B.Sc Mathematics Honours 7th Semester
Courses in this semester
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7th Semester syllabus (NEP)
Unit-wise syllabus from the Odisha State Higher Education Council model curriculum. Tap a paper to open it.
Core XVI · 4 creditsMeasure Theory & Integration
Course objective
The aim of this course is to provide a foundation for students to many branches of mathematics such as functional analysis, harmonic analysis, ergodic theory, and probability theory, etc. In this course, the students will be introduced to Lebesgue measure and integration, signed measures, Hahn-Jordan decomposition, Radon-Nikodym derivative and product measures.
Unit I
Lebesgue outer measure, measurable sets, Borel sets, regularity, measurable functions, Borel and Lebesgue measurability, non-measurable sets, integration of non-negative functions, simple functions, Lebesgue integration of simple function.
Unit II
Approximation of measurable functions by simple functions, Lebesgue integral of measurable functions and properties, Fatou's lemma, monotone convergence theorem, Lebesgue dominated convergence theorem, integration of series, Riemann and Lebesgue integrals, differentiation, Dini derivatives, Lebesgue differentiation theorem.
Unit III
Abstract measure spaces, measure and outer measure, extension of a measure, uniqueness of the extension, completion of a measure, integration with respect to a measure, modes of convergence, convergence in measure, almost uniform convergence, fundamental in measure convergence, Egorov's theorem.
Unit IV
Signed measure, absolute continuity, Hahn decomposition, Jordan decomposition, Lebesgue decomposition, Radon-Nikodym theorem, applications of Radon-Nikodym theorem, product measure, Fubini's theorem.
Learning outcomes
- Calculate Riemann-Stieltjes integrals and Lebesgue integrals of simple measurable functions.
- Know how to calculate Lebesgue integral of any measurable functions and learn how to apply monotone and dominated convergence theorems.
- Learn the concept of measure on abstract spaces and work on various modes of convergence of a sequence of measurable functions.
- Learn about complex measures, Radon-Nikodym derivatives and related results.
Books recommended
- G. De Barra: Measure Theory and Integration, New Age International, 1981.
- H. L. Royden: Real Analysis, Pearson, Fourth Edition, 2010.
Books for reference
- H. L. Royden and P. M. Fitzpatrick: Real Analysis, Fourth Edition, Pearson Asia Education Ltd and China Machine Press, 2010.
- C. D. Aliprantis, O. Burkinshaw: Principles of Real Analysis, Elsevier, 2011.
- J. Yeh: Real Analysis (Theory of Measure and Integration), 3rd Edition, World Scientific Publication, 2024.
Core XVII · 4 creditsAlgebra-III
Course objective
To present a systematic study of field theory and Galois theory.
Unit I
Basic theory of field extension, algebraic extension, classical straightedge and compass construction, splitting fields and algebraic closures.
Unit II
Separable and inseparable extension, cyclotomic polynomials and extensions, Galois theory, basic definitions, the fundamental theorem of Galois theory.
Unit III
Finite fields, composite extensions and simple extensions, cyclotomic extensions and abelian extensions over Q, Galois groups of polynomials over Q.
Unit IV
Solvability and radical extension, insolvability of quintic, computations of Galois group over Q.
Learning outcomes
- Understand the basic concept of field extension, and splitting fields.
- Understand the significance of separable extension, cyclotomic polynomials, Galois group.
- Understand the structures and properties of finite fields, composite extensions, simple extensions.
- Determine the Galois group of a polynomial and understand the conditions under which polynomial equations can be solved using radicals.
Books recommended
- D. S. Dummit, R. M. Foote, Abstract Algebra, Wiley-India edition, 2013.
- Joseph A. Gallian, Contemporary Abstract Algebra (4th Edition), Narosa Publishing House, New Delhi, 1999 (IX Edition 2010).
Books for reference
- John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002.
- I. N. Herstein, Topics in Algebra, Wiley Eastern Limited, India, 1975.
Core XVIII · 4 creditsTopology
Course objective
This is an introductory course in Topology. The objective of this course is to have knowledge on topological spaces, continuity, connectedness, compactness and separation axioms. Topology on quotient spaces, product spaces and metric spaces are also discussed. The student will also learn basic ideas of algebraic topology such as homotopy, fundamental groups and covering spaces.
Unit I
Cartesian product of a family of sets, Axiom of Choice and its equivalents (without proof), topological spaces, examples, open sets, closed sets, basis and subbasis for a topology, closure and interior of sets, subspace topology, order topology, continuous functions, homeomorphisms, product topology, quotient topology.
Unit II
Metric topology, standard topology, uniform topology, lower limit topology, connectedness, examples, local connectedness, path-connectedness, connected subsets of real line, compact spaces, examples, locally compact spaces, sequential compactness, limit point compactness, compact subsets of real line.
