Differential Geometry
Teacher
Dr Tumbanath Samantara
Category
Core Courses
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Course Name : Differential Geometry
Code(Credit) : CUTM1537(3-0-1)
Course Objectives
- To study about different geometrical figure and their representation in mathematical equation
- To study about notations and operations of Tensor.
Course Outcomes
- Upon successful completion of this course,students will be able to
- write equation of normal , binormal and tangent to a curve.
- able to understand tensorial expressions .
| COs | Course outcomes | Mapping Cos with POs (High-3, Medium-2, Low-1) |
| CO1 | · Write equation of normal , binormal and tangent to a curve.
|
PO1(3), PO2(2), PO9(3) |
| CO2 | · Able to understand tonsorial expressions.
|
PO1(3), |
- write equation of normal , binormal and tangent to a curve.
- able to understand tensorial expressions .
Course Syllabus
Module-I
Introduction to Differential Geometry,Introduction to Differential Geometry,Osculating plane and Rectifying Plane
Project 1:.finding the direction of tangent , normal and binormal at any point of curve
Module-II
Curvatures of a curve at a point,Torsion of a curve at a point,Expression of Curvature and Torsion in terms of arc length parameter,Expression of Curvature and Torsion in terms of arbitrary parameter
Project 2: Compute the Curvature of an ellipse.
Module-III
Spherical Indicatrix, Evolutes , Involutes
Project 3: Determine the evolutes of the given curve.
Module-IV
Betrand Curve,Osculating Spheres , Osculating circles.
Project-4 : Show that the tangent to the locus of osculating sphere passes through the centre of the Oscilating Circle.
Module-V
Surface: Tangent planes and Normals,The two fundamental forms
Project 5: Find the normal to a given surface
Module-VI
Tensor : Definitions and explanations,Vector Space,Free systems, Basis and Dimension,Suffix Conventions,Transformation law for change of Basis Vectors and Components ,Dual Spaces
Module-VII
Transformation law for change of Basis in dual Space,Isomorphism,Tensor Product of Vector Spaces, Real Valued Bilinear Functions,Special Tensors
Project-6: Show that the velocity of a fluid at any point is component of a contravariant vector
BOOK PRESCRIBED
1. A text book of vector calculus-Shanti Narayana and J.N.Kapoor
Chapters: II and III
2. An Introduction to Differential Geometry by T.G. Willmore -Oxford University Press (1983)
Chapters : V
BOOK FOR REFERENCE
1.Differential Geometry-P.P.Gupta,G.S.Malik, S.K.Pundir
2.Tensor Analysis- Edward Nelson( Princeton University Press & University of Tokyo Press),1967
3.Introduction to Tensor Analysis and the Calculus of Moving Surfaces-Pavel Grinfeld, Springer
Session Plan
Session 1
Introduction to Differential Geometry
Session 6&7
Project- 1: Finding the direction of tangent , normal and binormal at any point of the curve
Session 10:
Expression of Curvature and Torsion in terms of arc length parameter
Session 11:
https://www.youtube.com/watch?v=s6RFFqt-Vfw
Expression of Curvature and Torsion in terms of arbitrary parameter
Session 12&13:
Project-2: Compute the Curvature of an ellipse.
Session 17&18:
Project 3: Determine the evolutes of the given curve.
Session 19:
Betrand Curve
Session 22&23:
Project-4 : Show that the tangent to the locus of osculating sphere passes through the centre of the Oscilating Circle.
Session 27
The Second Fundamental Forms
Session 28&29:
Project-5 :Find the normal to a given surface
Session 31:
Vector Spaces
Session 34:
Transformation law for change of Basis Vectors and Components
Session 36:
Transformation law for change of Basis in dual Space
Session 38:
Real Valued Bilinear Functions
Session- 41&42
Project-6: Show that the velocity of a fluid at any point is component of a contravariant vector
Case Studies
Case Studies
Our Main Teachers
Dr. Tumbanath Samantara currently works as Associate Professor & HoD, Department of Applied Mathematics, , School of Applied Sciences, Centurion University of Technology and Management, (Bhubaneswar Campus) Odisha. He has completed his Ph.D. in the field of Fluid Dynamics. He has more than 10 years of research experience and 19 years of teaching experience. During […]