__MA3151 __

__MATRICES AND CALCULUS__

** **

**COURSE OBJECTIVES:**

• To develop the use of matrix algebra
techniques that is needed by engineers for practical applications.

• To familiarize the students with
differential calculus.

• To familiarize the student with functions
of several variables. This is needed in many branches of engineering.

• To make the students understand various
techniques of integration.

• To acquaint the student with mathematical
tools needed in evaluating multiple integrals and their applications.

**UNIT - I **

**MATRICES**

Eigenvalues
and Eigenvectors of a real matrix – Characteristic equation – Properties of
Eigenvalues and Eigenvectors – Cayley - Hamilton theorem – Diagonalization of
matrices by orthogonal transformation – Reduction of a quadratic form to
canonical form by orthogonal transformation – Nature of quadratic forms –
Applications: Stretching of an elastic membrane.

**UNIT - II **

**DIFFERENTIAL CALCULUS**

Representation
of functions - Limit of a function - Continuity - Derivatives - Differentiation
rules (sum, product, quotient, chain rules) - Implicit differentiation -
Logarithmic differentiation - Applications: Maxima and Minima of functions of
one variable.

**UNIT - III **

**FUNCTIONS OF SEVERAL VARIABLES**

Partial
differentiation – Homogeneous functions and Euler’s theorem – Total derivative
– Change of variables – Jacobians – Partial differentiation of implicit
functions – Taylor’s series for functions of two variables – Applications :
Maxima and minima of functions of two variables and Lagrange’s method of
undetermined multipliers.

**UNIT - IV **

**INTEGRAL CALCULUS**

Definite
and Indefinite integrals - Substitution rule - Techniques of Integration:
Integration by parts, Trigonometric integrals, Trigonometric substitutions,
Integration of rational functions by partial fraction, Integration of irrational
functions - Improper integrals - Applications: Hydrostatic force and pressure,
moments and centres of mass.

**UNIT - V **

**MULTIPLE INTEGRALS**

Double
integrals – Change of order of integration – Double integrals in polar
coordinates – Area enclosed by plane curves – Triple integrals – Volume of
solids – Change of variables in double and triple integrals – Applications:
Moments and centres of mass, moment of inertia.

**TOTAL:
60 PERIODS**

** **

**COURSE OUTCOMES:**

At
the end of the course the students will be able to

• Use the matrix algebra methods for
solving practical problems.

• Apply differential calculus tools in
solving various application problems.

• Able to use differential calculus ideas
on several variable functions.

• Apply different methods of integration in
solving practical problems.

• Apply multiple integral ideas in solving
areas, volumes and other practical problems.

**TEXT BOOKS:**

1.
Kreyszig.E, "Advanced Engineering Mathematics", John Wiley and Sons,
10th Edition, New Delhi, 2016.

2.
Grewal.B.S., “Higher Engineering Mathematics”, Khanna Publishers, New Delhi,
44th Edition, 2018.

3.
James Stewart, "Calculus: Early Transcendentals", Cengage Learning,
8th Edition, New Delhi, 2015. [For Units II & IV - Sections 1.1, 2.2, 2.3,
2.5, 2.7 (Tangents problems only), 2.8, 3.1 to 3.6, 3.11, 4.1, 4.3, 5.1 (Area
problems only), 5.2, 5.3, 5.4 (excluding net change theorem), 5.5, 7.1 - 7.4
and 7.8].

**REFERENCES:**

1.
Anton. H, Bivens. I and Davis. S, " Calculus ", Wiley, 10th Edition,
2016

2.
Bali. N., Goyal. M. and Watkins. C., “Advanced Engineering Mathematics”,
Firewall Media (An imprint of Lakshmi Publications Pvt., Ltd.,), New Delhi, 7th
Edition, 2009.

3.
Jain. R.K. and Iyengar. S.R.K., “Advanced Engineering Mathematics”, Narosa
Publications, New Delhi, 5th Edition, 2016.

4.
Narayanan. S. and Manicavachagom Pillai. T. K., “Calculus" Volume I and
II, S. Viswanathan Publishers Pvt. Ltd., Chennai, 2009.

5.
Ramana. B.V., "Higher Engineering Mathematics", McGraw Hill Education
Pvt. Ltd, New Delhi, 2016.

6.
Srimantha Pal and Bhunia. S.C, "Engineering Mathematics” Oxford University
Press, 2015.

7.
Thomas. G. B., Hass. J, and Weir. M.D, "Thomas Calculus ", 14th
Edition, Pearson India, 2018.