Course Syllabus
Textbook: Textbook: O'Neill, Barrett : Elementary differential geometry. Academic Press, 2nd edition (1997) ; ISBN: 0125267452
Objectives of the course: This is
an introductory course on the geometry of curves and
surfaces and higher dimensional submanifolds in Euclidian spaces.
This course is also meant to provide the student an insight
that will help visualize and better understand facts of linear
algebra, multivariate calculus and differential equations, and
physics and astronomy .
The differential calculus has two fundamental continuations:
differential equations and differential geometry. These two
theories are intertwined and are both needed in the comprehension
of continuous phenomena. This course is meant to fill a gap
in the mathematics curriculum in our college, and help the
student improve his knowledge of mathematics and its applications
as a whole.
Course description: Curves in plane and in space, the Frenet equations of curve in general position. Surfaces in parametric and implicit form. Parallel transport and geodesics. The Gauss and map, the shape operator and the total curvature of a surface. The Gauss-Codazzi equations. Surfaces of constant curvature and non-Euclidean geometry. Mean curvature and minimal surfaces. Focal points and caustics. The Gauss Bonnet Theorem. Submanifolds in the Euclidian space and their tangent spaces. Vector fields, and differential forms and the theorems of Frobenius and Stokes. Applications to Physics.
Technology: Maple, a powerful mathematical software will be utilized . Here is a typical example of a closed surface in differential geometry obtained with Maple:
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