University Of Pune M.Sc. Maths Syllabus

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Here I am providing the M.Sc. Part-1 Maths Syllabus of University Of Pune which you are looking for .

Real Analysis
Advanced Calculus
Linear Algebra
Number Theory
Ordinary Differential
Equations

MT-501: Real Analysis
1. Metric Spaces, Normed Spaces, Inner Product Spaces:
Definitions and examples, Sequence Spaces, Function Spaces, Dimension.
2. Topology of Metric Spaces:
Open, Closed and Compact Sets, the Heine-Borel and Ascoli-Arzela’ Theorems,
Separability, Banach and Hilbert Spaces.
3. Measure and Integration :
Lebesgue Measure on Euclidean Space, Measurable and Lebesgue Integrable Functions, The
Convergence Theorems, Comparison of Lebesgue Integral with Riemann Integral, General
Measures and the Lebesgue LP-Space.
4. Fourier Analysis in Hilbert Space :
Orthonormal Sequences, Bessel’s Inequality, Parseval’s Theorem, Riesz-Fischer Theorem,
Classical Fourier Analysis.
5. Weierstrass Approximation Theorem, Generalised Stone-Weierstrass Theorem, Baire
Category Theorem and its Applications, Contraction Mapping.

Text Book: Karen Saxe : Beginning Functional Analysis
(Springer International Edition)
Chapters: Chapters 1 to 4 and 6.1, 6.2, 6.5

Reference Books:
1. N. L. Carothers: Real Analysis (Cambridge University Press)
2. T. M. Apostol: Mathematical Analysis (Narosa Publishing)
3. S. Kumaresan: Topology of Metric Spaces (Narosa Publishing)
4. G. F. Simmons: Introduction to Topology and Modern Analysis. (Mc-Graw Hill)
5. W. Rudin : Principles of Mathematical Analysis. (Mc-Graw Hill)

MT - 502: Advanced Calculus
1. Derivative of a scalar field with respect to a vector, Directional derivative, Gradient of a
scalar field, Derivative of a vector field, Matrix form of the chain rule, Inverse function
theorem and Implicit function theorem.
2. Path and line integrals, The concept of a work as a line integral, Independence of path, The
first and the second fundamental theorems of calculus for line integral, Necessary condition
for a vector field to be a gradient.
3. Double integrals, Applications to area and volume, Green's Theorem in the plane, Change of
variables in a double integral, Transformation formula, Change of variables in an n-fold
integrals.
4. The fundamental vector product, Area of a parametric surface, Surface integrals, The
theorem of Stokes, The curl and divergence of a vector field, Gauss divergence theorem,
Applications of the divergence theorem.

Text Book:
T. M. Apostol: Calculus vol. II (2nd edition)(John Wiley and Sons,Inc.)
Chapter 1 : Sections 81 to 8.22
Chapter 2: Sections 10.1 to 10.11 and 10.14 to 10.16
Chapter 3: Sections 11.1 to 11.5 and 11.19 to 11.22 and 11.26 to 11.34.
Chapter 4: Sections 12.1 to 12.15, 12.18 to 12.21
For Inverse function theorem, Implicit function theorem refer the book ‘Mathematical Analysis’
by T. M. Apostol.

Reference Books :
1. T. M. Apostol: Mathematical Analysis (Narosa publishing house)
2. W. Rudin: Principles of Mathematical Analysis.(Mc-Graw Hill)
3. Devinatz: Advanced Calculus

MT-503 : Linear Algebra
Revision – Matrices, Determinants, Polynomials. (Chapter 1 of the Text Book).
1. Vector Spaces
Subspaces
Basis and dimension
Linear Transformations
Quotient spaces
Direct sum
The matrix of a linear transformation
Duality
2. Canonical Forms
Eigenvalues and eigenvectors
The minimal polynomial
Diagonalizable and triangulable operators
The Jordan Form
The Rational Form
3. Inner Product Spaces
Inner Products
Orthogonality
The adjoint of a linear transformation
Unitary operators
Self adjoint and normal operators
Polar and singular value decomposition
4. Bilinear Forms
Definition and examples
The matrix of a bilinear form
Orthogonality
Classification of bilinear forms

Text Book: - Vivek Sahai, Vikas Bist : Linear Algebra (Narosa Publishing House).
Chapters : 2 to 5

Reference Books:
i) K. Hoffman and Ray Kunje : Linear Algebra (Prentice - Hall of India private Ltd.)
ii) M. Artin : Algebra (Prentice - Hall of India private Ltd.)
iii) A.G. Hamilton : Linear Algebra (Cambridge University Press (1989))
iv) N.S. Gopalkrishanan : University algebra (Wiley Eastern Ltd.)
v) J.S. Golan : Foundations of linear algebra (Kluwer Academic publisher (1995) )
vi) Henry Helson : Linear Algebra (Hindustan Book Agency (1994) )
vii) I.N. Herstein : Topics in Algebra, Second edition (Wiley Eastern Ltd.)

MT-504 : Number Theory
1. Revision :- Divisibility in integers, Division algorithm, G.C.D., L.C.M. Fundamental
theorem of arithmetic. The number of primes. Mersene numbers and Fermat's numbers.
2. Congruences :- Properties of congruence relation. Resicle classes their properties Fermat's
and Euler's theorems. Wilson's Theorem. The congruence X2 = -1 (mod p) has solution iff p
is the form 4n+1 where p is prime. Linear congruences of degree one. Chinese remainder
Theorem.
3. Arithmetic functions : Euler function, Greatest integer function, Divisor function (n),
Mobius
function (n). Properties and their inter relation.
4. Quadratic Reciprocity :- Quadratic residue, Legendre's symbol, Its properties, Quadratic
reciprocity law, Jacobi symbol, Its properties. Sums of Two Squares.
5. Some Diophantine Equations :
The equation ax + by = c , simultaneous linear equations.
6. Algebraic Numbers :- Algebraic Numbers, Algebraic number fields. Algebraic integers,
Quadratic fields. Units in Quadratic fields. Primes in Quadratic fields. Unique factorization
Primes in quadratic fields having the unique factorization property.

Text Book :- Ivan Nivam & H.S. Zuckerman, An introduction to number theory (Wiley Eastern
Limited)

Sections: 2.1 to 2.4, 3.1 to 3.3, 3.6, 4.1 to 4.4, 5.1, 5.2, and 9.1 to 9.9

Reference Books :-
1. T.M. Apostol, An Introduction to Analytical Number Theory (Springer International
Student's Edition)
2. David M Burton, Elementary Number Theory (Universal Book Stall, New Delhi)
3. S. G. Telang, Number Theory (Tata Macgrow Hill)
4. G. H. Hardy and E. M. Wright, Introduction to Number Theory (The English language book
society and oxford university press)

For information , here is the attachment

address
Savitribai Phule Pune University
Ganeshkhind, Pune, Maharashtra 411007 ‎
020 2560 1099 ‎

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