Differential Geometry

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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

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

Session 1

Introduction to Differential Geometry

Osculating plane

Rectifying Plane

Session 5:

Serret-Frenet Formulae

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

Session 9:

Torsion of a curve at a point

Session 10:

Expression of Curvature and Torsion in terms of arc length parameter

Session 11:

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,

Evolutes

Involutes

Session 17&18:

Project 3: Determine the evolutes of the given curve.

Betrand Curve

Session 20:

Osculating Spheres

Session 21:

Osculating circles

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

Surface: Normals

Session 26

The First  Fundamental Forms

Session 27

The Second  Fundamental Forms

Session 28&29:

Project-5 :Find the normal to a given surface

Session 30 :

Tensor : Definitions and explanations

Vector Spaces

Session 32:

Free systems, Basis and Dimension

Session 33:

Suffix Conventions

Session 34:

Transformation law for change of Basis Vectors and Components

Dual Spaces

Session 36:

Transformation law for change of Basis in dual Space

Session 37:

Tensor Product of Vector Spaces.

Session 38:

Real Valued Bilinear Functions

Session 39:

Components of Tensor

Special  Tensors

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

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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 […]