Calculus Syllabus S1 2015-16

Course No.
Course Name
L-T-P-Credits
Year of Introduction
MA101
CALCULUS
3-1-0-4
2015
Course Objectives
In this course the students are introduced to some basic tools in Mathematics which are useful in modelling and analysing physical phenomena involving continuous changes of variables or parameters. The differential and integral calculus of functions of one or more variables and of vector functions taught in this course have applications across all branches of engineering. This course will also provide basic training in plotting and visualising graphs of functions and intuitively understanding their properties using appropriate software packages.
Syllabus
Single Variable Calculus and Infinite series, Three dimensional space and functions of more than one variable, Partial derivatives and its applications, Calculus of vector valued functions, Multiple Integrals, Vector Integration.
Expected outcome
At the end of the course the student will be able to model physical phenomena involving continuous changes of variables and parameters and will also have acquired basic training in visualising graphs and surfaces using software or otherwise. 
Text Book:
      Anton, Bivens and Davis, Calculus, John Wiley and Sons.
      Pal, S. and Bhunia, S. C., Engineering Mathematics, Oxford University Press, 2015.
      Thomas Jr., G. B., Weir, M. D. and Hass, J. R., Thomas’ Calculus, Pearson.
References:
      Bali, N. P. and Goyal, M., Engineering Mathematics, Lakshmy Publications. 
      Grewal, B. S., Higher Engineering Mathematics, Khanna Publishers, New Delhi.
      Jordan, D. W. and Smith, P., Mathematical Techniques, Oxford University Press.
      Kreyszig, E., Advanced Engineering Mathematics, Wiley India edition.
      Sengar and Singh, Advanced Calculus, Cengage Learning.
      Srivastava, A. C. and Srivasthava, P. K., Engineering Mathematics Vol. 1, PHI Learning Pvt.
Ltd.
Course Plan
Module
Contents
Hours
Sem. Exam Marks 
I
Single Variable Calculus and Infinite  series (Book I –sec.6.1, 6.4, 
6.8, 9.3, 9.5, 9.6,  9.8)
15 %
Introduction: Hyperbolic functions and inversesderivatives and integrals.  
3
Basic ideas of infinite series and convergence.  Convergence tests-comparison, ratio, root tests (without proof). Absolute convergence. Maclaurins series-Taylor series - radius of convergence
3





(For practice and submission as assignment only:
Sketching, plotting and interpretation of exponential, logarithmic and hyperbolic functions using suitable software. Demonstration of convergence of series by software packages)
3

II
Three dimensional space and functions of more than one variable        
(Book I – 11.7, 11.8, 13.1, 13.2)
15 %
Three dimensional space; Quadric surfaces, Rectangular, Cylindrical and spherical coordinates, Relation between coordinate systems.
Equation of surfaces in cylindrical and spherical coordinate systems.  
4
Functions of two or more variables – graphs of functions of two variables- level curves and surfaces –Limits and continuity.
2
(For practice and submission as assignment only:
Tracing of surfaces- graphing quadric surfaces- graphing functions of two variables using software packages)
2
FIRST INTERNAL EXAM
III
Partial derivatives and its applications(Book I –sec. 13.3 to 13.5 and
13.8)
15 %
Partial derivatives - Partial derivatives of functions of more than two variables - higher order partial derivatives - differentiability, differentials and local linearity.
4
The chain rule - Maxima and Minima of functions of two variables - extreme value theorem (without proof)relative extrema.
5
IV
Calculus of vector valued functions(Book I-12.1-12.6, 13.6,13.7)
15 %
Introduction to vector valued functions - parametric curves in 3-space. Limits and continuity - derivatives - tangent lines - derivative of dot and cross productdefinite integrals of vector valued functions.
2
Change of parameter - arc length - unit tangent - normal - velocity - acceleration and speed - Normal and tangential components of acceleration.
2
Directional derivatives and gradients-tangent planes and normal vectors.
2
(For practice and submission as assignment only:
Graphing parametric curves  and surfaces using software packages)
4
SECOND INTERNAL EXAM
V
Multiple  integrals (Book I-sec. 14.1, 14.2, 14.3, 14.5, 14.6, 14.7)
20 %
Double integrals - Evaluation of double integrals - Double integrals in non-rectangular coordinates - reversing the order of integration.
3
Area calculated as double integral - Double integrals in polar coordinates. 
2
Triple integrals - volume calculated as a triple integral -


triple integrals in cylindrical and spherical coordinates.
2

Converting triple integrals from rectangular to cylindrical coordinates - converting triple integrals from rectangular to spherical coordinates  - change of variables in multiple integrals - Jacobians (applications of results only)
3
VI
Vector integration(Book I sec. 15.1, 15.2, 15.3, 15.4, 15.6, 15.7, 15.8)
20 %
Vector and scalar fields- Gradient fields – conservative fields and potential functions – divergence and curl - the
∇      ∇2  operator - the Laplacian
3
Line integrals - work as a line integral- independence of path-conservative vector field.
3
Green’s  Theorem (without proof- only for simply connected region in plane),  surface integrals – Divergence Theorem (without proof) , Stokes’ Theorem
(without proof)


(For practice and submission as assignment only:
graphical representation of vector fields using software packages)
Green’s  Theorem (without proof- only for simply connected region in plane),  surface integrals – flux integral - Divergence Theorem (without proof) , Stokes’
Theorem (without proof)
(For practice and submission as assignment only:
graphical representation of vector fields using software packages )
4

END SEMESTER EXAM
Open source software packages such as gnuplot, maxima, scilab, geogebra or R may be used as appropriate for practice and assignment problems.
TUITORIALS: Tutorials can be ideally conducted by dividing each class in to two groups. Prepare necessary materials from each module that are to be taught using computer. Use it uniformly to every class.  

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