Sub Code:MA3151
Sub Name: 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.
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
- Kreyszig.E, "Advanced Engineering Mathematics", John Wiley and Sons, 10th Edition, New Delhi, 2016.
- Grewal.B.S., “Higher Engineering Mathematics”, Khanna Publishers, New Delhi, 44th Edition, 2018.
- 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
- Anton. H, Bivens. I and Davis. S, " Calculus ", Wiley, 10th Edition, 2016
- Bali. N., Goyal. M. and Watkins. C., “Advanced Engineering Mathematics”, Firewall Media (An imprint of Lakshmi Publications Pvt., Ltd.,), New Delhi, 7th Edition, 2009.
- Jain. R.K. and Iyengar. S.R.K., “Advanced Engineering Mathematics”, Narosa Publications, New Delhi, 5th Edition, 2016.
- Narayanan. S. and Manicavachagom Pillai. T. K., “Calculus" Volume I and II, S. Viswanathan Publishers Pvt. Ltd., Chennai, 2009.
- Ramana. B.V., "Higher Engineering Mathematics", McGraw Hill Education Pvt. Ltd, New Delhi, 2016.
- Srimantha Pal and Bhunia. S.C, "Engineering Mathematics” Oxford University Press, 2015.
- Thomas. G. B., Hass. J, and Weir. M.D, "Thomas Calculus ", 14th Edition, Pearson India, 2018.
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