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 ObjectivesThe 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

- Space curve and its class
- Tangent to the space curve
- Order of contact
- Osculating plane
- Fundamentals on space curve
- Curvature, torsion and screw curvature
- Intrinsic equation

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

- Cylindrical helix
- Osculating circle and osculating sphere
- Evolute and involutes
- Bertrand curves and its properties

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

4. Instructional TechniquesThe 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 activitiesActivities | 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)

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