Syllabus (MATHEMATICS)
Course Type: MAJ-16
Semester: 8
Course Code: BMTMMAJ16T
Course Title: Elementary Differential Geometry
(L-P-Tu): 3-0-1
Credit: 4
Practical/Theory: Theory
Course Objective: To be done.
Learning Outcome: Course Outcomes (CO): The whole course will have the following outcomes: Upon successful completion, the students will be able to CO1: describes a multilinear relationship between sets of algebraic objects related to a vector space. CO2: do computations w
Syllabus:
Elementary Differential Geometry [Credit: 3, 45L]
Tensors:
- Definition of a tensor as a generalization of vectors in a vector space V. Tensor and their transformation laws, Covariant and contravariant vectors to Covariant and contravariant tensors, Tensor algebra, Contraction, Quotient law, Reciprocal tensors, Kronecker delta, Symmetric and skew-Symmetric tensors, fundamental metric tensor, Riemannian space, Christoffel symbols and their transformation law, Covariant differentiation of vector and tensor.
- Curves in Space: level curves and parametrized curves in Rn
, arc length of curve, reparametrization, plane curves and space curves and their curvature and properties and tangent and normal at a point on the curves. Torsion of space curves, Helix, Serret- Frenet formulae for curves in space. Simple closed curves with periods. Isoperimetric inequality, Intrinsic differentiation and Curvilinear coordinates Geodesic. - Surfaces: Surfaces and its parametric representation, Regular surfaces and example. Tangent, normal and orientability of surfaces. Smooth functions on a surface, Differential of a smooth function defined on a surface. Angle between two curves on a surface, The first and second fundamental form of surface, Geodesic curvature, normal curvature and principal curvature of a surface curve, Meusnier’s theorem, The third fundamental form, Gaussian and mean curvature, Riemann curvature tensor and Ricci tensor. Riemannian metric, Isometry of surfaces, Developable surfaces, Weingarten formula, Equation of Gauss and Codazzi.
Reading References:
- M. C. Chaki; Tensor Analysis; The Calcutta Publishers.
- B. Spain, Tensor Calculus: A Concise Course, Dover Publication, 2003.
- U.C. De, A. A. Shaikh, J. Sengupta; Tensor Calculus; Narosa.
- M. Majumder, A. Bhattacharyya; Differential Geometry; Books & Allied Pub.
- I. S. Sokolnikoff; Tensor Analysis; Wiley, New York; 1951.
Basic Features
Undergraduate degree programmes of either 3 or 4-year duration, with multiple entry and exit points and re-entry options, with appropriate certifications such as:
- UG certificate after completing 1 year (2 semesters with 40 Credits + 1 Summer course of 4 credits) of study,
- UG diploma after 2 years (4 semesters with 80 Credits + 1 Summer course of 4 credits) of study,
- Bachelor’s degree after a 3-year (6 semesters with 120 credits) programme of study,
- 4-year bachelor’s degree (Honours) after eight semesters (with 170 Credits) programme of study.
- 4-year bachelor’s degree (Honours with Research) if the student completes a rigorous research project (of 12 Credits) in their major area(s) of study in the 8th semester.
Note: The eligibility condition of doing the UG degree (Honours with Research) is- minimum75% marks to be obtained in the first six semesters.
- The students can make an exit after securing UG Certificate/ UG Diploma and are allowed to re-enter the degree programme within three years and complete the degree programme within the stipulated maximum period of seven years.
