# Differential Geometry

Teacher

Category

### Course Attendees

Still no participant

Still no reviews

# 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.

## 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

https://www.youtube.com/watch?v=JCor1st0d2E https://www.youtube.com/watch?v=tKnBj7B2PSg https://www.youtube.com/watch?v=NKlfCLsFzUY

https://www.powershow.com/view1/237cec-ZDc1Z/Differential_Geometry_with_Applications_powerpoint_ppt_presentation

## 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

https://www.youtube.com/watch?v=mvvCmwALmLw&t=504s

## Session 5:

Serret-Frenet Formulae

https://www.youtube.com/watch?v=1T58bK-WFsQ

## Session 6&7

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

## Session 8:

Curvatures of a curve at a point

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

## Session 9:

Torsion of a curve at a point

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

## Session 10:

Expression of Curvature and Torsion in terms of arc length parameter

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

## 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 14:

Spherical Indicatrix,

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

## Session 15

Evolutes

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

## Session 16:

Involutes

https://www.youtube.com/watch?v=-V8mfHcnf08

## Session 17&18:

Project 3: Determine the evolutes of the given curve.

Betrand Curve

## Session 20:

Osculating Spheres

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

## Session 21:

Osculating circles

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

## Session 22&23:

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

## Session 24

Surface: Tangent planes

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

## Session 25:

Surface: Normals

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

## Session 26

The First  Fundamental Forms

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

## Session 27

The Second  Fundamental Forms

## Session 28&29:

Project-5 :Find the normal to a given surface

## Session 30 :

Tensor : Definitions and explanations

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

Vector Spaces

## Session 32:

Free systems, Basis and Dimension

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

## Session 33:

Suffix Conventions

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

## Session 34:

Transformation law for change of Basis Vectors and Components

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

## Session 35:

Dual Spaces

https://www.youtube.com/watch?v=46AJDYEQ4VA

## Session 36:

Transformation law for change of Basis in dual Space

## Session 37:

Tensor Product of Vector Spaces.

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

## Session 38:

Real Valued Bilinear Functions

## Session 39:

Components of Tensor

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

## Session 40:

Special  Tensors

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

## Session- 41&42

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

## Case Studies

### Dr Tumbanath Samantara

##### Teaching
VIEW PROFILE

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 […]