Calculus and Analytical Geometry

Course Title: Calculus and Analytical Geometry
Course no: MTH-104                                                                                                                        Full Marks: 80+20
Credit hours: 3                                                                                                                                 Pass Mark: 32+8
Nature of Course: Theory

Course Synopsis: Preliminaries revision of differentiation and integration; Techniques of integration infinite series; Vectors and analytical geometry in space (differential geometry). Vector valued functions. Multivariable functions and partial derivatives. Multiple integrals and integration in vector fields. Partial derivatives; Equations of First Partial Derivatives.

Goal: This course aims at providing students with some advanced topics in undergraduate calculus and fundamental concepts of partial differentiation and P.D.E of second order. It is assured that a student who has done Certificate Level papers in mathematics will be able to study this course.

Course Contents: 

Unit 1. Topics in Differential Calculus and Integral Calculus                              8 Hrs.

• Functions and Graphs      
• Extreme values of functions; graphing of derivatives     
• Mean value integers                     
• Definite integers, Properties and application, Mean value theory for definite integers  
• Fundamental theory of Integral Calculus and application, Improper integrals

Unit  2. Infinite Series                                                                                               5 Hrs.

• Infinite sequence and sequence of convergence and divergence
• Integral test, comparison test, ratio and root test
• Absolute and conditional convergence
• Power series, Taylor and Maclaurin series, convergence of Taylor series

Unit 3. Conic Section                                                                                                3 Hrs.

• Classifying conic sections by eccentricity           
• Plane curves, parametric and polar equations, integration in polar coordinates

Unit 4. Vectors and Vectors Valued Functions                                                      6 Hrs.

• Vectors in the space         
• Lines and planes in space 
• Cylinders and Quadric surfaces   
• Cylindrical and Spherical Coordinates    
• Vector valued functions and space curves          
• Unit tangent vector, curvature and torsion and TNB system      

Unit 5. Multiple Integrals                                                                                         5 Hrs.

• Double integrals in rectangular polar coordinates
• Finding areas, moments and centre of mass
• Triple integrals in rectangular coordinates and application          
• Substitutes in multiple integrals   

Unit 6. Multivariate Calculus                                                                                   9 Hrs.

• Functions, limits and continuity of two or more variables          
• Partial derivatives
• Differentiability, Differentials, Total Differential Coefficients  
• Directional derivatives and gradient vectors       
• Extreme values     
• Lagrange Multiplies         

Unit 7. Partial Differential Equations                                                                     9 Hrs.

• Review of Ordinary Differential Equations        
• Analysis of P.D.E of 1st and 2nd order     
• Linear equations of the 1st order and the general solutions                     
• P.D.E of 2nd order, its derivation and basic concepts      
• Solution of general P.D.E with constant coefficients, complimentary solution and integral solution    
• Wave equations and heat equations and their solutions (Chapter II, Section 11.1, 11.2, 11.4, 11.5). Erwin and Kreyszig. 8th edition, John-Wiley Publications.

Text Books

Thomas and Fenns: Calculus and Analytical Geometry, 9th Edition, 2004. (Thomas, Jr. G. B., and Finney, Ross L. Publisher: Pearson Education Pvt. Ltd.
Kreyszig, Erwin, Advanced Engineering Mathematics, John- Wiley & Sons (1991). 5th Edition.

References

E.W. Swokowski, Calculus with Analytical Geometry, Second Alter Edition.
Sneddan Ian- Elements of Partial Differential Equations.

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