Differential Geometry ~ Bed Prasad Dhakal

Differential Geometry

Course title: Differential Geometry Nature of the course: Theory
Course no.: Math Ed.537 Credit hours: 3
Level: M.Ed. Teaching hours: 48
Semester: Third

1. Course Description

This course is designed to provide wider knowledge and skills on differential geometry for Math Educators. It comprises a range of skills varies from curves in space to intrinsic and extrinsic properties on surface. This course deals with curves and surfaces in 3-space using the tools of calculus and linear algebra. Topics covered in this course includes local and global properties of curves and surfaces. The course is divided in five major units. It starts with curves in space and then introduce some special curves. Then the course introduces surface and its fundamental form. Finally, the course deals with intrinsic and extrinsic properties on surface.

2. General Objectives

The general objectives of this course are as follows:

To utilize the concept of a space curve and its types in problem solving
To apply basic results of surface to solve related problems
To interpret the fundamental forms of surface
To explore and prove local properties on surface
To calculate and apply fundamental coefficients of surface in problem solving

3. Specific Objectives and Contents

Unit I: Curves in Space (10)

Learning Outcomes

To understand curves in space, and to find its class
To define, derive, and compute tangent line and its related theorems
To compute order of contact between curve and surface and apply it in problem solving
To explain osculating plane, derive its equation and apply it in problem solving and theorem proof
To analyze fundamentals of space curve and derive its equations
To define curvature and torsion and apply it in problem solving and theorem proof
To state and prove fundamental theorem of space curve

Learning Content

Unit II: Special Curves (9)

Learning Outcomes

To explain helix and prove its related theorems
To define osculating circle and analyze its properties
To analyze osculating sphere and prove its properties and related theorems
To compare Evolute and involute and compute its curvature and torsion
To state Bertrand curves and prove its properties

Learning Content

Unit III: Surface (10)

Learning Outcomes

To define surface and find its class
To analyze regular point, singular point
To explain parameter transformation and prove its geometric significance
To analyze tangent plane and normal line and use it in problem solving and theorem proof
To explain family of surface, and evaluate characteristic line, envelope, characteristic point and edge of regression, and to prove related theorems
To compare ruled surface and its kinds and use it in problem solving and theorem proof
To explore developable surface associated with space curves and prove related theorems

Learning Content

Surface and its Class
Regular and Singular Point
Transformation and its geometric significance
Tangent plane and normal line
Family of surface
The ruled surface
Developable surface

Unit IV: Fundamental Forms (9)

Learning Outcomes

To interpret first and second fundamental forms geometrically and apply them in proving theorems
o calculate first and second fundamental coefficients of surface
To prove Weingarten equations
To explain direction component and direction coefficient
To define family of curves and its differential equation
To explore orthogonal trajectories and its differential equation
To compare double family of curves and its orthogonality

Learning Content

Fundamental forms of surface
Fundamental coefficients of surface
Weingarten equations
Direction coefficients and related results
Family of curves
Orthogonal trajectories
Double family of curves

Unit V: Properties on Surface (10)

Learning Outcomes

To define intrinsic and non-intrinsic properties on surface
To explain normal curvature, principal curvature, normal section, principal section, principal direction and derive its differential equation
To prove Meusnier’s theorem
To define developable surface and prove related theorems
To understand line of curvatures and prove related theorems
To state and prove Rodrigue’s formula, Monge’s theorem, Euler’s theorem, Joachimsthal’s theorem
To compute conjugate direction and prove related theorems
To define asymptotic lines and prove related theorems
To explore fundamental equation and compute christoffel coefficients
To state and derive Gauss characteristic equation
To derive Mainardi-codazzi equation

Learning Content

Local non-intrinsic property of surface
Normal curvature and related theorems
Meusnier’s theorem
Developable surface
Minimal surface
Line of curvature and its properties
Rodrigue’s formula, Monge’s theorem, Euler’s theorem,
Joachimsthal’s theorem
Conjugate direction and its properties
Asymptotic lines and related theorems
The fundamental equation of surface theorem
Gauss characteristic equation
Mainardi-codazzi equation

Note: The figures in the parentheses indicate approximate teaching hours allocated for respective units.

4. Instructional Techniques

The instructor will select the method or methods of instruction most suitable for a topic. It is quite acceptable to select more than one method and combine them into a single period of instruction whenever it is needed. The general and specific instructional techniques are described below.

