Title: Advanced Calculus (Code-09050301)
B.Sc. (Hons), Semester-III
S.No. |
Topic |
Domain |
Hours as per UGC |
1 |
Continuity, Sequential Continuity, properties of continuous functions, Uniform continuity, Differentiability, Chain rule of differentiability. Rolle’s Theorem and Lagrange’s mean value theorem and their geometrical interpretations. Taylor’s Theorem with various forms of remainders, Darboux Theorem (Intermediate value theorem) for derivatives, Indeterminate forms. |
Must know |
14 |
2 |
Limit and continuity of real valued functions of two variables. Partial differentiation. Total Differentials; Composite functions & implicit functions. Change of variables. Homogenous functions & Euler’s theorem on homogeneous functions. Taylor’s theorem for functions of two variables. |
Must know |
12 |
3 |
Differentiability of real valued functions of two variables. Schwarz and Young’s theorem. Implicit function theorem. Maxima, Minima and saddle points of two variables. Lagrange’s method of multipliers. |
Must know |
10 |
4 |
Curves: Tangents, Principal normals, Binormals, Serret-Frenet formulae. Locus of the centre of curvature, Spherical curvature, Locus of centre of Spherical curvature, Involutes, evolutes, Bertrand Curves. Surfaces: Tangent planes, one parameter family of surfaces, Envelopes. |
Nice to know |
14 |
Books Recommended:
- Goldberg, R.R. Real Analysis. New Delhi : Oxford & IBH, 1970.
- Gorakh Prasad, Differential Calculus. Allahabad : Pothishala.
- Malik, S.C. Mathematical Analysis. New Delhi: Wiley Eastern.
- Shanti Narayan. A Course in Mathematical Analysis. New Delhi: S.Chand.
- Spiegel, Murray, R. Theory and Problems of Advanced Calculus. New York: Schaum Publishing.