Hey Guys, Welcome to our website, where I discuss the M.S.C syllabus related to Real analysis. This is mainly for WBSU, but similar to other universities also. This gives you an overall view of the M.sc course.
After that, I mention some useful books suggested by my professors, which will give you a guide in m.sc and various competitive exams.
In postgraduate mathematics, Real Analysis appears in 1st semester in various colleges under WBSU. But the other college under WBSU may not follow the syllabus properly. Total marks of algebra for 1st semester is 50, during 6 months of 1st semester this 50 marks subdivided into internal three topics which are-
- Midterm exam: It will occur generally on the head of 3 months of 1st semester. This mod term consists of 5 marks. But teachers took 20 or 15 marks exam and make a percent out of 5. For example, if you get 10 out of 20 then you get 2.5 out of 5.
- Student Lecture/viva: It consists of 5 marks. Students choose a topic sweetly for them and make a presentation with the help of PowerPoint, python, MathType e.t.c on the computer and present in front of the teacher. The teacher can ask questions during the presentation.
- The final exam consists of 40 marks for 2 hours.
- Syllabus Of WBSU Real Analysis for M.S.C:-
Lebesgue measure: Definition of Lebesgue outer measure and inner measure. Measurable functions: Definition on a measurable set in R and basic properties. Simple functions.
Functions of bounded variation: Definition and basic properties, Lipschitz condition, Jordan decomposition, Nature of points of discontinuity, Nature of points of non-differentiability.
Absolutely continuous functions: Definition and basic properties, Deduction of the class of all absolutely continuous functions as a proper subclass of all functions of bounded variation; Characterization of an absolutely continuous function in terms of its derivative vanishing almost everywhere.
Riemann-Stieltjes integral: Existence and basic properties, Integration by parts, integration of a continuous function with respect to a step function, Convergence theorems in respect of integrand.
Differentiation on R^n: Directional derivatives and continuity, the total derivative and continuity, total derivative in terms of partial derivatives, the matrix transformation of T: R^n → R^m. The Jacobian matrix.
The chain rule and its matrix form. Mean value theorem for vector-valued function. Mean value inequality. 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 is an application of the inverse function theorem.
Extremum problems with side conditions. Lagrange’s necessary conditions as an application of inverse function
theorem.
Integration on R^n: Integral of the function f: A → R where A ⊆ R^n is a closed rectangle. Conditions of integrability. Integral of the function f: C → R where C ⊆ R^n is not a rectangle. Concept of Jordan measurability of a set in R^n. Fubini’s theorem
for integral of the function f: A × B → R, where A ⊆ R^n,
B ⊆ R^m are closed rectangles. Fubini’s theorem for integral of f: C → R, where C ⊆ A × B. Formula for change of variables in an
integral in R^n.
- WBSU Previous Year Question:
- Some Useful Books:
1)Mathematical Analysis by Apostol:
2)Calculus by T.M Apostol Volume-1:
3)Calculus by T.M Apostol Volume-2:
4)Real and Complex Analysis by Walter Rudin:
5)Real Analysis by Santi Narayan:
6)Real Analysis by N.P Bali:
7)Real Analysis by S.K. Mapa:
For more about Real analysis, you can comment below.
Visit,
for postgraduate free textbook pdf of various topics for the 2nd and 3rd semester respectively.
Earn Money Online: Click NowWant to make a world-class business card: Visit now (Starting from Rs.350 only)
Lebesgue measure: Definition of Lebesgue outer measure and inner measure. Measurable functions: Definition on a measurable set in R and basic properties. Simple functions.
Functions of bounded variation: Definition and basic properties, Lipschitz condition, Jordan decomposition, Nature of points of discontinuity, Nature of points of non-differentiability.
Absolutely continuous functions: Definition and basic properties, Deduction of the class of all absolutely continuous functions as a proper subclass of all functions of bounded variation; Characterization of an absolutely continuous function in terms of its derivative vanishing almost everywhere.
Riemann-Stieltjes integral: Existence and basic properties, Integration by parts, integration of a continuous function with respect to a step function, Convergence theorems in respect of integrand.
Differentiation on R^n: Directional derivatives and continuity, the total derivative and continuity, total derivative in terms of partial derivatives, the matrix transformation of T: R^n → R^m. The Jacobian matrix.
The chain rule and its matrix form. Mean value theorem for vector-valued function. Mean value inequality. 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 is an application of the inverse function theorem.
Extremum problems with side conditions. Lagrange’s necessary conditions as an application of inverse function
theorem.
Integration on R^n: Integral of the function f: A → R where A ⊆ R^n is a closed rectangle. Conditions of integrability. Integral of the function f: C → R where C ⊆ R^n is not a rectangle. Concept of Jordan measurability of a set in R^n. Fubini’s theorem
for integral of the function f: A × B → R, where A ⊆ R^n,
B ⊆ R^m are closed rectangles. Fubini’s theorem for integral of f: C → R, where C ⊆ A × B. Formula for change of variables in an
integral in R^n.
- WBSU Previous Year Question:
- Some Useful Books:
1)Mathematical Analysis by Apostol:
2)Calculus by T.M Apostol Volume-1:
3)Calculus by T.M Apostol Volume-2:
4)Real and Complex Analysis by Walter Rudin:
5)Real Analysis by Santi Narayan:
6)Real Analysis by N.P Bali:
7)Real Analysis by S.K. Mapa:
For more about Real analysis, you can comment below.
Visit,
for postgraduate free textbook pdf of various topics for the 2nd and 3rd semester respectively.
Earn Money Online: Click Now
Want to make a world-class business card: Visit now (Starting from Rs.350 only)
