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ME010303 – MULTIVARIATE CALCULUS AND INTEGRAL
TRANSFORMS
5 Hours/Week ( Total Hours : 90) 4 Credits
Text 1: Tom Apostol, Mathematical Analysis, Second edition, Narosa Publishing
House.
Text 2: Walter Rudin, Principles of Mathematical Analysis, Third edition –
International Student Edition.
Module 1: The Weirstrass theorem, other forms of Fourier series, the Fourier integral
theorem, the exponential form of the Fourier integral theorem, integral
transforms and convolutions, the convolution theorem for Fourier transforms.
(Chapter 11 Sections 11.15 to 11.21 of Text 1) (20 hours.)
Module 2: Multivariable Differential Calculus
The directional derivative, directional derivatives and continuity, the total
derivative, the total derivative expressed in terms of partial derivatives, An
application of complex valued functions, the matrix of a linear function, the
Jacobian matrix, the matrix form of the chain rule. Implicit functions and
extremum problems, the mean value theorem for differentiable functions,
(Chapter 12 Sections. 12.1 to 12.11 of Text 1) (22 hours.)
Module 3: A sufficient condition for differentiability, a sufficient condition for
equality of mixed partial derivatives, functions with nonzero Jacobian
determinant, the inverse function theorem ,the implicit function theorem,
extrema of real valued functions of one variable, extrema of real valued
functions of several variables.
Chapter 12 Sections. 12.12 to 12.13 of Text 1
Chapter 13 Sections. 13.1 to 13.6 of Text 1 (28 hours.)
Module 4: Integration of Differential Forms
Integration, primitive mappings, partitions of unity, change of variables,
differential forms.
(Chapter 10 Sections. 10.1 to 10.14 of Text 2) (20 hrs)
References:
 Limaye Balmohan Vishnu, Multivariate Analysis, Springer.
 Satish Shirali and Harikrishnan, Multivariable Analysis, Springer
LEARNING OBJECTIVE:
They study weirstrass theorem , fourier integral theorem ,convolution theorem & its applications, multivariate differential calculus, sufficient condition of differenciability, equality of mixed fractions, & extrema of real valued functions with several variables.
LEARNING OUTCOMES
 Explain the fundamental concepts of real analysis and their role in modern mathematics and applied contexts
 Demonstrate accurate and efficient use of Real analysis techniques
 Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from Real analysis
 Students get familiarized with the basic results
in the analysis of functions of several variables

Module .1
The Weirstrass theorem, other forms of Fourier series, the Fourier integral theorem, the exponential form of the Fourier integral theorem, integral transforms and convolutions, the convolution theorem for Fourier transforms.

Module .2
Multivariable Differential Calculus The directional derivative, directional derivatives and continuity, the total derivative, the total derivative expressed in terms of partial derivatives, An application of complex valued functions, the matrix of a linear function, the Jacobian matrix, the matrix form of the chain rule. Implicit functions and extremum problems, the mean value theorem for differentiable functions,

Module .3
A sufficient condition for differentiability, a sufficient condition for equality of mixed partial derivatives, functions with nonzero Jacobian determinant, the inverse function theorem ,the implicit function theorem, extrema of real valued functions of one variable, extrema of real valued functions of several variables.

Module.4
Integration of Differential Forms Integration, primitive mappings, partitions of unity, change of variables, differential forms.
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