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Title- Advanced Calculus (09050301)

B.Sc. (HONS) – 3rd semester

S.N Unit Content of the topics Learning objects Teaching Guidelines Methodology Time (Hrs)
1 one 1.    Continuity, properties of continuous functions,

2.    Sequential Continuity,

3.    Uniform continuity,

4.    Chain rule of differentiability.

5.    Mean value theorems; Rolle’s Theorem and Lagrange’s mean value theorem and their geometrical interpretations.

6.    Taylor’s Theorem with various forms of remainders,

7.    Darboux intermediate value theorem for derivatives,

8.    Indeterminate forms.

Students will be able to

1.    Continuity, Sequential Continuity, properties of continuous functions,

2.    Uniform continuity least upper bound, greatest lower bound of a set,

3.    Mean value theoremsTaylor’s Theorem

4.    Darboux theorem for derivatives,

Lecture should be effective so that student can grasp the topics easily. 1. Lecture.

2.  Seminar.

3. Discussion/Interaction with Students.

4. Assignment.

5.  Discussion on Assignment.

6. Evaluation of Assignment.

15
2 Two 1.    Limit and continuity of real valued functions of two variables.

2.    Partial differentiation.

3.    Total Differentials;

4.    Composite functions & implicit functions.

5.    Change of variables.

6.    Homogenous functions

7.    Euler’s theorem on homogeneous functions.

8.    Taylor’s theorem for functions of two variables

Students will be able to

1.    Limit and continuity of real valued functions of two variables. Partial differentiation.

2.    Homogeneous equations with constant co-efficient,

3.    Differentials; Composite functions & implicit functions.

4.    Taylor’s theorem

Lecture should be effective so that student can grasp the topics easily. 1.    Lecture.

2.     Seminar.

3.    Discussion/Interaction with Students.

4.    Assignment.

5.    Discussion on Assignment.

6.    Evaluation of Assignment

15
3 Three 9.    Differentiability of real valued functions of two variables.

10. Schwarz and Young’s theorem. Implicit function theorem.

11. Maxima, Minima and saddle points of two variables.

12. Lagrange’s method of multipliers.

To make students familiar with

1.    Differentiability of real valued functions of two variables. Schwarz and Young’s theorem.

2.     Maxima, Minima Lagrange’smethod of multipliers, Involutes, evolutes,

Lecture should be effective so that student can grasp the topics easily.

 

1.    Lecture.

2.    Seminar.

3.    Discussion

4.     Assignment.

5.    Discussion

6.    Evaluation

15
4 Four 13. Curves: Tangents, Principal Normals, Binormals,

14. Serret-Frenet formulae.

15. Locus of the centre of curvature, Spherical curvature, Locus of centre of Spherical curvature,

16. Involutes, evolutes, Bertrand Curves. Tangent planes,

17. One parameter family of surfaces, Envelopes.

 

1.       Bertrand Curves. Surfaces: Tangent planes,

2.       Envelopes. Involutes, evolutes,

Lecture should be effective so that student can grasp the topics easily.

 

7.    Lecture.

8.    Seminar.

9.    Discussion/ Assignment.

10. Discussion on Assignment.

Book Recommended:

  1. R. Goldberg : Real Analysis, Oxford & I.B.H. Publishing Co., New Delhi, 1970
  2. Gorakh Prasad : Differential Calculus, Pothishala Pvt. Ltd., Allahabad
  3. C. Malik : Mathematical Analysis, Wiley Eastern Ltd., Allahabad.
  4. Shanti Narayan : A Course in Mathemtical Analysis, S.Chand and company, New Delhi
Admissions Open 2019-20