4.1 General Instructional Techniques:

The following general method of instruction will be adopted according to the need and nature of the lesson:

Lecture
Demonstration
Discussion
Group work

4.2 Specific Instructional Techniques

Multimedia presentation
Project work
Group Discussion

5. Evaluation

5.1 Internal Evaluation (40%)

Internal evaluation will be conducted by course teacher based on following activities

Activities	points
Attendance	5 marks
Participation in learning activities	5 marks
First assessment (assignment)	10 marks
Second assessment (written test)	10 marks
Third assessment (written test)	10 marks
Total	40 marks

5.2 External Examination (60%)

Examination Division, Office of the Dean, Faculty of Education will conduct final examination at the end of the semester. The number of questions and their types along with their marks allocated to each type will be as follows:

type	items	points
Objective questions (multiple choice)	(10 x 1)	10 marks
Short answer questions 6 with 2 OR questions	(6 x 5)	30 marks
Long answer questions 2 with 1 OR question	(2 x10)	20 marks
Total		60 marks

6. Recommended and References

6.1 Recommended Books

Gupta, P. P., Mallik, G. S & Pundir, S. K., (2011). Differential geometry. Meerut: Meerut Pragati Prakashan. (Units I -V)
Koirala S. P, & Dhakal B. P. (2068). Differential geometry. Sunlight Publication, Kirtipur, Nepal. (Units I -V)

6.2 Reference Books

Carmo, M. P. (1976) Differential geometry of curves and surfaces. Englewood Cliffs, NJ: Prentice-Hall (Units I & II)
Lal, B., (1969). The three-dimensional differential geometry. Delhi: Atma Ram and Sons. . (Units I-II)
Lipschutz, M. M., (2005). Theory and problems of differential geometry- Schaum’s outline series. Delhi: Tata McGraw-Hill Publishing Company Ltd. 6.
Wilmore, T. J., (2006). An introduction to differential geometry. Delhi: Oxford University Press.. (Units I -V)

Curriculum Implementation Plan

CREDIT HOURS [3]

Upon successful completion of this course, you will earn 3 semester hour of college credit, achieved through [16T+32S] 48 hours of student work. Student work may include, but is not limited to, direct or indirect faculty instruction, listening to class lectures or webinars (synchronous or asynchronous), attending a study group that is assigned by the institution, submitting an academic assignment (quiz, discussion, projects), taking an exam, laboratory work, externship or internship. It also includes preparation time such as for homework, reading and study time.

ASSIGNMENT POLICY

Students are expected to meet all deadlines relative to Quiz, discussions and assignments. The late submission will receive a 10% deduction on full marks.

CIP Summary || Math Ed 537

Week	Date	Unit	Lesson	Activity	Assignment
Week 1	2-16-18	1	Space curve	QDA	Assignment
Week 2	3-12-14	1	Osculating plane	QDA	Discussion
Week 3	3-19-21	1	Curvature/Torsion	QDA	Quiz
Week 4	3-26-28	2	Helix	QDA	Assignment
Week 5	4-2-4	2	Osculating Circle/Sphere	QDA	Discussion
Week 6	4-9-11	2	Involute/Involute Bertrand Curves	QDA	Quiz
Week 7	4-16-18	3	Surface and Class Regular and Singular Point Transformation and its geometric significance Tangent plane and Normal line	QDA	Assignment
Week 8	4-23-25	3	Family of surface	QDA	Discussion
Week 9	4-30-32	3	Ruled and Developable	QDA	Quiz
Week 10	5-5-7	4	Fundamental forms of surface coefficients	QDA	Assignment
Week 11	5-12-14	4	Weingarten equations Direction coefficients and related results Family of curves	QDA	Discussion
Week 12	5-19-20	4	Orthogonal trajectories Double family of curves	QDA	Quiz
Week 13	5-26-28	5	Properties	QDA	Assignment
Week 14	6-2-4	5	Developable and Minimal	QDA	Discussion
Week 15	6-9-11	5	Theorems 1. Rodrigue’s formula, Monge’s theorem, Euler’s theorem 2. Joachimsthal’s theorem 3. The fundamental equation of surface theorem 1. Gauss characteristic equation 2. Mainardi-codazzi equation	QDA	Quiz
Week 16	6-16-18	Review

QDA: Quiz, Discussion, Q/A demonstration

Lesson 1

LESSON 1 LEARNING OBJECTIVES (LO’s)

After completing this Lesson, students should be able to understand the following concepts:
 Define curve in space
 Give explicit and implicit example of space curve
 Execute and identify the class of space curve

LESSON 1 READING ASSIGNMENT

TEXTBOOK
 Chapter 1: Curves in Space
INSTRUCTOR WEEKLY VIDEO
 Review the instructor’s video for this Lesson.

LESSON 1 LESSON ACTIVITY

1. Write five example of space curve in parametric form
2. Write five example of space curve in implicit form

LESSON 1 Assignment

1. Write five example of space curve whose class are respectively 1,2,3,4,5.