Course Description: This course covers sequences, series, power series, and ; differential and integrals calculus of function of several variables and their applications, and multiple integral, ordinary differential equation and Laplace transforms
Course objective: to equip students with mathematical tools of developing mathematical models of physical problems
Course Contents
Chapter 1: Sequence ,Series and power series
Chapter I: part I: Sequence and Series
1.1. Definition and Types of sequence
1.2. Convergence properties of sequences
1.3. Subsequence and limit points
1.4. Definition of infinite series
1.5. Convergence and divergence properties of serieses
1.6. Non negative term series
1.7. Test of convergence (integral, comparison, ratio and root tests)
1.8. Alternating series and alternating series test
1.9. Absolute and conditional convergence
1.10. Generalized convergence tests
Chapter 1: part II:Power Series
1.2.1 Definition of power series at any and
1.2.2 Convergence and divergence, radius and interval of convergence
1.2.3 Algebraic operations on convergent power series
1.2.4 Differentiation and integration of power series
1.2.5 Taylor series; Taylor polynomial and application
Chapter 2: Differential Calculus of Function of Several Variables
2.1. Notations, examples, level curves and graphs
2.2. Limit and continuity
2.3. Partial derivatives; tangent lines, higher order partial derivatives.
2.4. Directional derivatives and gradients
2.5. Total differential and tangent planes
2.6. Applications: tangent plane approximation of values of a function
2.7. The chain rule, implicit differentiation
2.8. Relative extrema of functions of two variables
2.9. Largest and smallest values of a function on a given set
2.10. Extreme values under constraint conditions: Lagrange’s multiplier
Chapter 3: Multiple Integrals
3.1. Double integrals and their evaluation by iterated integrals
3.2. Double integrals in polar coordinates
3.3. Application: Area, center of mass of plane region, surface area
3.4. Triple integrals in cylindrical and spherical coordinates
3.5. Application: Volume, center of mass of solid region
Chapter 4: ordinary differential equations
part I: ordinary differential equations of first order
4.1.1 Definition of ODE and examples
4.1.2 Method of separable of variables
4.1.3 Homogenous equations
4.1.4 Exact non exact equation s and integrating factor
4.1.5 Linear equations
Part II: Ordinary linear differential equation of second order
4.2.1 Definition of SOODE
4.2.2 Constant coefficient SOODE
4.2.3 Homogenous and non-homogenous SOODE
4.2.4 Method of solving homogenous and non-homogenous SOODE
Chapter 5: Laplace T transformation
5.1 Definition of Laplace transformation and some examples
5.2 Existences’ of the Laplace transformation
5.3 Laplace transformation of derivatives and integrals
5.4 Solving ODE by using Laplace transform
- Teacher: Nure Amin