the Plane. Using Green’s Theorem to Evaluate Line Integrals.
6.3: Surface Area and Surface Integrals. Surface Area. Surface Integrals. Parametrized Surfaces.
6.4:Stokes' Theorem.
6.5: The Divergence Theorem.

Course Reading List and References :
- Advanced Calculus 3rd Edition by Taylor Angus & Wiley. Fayez, 1955.
- Calculus, Haward Anton, Eighth Edition, 2005.
- Schaum's Outline Theory and Problems in Advanced Calculus Robert Wrede, Murray R. Spiegel. 2002 by The McGraw -Hill Companies.
- Thomas’ Calculus, 2005 Pearson Education, Inc., publishing as Pearson Addison – Wesley.
-Calculus, Haward Anton, Eighth Edition, 2005

Axiomatic Geometry

Units : 6
Lecturer : Nechirvan Badal Ibrahim

Description :

CHAPTER ONE: Essential of Geometry

1.1. Undefined Terms

1.2. Midpoint and Bisectors

1.3. Line Segments, Rays and Angles

1.4. Triangles

CHAPTER TWO: Axiom Systems

2.1. Axioms of connection

2.2. Axioms of Order

2.3. Consequences of the axioms of connection and order

2.4. Axiom of Parallels (Euclid’s axiom)

3.5. Axioms of Congruence

CHAPTER THREE: Congruence of Line Segments, Angles and Triangles:

3.1. Postulates of Lines, Line Segments and Angles

3.2. Proving Theorems about Angles

3.3. Congruent Polygons and Corresponding Parts

3.4. Proving Triangles Congruent Using Side, Angle, Side

3.5. Proving Triangles Congruent Using Angle, Side, Angle

3.6. Proving Triangles Congruent Using Side, Side, Side

3.7. Pythagoreans Theorem

CHAPTER FOUR: Congruence  on Based Triangles

4.1 Altitude of a Triangle

4.2 Median of  a Triangle

4.3 Using Congruent Triangle to Prove Line Segments Congruent

4.4 Using Congruent Triangle to Prove Angles Congruent

4.5 Using Two Pairs of Congruent Triangles

4.6 Ratio and Proportion

4.7 Similar Triangles

CHAPTER FIVE:   Euclidean Geometry, Area and Circles

5.1. Arcs and Angles

5.2 Arcs and Chords

5.3. Inscribed Angles and Their Measures

5.4. Tangents and secants

5.5. Measures of tangent Segments, Chords and Secant segments

5.6. Circle in the Coordinate Plane

CHAPTER SIX: Constructions, Hyperbolic Geometry

CHAPTER SIVEEN: Spherical Geometry

Linear Algebra
Computer Application - Matlab

Units : 6
Lecturer : Huda Younus Najm

Description :

MATLAB (matrix  laboratory)   is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation. Typical uses include:  Math and computation , Algorithm development, Modeling, simulation, and prototyping, Data analysis, exploration, and visualization, Scientific and engineering graphics, Application development, including Graphical User Interface building .

MATLAB is a programming language developed by MathWorks. It started out as a matrix programming language where linear algebra programming was simple. It can be run both under interactive sessions and as a batch job. 

This course  gives you aggressively a gentle introduction of MATLAB programming language.  It  is  designed  to  give  students  fluency  in  MATLAB  programming language. Problem-based MATLAB examples have been given in simple and easy way to make your learning fast and effective. This course has been prepared for the beginners to help them understand basic to advanced functionality of MATLAB. It allows matrix manipulations; plotting of functions and data; implementation of algorithms; creation of user interfaces; interfacing with programs written in other languages,  including  C,  C++,  Java,  and  FORTRAN;  analyze  data;  develop algorithms; and create models and applications.  It  has  numerous  built-in  commands  and  math  functions  that  help  you  in mathematical calculations, generating plots, and performing numerical methods.