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 equations
  • To study about  notations and operations of  Tensor.

Learning 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 .

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

https://www.youtube.com/watch?v=S0_qX4VJhMQ

Session 2 :

Fundamental Triads, Curves

https://www.youtube.com/watch?v=4fB0VfKZRXM

Session 3 :

Osculating plane

https://www.youtube.com/watch?v=mvvCmwALmLw

Session 4:

Rectifying Plane

Session 5:

Serret-Frenet Formulae

Session 6:(2hr)

Project- 1: Finding the direction of tangent , normal and binormal at any point of    the curve

Session 7:

Curvatures of a curve at a point

https://www.youtube.com/watch?v=ugtUGhBSeE0

Session 8:

Torsion of a curve at a point

https://www.youtube.com/watch?v=VIqA8U9ozIA

Session 9:

Expression of Curvature and Torsion in terms of arc length parameter

https://www.youtube.com/watch?v=AE8NVTyESd8

Session 10:

Expression of Curvature and Torsion in terms of arbitrary parameter

Session 11:(2 hr)

Project-2: Compute the Curvature of an ellipse.

Session 12:

Spherical Indicatrix,

https://www.youtube.com/watch?v=1NMpdrFrphA

Session 15: (2hr)

Project 3: Determine the evolutes of the given curve.

Session 16:

Betrand Curve

Session 17:

Osculating Spheres

https://www.youtube.com/watch?v=bAmzhP6622g

Session 18:

Osculating circles

https://www.youtube.com/watch?v=qfE2nTaLxD8

Session 19: (2hr)

Project-4 : Show that the tangent to the locus of osculating sphere passes through  the centre of the Oscilating Circle.

Session 20

Surface: Tangent planes

https://www.youtube.com/watch?v=UfnpcOcZAO0

Session 21:

Surface: Normals

https://www.youtube.com/watch?v=mH9aFGUZdug

Session 22

The First  Fundamental Forms

https://www.youtube.com/watch?v=yUgbTSvMREg

Session 23

The Second  Fundamental Forms

Session 24:(2hr)

Project-5 :Find the normal to a given surface

Session 25 :

Tensor : Definitions and explanations

https://www.youtube.com/watch?v=f5liqUk0ZTw

Session 26:

Vector Spaces

Session 27:

Free systems, Basis and Dimension

https://www.youtube.com/watch?v=4C9GKyfUQkc

Session 28:

Suffix Conventions

https://www.youtube.com/watch?v=zG44Y1JIAts

Session 29:

Transformation law for change of Basis Vectors and Components

https://www.youtube.com/watch?v=j6DazQDbEhQ

Session 31:

Transformation law for change of Basis in dual Space

Session 32:

Tensor Product of Vector Spaces.

https://www.youtube.com/watch?v=tpL95Sd7zT0

Session 33:

Real Valued Bilinear Functions

Session 34:

Components of Tensor

https://www.youtube.com/watch?v=kUYjKzU6e60

Session 35:

Special  Tensors

https://www.youtube.com/watch?v=15lOIXgFkPE

Session- 36 (2hr)

Project-6: Show that the velocity of a fluid at any point is component of a contravariant vector

Case Studies

Case Studies

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    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 […]