Course
No.

Course
Name

LTPCredits

Year of
Introduction


MA101

CALCULUS

3104

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 testscomparison, ratio, root
tests (without proof). Absolute convergence. Maclaurins seriesTaylor 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 I12.112.6,
13.6,13.7)

15
%


Introduction to vector valued
functions  parametric curves in 3space. 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 gradientstangent 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 Isec. 14.1, 14.2, 14.3, 14.5, 14.6, 14.7)

20
%


Double integrals  Evaluation of
double integrals  Double integrals in nonrectangular 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 pathconservative 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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