# Multivariate and Integral Transforms Maya K.
Mathematics
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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 non-zero 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:-

1. Limaye Balmohan Vishnu, Multivariate Analysis, Springer.
2. 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,

No items in this section
• ##### Module .3

A sufficient condition for differentiability, a sufficient condition for equality of mixed partial derivatives, functions with non-zero 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.

No items in this section
• ##### Module.4

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

No items in this section Maya K.
Maya.K, M.Sc, B.Ed Lecturer, Department of Mathematics Joined St. Paul's college in 1996. 19 years of teaching experience in both UG and PG and 6 years of service as Govt. HSST ( on work arrangement from the college) . Area of interest is Analysis. Published two international papers in that area.

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