AMAT 12242: Vector Methods in Geometry

Course Code :  AMAT 12242

Title               :  Vector Methods in Geometry

Pre-requisites          :  AMAT 11223

 Learning Outcomes  :

At the end of the course the student will be able to 

1. state the standard vector forms of lines and planes
2. calculate the tangent, normal, binormal vectors of curves and the characteristics of curves such as
curvature, torsion and radius of curvature
3. explain osculating, normal ad rectifying planes of curves
4. apply Frenet-Serret formulae
5. calculate unit vectors and scalar factors of curvilinear coordinates
6. describe the form of differential equations in curvilinear coordinates
7. calculate tangent plane and normal vectors of surfaces.

Course Content  :

Lines and planes: 

• Vector form of lines and planes, Parameterized curves. Condition for planarity of two lines, bisecting the angle, between two given planes

      Curves and Surfaces: 

• Tangent, Normal and Binormal vectors, Curvature, torsion and radius of curvature, Frenet-Serret formulae, Osculating Plane, Normal Plane, Rectifying Plane

      Curvilinear Coordinates: 

• Scalar Factors and unit vectors for curvilinear coordinates, Differential operators in curvilinear coordinates, Length element, surface elements and volume elements in curvilinear coordinates, Scalar Factors and unit vectors for spherical and cylindrical coordinate systems.

      Theory of Surfaces: 

• Concept of a surface, Implicit Equation of a Surface, Parametric Equation, Parametric Curves, Curves on a Surface, Tangent Plane, Normal Vector, Surface of Revolution.

Method of Teaching and Learning : A combination of lectures and tutorial discussions.

Assessment          :  Based on tutorials, tests and end of course examination.

Recommended Reading:

1. Shanthi Narayan, Vector Algebra, S. Chands and Company, 2005

2. L, Brand. Vector Analysis. Courier Corporation, 2012

3. R. Shorter. Problems and Worked Solutions in Vector Analysis. Courier Corporation, 2014