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SILVER OAK UNIVERSITY
Institute of Science (05)
Programme Name: Master of Science (Mathematics)
Subject Name: Measure Theory
Subject Code: 1050277107
Semester: II
Prerequisite:
Knowledge of Real Analysis.
Objective:
To introduce the concept of measure on the Real line and discuss Lebesgue theory on the Real
line.
Teaching and Examination Scheme:
Teaching Scheme Evaluation Scheme
Contact Theory Practical Total
L T P Hours Credit CIE ESE CIE ESE Marks
(TH) (TH) (PR) (PR)
4 1 0 5 5 40 60 - - 100
Content:
Unit Contents Teaching Weightage
No. Hours %
1. Algebra and σ-algebra of sets, Borel sets in ℝ, Lebesgue outer 14 25 %
measure in ℝ, measurable sets and Lebesgue measure on ℝ, non-
measurable set, measurable functions.
2. Littlewood's three principles, Egoroff's theorem, the Lebesgue 16 25 %
integral of a measurable simple function vanishing outside a set
of finite measure, the Lebesgue integral of a bounded function
over a set of finite measure, comparison of Riemann and
Lebesgue integral, bounded convergence theorem. Lebesgue
integral of a nonnegative measurable function.
3. Fatou's lemma and monotone convergence theorem, Beppo- 14 25 %
Levis theorem, general Lebesgue integral, dominated
convergence theorem, convergence in measure, relation with
convergence a.e.
4. Vitali’s theorem (statement only), functions of bounded variation, 16 25 %
Jordan's lemma, differentiation of an integral, continuity and
bounded variation of indefinite integral, absolute continuity of
indefinite integral, different forms of fundamental theorem of
integral calculus, relation between indefinite integral and absolute
continuity.
Course Outcome:
After the successful completion of the course, students will be able to
Sr. No. CO statement Unit No
CO-1 Understand Lebesgue measure on ℝ, measurable functions 1
CO-2 Do Littlewood's three principles, Lebesgue integral of a nonnegative 2
measurable function.
CO-3 Learn about Fatou's lemma and its convergence theorem 3
CO-4 Learn about different forms of fundamental theorem of integral calculus 4
Teaching & Learning Methodology: -
The various methods or tools follows by the faculties to teach the above subject are:
1. With the aid of multi-media projector, black board, Chalk etc.
2. Lectures with discussion, question and answer sessions, informal quizzes.
3. E-sources for the virtual learning environment.
4. Model based learning.
Books Recommended: -
1. “Real Analysis (Third Edition)”, Royden H. L., Mac Millan, 1998.
2. “An Introduction to Measure and Integration”, Rana, I. K., Narosa Publ. House, New
Delhi, 1997.
3. “Introduction to Measure Theory”, De Barra G, Van Nostrand Reinhold Co., 1974.
List of Open Source Software/learning website:
• http://silveroakuni.ac.in/video-lecture
• https://nptel.ac.in/courses/
• https://swayam.gov.in/
• https://mathworld.wolfram.com/
• https://www.khanacademy.org/
SILVER OAK UNIVERSITY
Institute of Science (05)
Programme Name: Master of Science (Mathematics)
Subject Name: Partial Differential Equations
Subject Code: 1050277108
Semester: II
Prerequisite:
Multi-variable calculus, Ordinary differential equations, Linear Algebra.
Objective:
The objective of this course is to introduce partial differential equations, particularly the second
order equations of mathematical physics.
Teaching and Examination Scheme:
Teaching Scheme Evaluation Scheme
Contact Theory Practical Total
L T P Hours Credit CIE ESE CIE ESE Marks
(TH) (TH) (PR) (PR)
4 1 0 5 5 40 60 - - 100
Content:
Unit Contents Teaching Weightage
No. Hours %
1. Origin of first order partial differential equation, solution of first 15 25 %
order partial differential equation using Lagrange’s method, non-
linear first order partial differential equations: compatible system
of first order partial differential equations, solution by Charpit’s
method and Jacobi’s method.
2. Origin of second order partial differential equations, linear second 15 25 %
order partial differential equations with constant coefficients,
solutions for F(D, D’)z = f(x, y) to be polynomial, exponential,
sin/cos functions, general method for homogeneous equations.
3. Second order partial differential equations with variable 15 25 %
coefficients, solution by method of changing variables u = logx, v
= logy for special type of equations, Separation of variable
method: solution of three special equations – Laplace, wave and
diffusion equation, solution of these equations in cartesian and
polar coordinate systems.
4. Second order partial differential equations with variable 15 25 %
coefficients, solution by method of changing variables u = logx, v
= logy for special type of equations, Separation of variable
method: solution of three special equations – Laplace, wave and
diffusion equation, solution of these equations in cartesian and
polar coordinate systems.
Course Outcome:
After the successful completion of the course, students will be able to
Sr. No. CO statement Unit No
CO-1 Classify the partial differential equations its formation. 1
Find solution of linear and non-linear first order PDE.
Obtain solution by Charpit’s method and Jacobi’s method.
CO-2 Classify the second order partial differential equations. 2
Find solution of second order homogeneous PDE.
CO-3 Find solution of second order homogeneous PDE by method of changing 3
variables.
CO-4 Understand Separation of variable method and solution of three special 4
equations – Laplace, wave and diffusion equation,
Teaching & Learning Methodology: -
The various methods or tools follows by the faculties to teach the above subject are:
1. With the aid of multi-media projector, black board, Chalk etc.
2. Lectures with discussion, question and answer sessions, informal quizzes.
3. E-sources for the virtual learning environment.
4. Model based learning.
Books Recommended: -
1. “Elementary Course in Partial Differential Equations”, Amarnath T., Narosa Pub. House,
New Delhi, 1997.
2. “Elements of Partial Differential Equations”, Sneddon I. N., McGraw- Hill Pub. Co.,
1957.
Chapter 2: Section 9,10,11,13 and Chapter 3: Section 1,4,5,9,11
3. “Higher Engineering Mathematics”, Grewal B. S. and Grewal, J. S., Khanna Pub., New
Delhi, 2000.
4. “Advanced Differential Equations”, Raisinghania M. D., S. Chand & Co., 1995.
5. “Partial Differential Equations”, Phoolan Prasad and Ravindran R., Wiley Eastern.
List of Open Source Software/learning website:
• http://silveroakuni.ac.in/video-lecture
• https://nptel.ac.in/courses/
• https://swayam.gov.in/
• https://mathworld.wolfram.com/
• https://www.khanacademy.org/
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