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Course Description form
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Course Title
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English Code /No
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ARABIC code/no.
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credits
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Th.
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Pr.
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Tr.
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TCH
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Differential GEOMETRY
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math 463
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ر463
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3
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-
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-
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3
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Pre-requisites
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Math 312
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Brief contents, to be posted in university site and documents(4-5 lines):
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Manifolds- Differentiable manifolds – Differentiable mapping – Sub manifolds.
Tangent vector space- Tangent vector – Tangent space – Differential at a point. Tangent bundle- Vector fields on manifolds – Lie algebra structure. Cotangent bundle. Co vector fields – Tensor algebra. Exterior differential forms- Exterior form at a point – Differential forms on a manifold – Exterior differentiation – Orient able manifolds.
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Faculties and departments requiring this course (if any): Faculty of
Science/Department of Mathematics.
Objectives: Prefer In points
1- To provide the simple concept of the manifold and classifications, such as
Differentiable Manifold and Sub-manifold and highlighting the importance
of differential geometry in all the various sciences.
2 - Training the student to resolve the issues at Vector fields and tensor
algebra and differential forms on manifolds.
Contents: Prefer In points
1. Manifolds: Differentiable manifolds – Differentiable mapping – Sub
manifolds.
2. Tangent vector space: Tangent vector – Tangent space – Differential at
a point.
3. Tangent bundle: Vector fields on manifolds – Lie algebra structure.
4. Cotangent bundle: Co vector fields – Tensor algebra.
5. Exterior differential forms. : Exterior form at a point – Differential
forms on a manifold – Exterior differentiation – Orient able manifolds.
Course Outcomes:
A- Knowledge:
(Specific facts and knowledge of concepts, theories, formula etc.)
After studying this course a student will be able to recognize the differentiable manifolds and exterior differential forms. Also, the student be able to deal with formulas differential external models describe the application in various applied sciences, especially mechanics special and general relativity.
B-Cognitive Skills:
(Thinking, problem solving)
1. The students acquire the ability to find mathematical models that describe concrete world problems.
2. Applying the standard emanating criteria given in the course to much better understanding applied fields.
3. Recognizing the importance of mathematics as a wonderful and power tools to better understand what the students study
C- Interpersonal skills and responsibilities:
(group participation, leadership, personal responsibility , ethic and moral behavior, capacity for self directed learning)
The student will recognize that the various sciences form an integrated system and found on the other sides what help him to better understand his field and opens ways of cooperation with others from other specialties, and then recognizes the importance of teamwork.
D- Analysis and communication:
(communication, mathematical and IT skills)
Differential geometry highlights the importance of theoretical physics. This course help the student to better understand the physical phenomena through mathematical analysis.
Assessment methods for the above elements
According to the following elements:
Discussions, Homework, Periodic tests and final test
Text book: Only one
Book Title: Differential Geometry with Applications to Mechanics and Physics.
Publisher: Marcel Dekker, Inc.
Author: Y. Talpaert.
Year: 2000
Supplementary references
Book Title: Differentiable Manifolds: An Introduction.
Publisher: Van Nostrand Reinhold Company.
Authors: F. Brickell and R.S. Clark.
Year: 1970.
Book Title: An Introduction to Differential Manifolds and Riemannian Geometry.
Publisher: Academic Press, New York.
Author: W. Boothby.
Year: 2003.
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Time table for distributing Theoretical course contents
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Remarks
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Experiment
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weak
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Differentiable manifolds: Chart and local coordinates – Differentiable manifolds.
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1
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Differentiable mappings: Generalities of differentiable mappings – Particular differentiable mapping.
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2
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Sub manifolds: Sub manifolds on R^n – Sub manifolds of manifolds.
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3
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Tangent vectctor: Tangent curves – Tangent vector.
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4
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Tangent space: Definition of a tangent space – Basis of tangent space – change of basis.
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5
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Tangent bundle.
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6
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Vector fields on manifold: Definitions – Properties of vector fields.
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7
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Lie algebra structure: Bracket – Lie algebra – Lie derivative.
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8
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Tensor algebra: Tensor at a point – Tensor fields.
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9
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Exterior dorm at a point: Definition of a p-form – Exterior product of a p-form – Expression of a p-form – Exterior product of forms – Exterior algebra.
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10
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Differential forms on a manifold.
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11
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Exterior differentiation.
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12
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Riemannian manifolds.
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13
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Linear connection.
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14
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Solving some problems.
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15
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Final exam.
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