Unit III
Countability axioms, first and second countable spaces, separable and Lindelöf spaces, separation axioms, regular & completely regular space, normal spaces, Urysohn Lemma.
Unit IV
Urysohn metrization theorem, Tychonoff theorem, compactness in metric spaces, compact-open topology, homotopy of paths, fundamental group, covering space.
Learning outcomes
- Know the basics of topological spaces with examples and be able to construct new topologies using the idea of product topology, quotient topology, etc.
- Solve problems involving continuous maps, homeomorphisms between two spaces, connectedness and compactness.
- Learn examples and properties of Hausdorff, regular, normal, separable, first and second countable spaces.
- Understand more results in separation axioms and learn basic concepts of algebraic topology like homotopy, fundamental groups, and covering spaces.
Books recommended
- J. R. Munkres, Topology: A First Course, Pearson, 2nd edition, 2000.
- M. A. Armstrong, Basic Topology, Springer, 1983.
Books for reference
- K. D. Joshi, Introduction to General Topology, Wiley Eastern Limited.
- T. S. Singh, Elements of Topology, CRC Press (special Indian Edition), 2015.
- O. Viro, O. Ivanov, V. Kharlamov and N. Netsvetaev, Elementary Topology, Problem Text Book, American Mathematical Society, 2008.
Core XIX · 4 creditsMathematical Methods
Course objective
The objective of this course is to prepare a student in basics of integral transforms, integral equations and calculus of variations. These tools have applications in other science and engineering fields and are necessary to understand.
Unit I
Laplace transforms: definitions, properties, Laplace transforms of some elementary functions, convolution theorem, inverse Laplace transformation and applications. Fourier transforms: definitions, properties, Fourier transforms of some elementary functions, convolution, Fourier transform of derivatives, Fourier transforms as a limit of Fourier series.
Unit II
Volterra Integral Equations: basic concepts, relationship between linear differential equations and Volterra integral equations, resolvent kernel of Volterra integral equations, solution of integral equations by resolvent kernel, the method of successive approximations, convolution type equations, solutions of integro-differential equations with the aid of Laplace transforms.
Unit III
Fredholm integral equations: Fredholm equations of the second kind, fundamentals, iterated kernel, constructing the resolvent kernel with the aid of iterated kernels, integral equations with degenerate kernels, characteristic numbers and eigenfunctions, solution of homogeneous integral equations with degenerate kernel, non-homogeneous symmetric equations, Fredholm alternatives.
Unit IV
Calculus of variations: extremal of functional, the variation of a functional and its properties, Euler's equations, field of extremals, sufficient conditions for the extremum of a functional, conditional extremum, moving boundary problem, discontinuous problems, one-sided variations, Ritz method.
Learning outcomes
- Calculate Laplace transform, Fourier transform and apply them in areas of differential equations immediately.
- Find solutions of Volterra integral equations.
- Find solutions of Fredholm integral equations.
- Use methods in calculus of variations to solve extremal problems in differential equations and physics.
Books recommended
- A. J. Jerri, Introduction to Integral Equations with Applications, John Wiley & Sons, Inc., 1999.
- Lokenath Debnath, Integral Transforms and Their Applications, CRC Press, New York.
- A. S. Gupta, Calculus of Variations with Applications, PHI Pvt. Ltd., New Delhi.
Books for reference
- I. Sneddon, The Use of Integral Transformations, Tata McGraw Hill, 1972.
- Murray R. Spiegel, Laplace Transforms, Schaum's Series, 1965.
- Gelfand and Fomin, Calculus of Variations, Dover Publications, 2003.
- Krasnov, Problems and Exercises in Calculus of Variations, Mir Publishers, 1970.
- Ram P. Kanwal, Linear Integral Equations, Academic Press, 2013.
Previous year questions (PYQ)
PYQs for this semester will be uploaded soon. Ask on WhatsApp for PYQ solutions in the AIFM app.
Demo classes
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What you get in the full course
Frequently asked questions
What is the syllabus of B.Sc Mathematics Honours 7th Semester in Odisha (NEP)?
It has 4 papers: Measure Theory & Integration, Algebra-III, Topology, Mathematical Methods. The unit-wise syllabus is on this page, and the syllabus PDF can be opened or downloaded.
What is the fee for the B.Sc Mathematics Honours 7th Semester course at AIFM EDUCATION?
The 7th Semester course is coming soon. Message AIFM EDUCATION on WhatsApp to be notified.
Where can I find B.Sc Mathematics Honours 7th Semester previous year question papers (PYQ)?
PYQs for this semester will be added to this page soon. PYQ solutions are available in the AIFM EDUCATION app.
Is this course for all universities in Odisha?
Yes. It follows the NEP 2020 model curriculum of the Odisha State Higher Education Council, used by universities across Odisha. Classes are explained in Odia, with notes in English.