The objective of the module is to introduce and develop the methods of vector analysis. These methods provide a natural aid to the understanding of geometry and some physical concepts. They are also a fundamental tool in many theories of Applied Mathematics
1.1 Vector Algebra and Geometry:Vector addition and scalar multiplication. Scalar and vector products. Equations of lines and planes. Curves and surfaces; parametric andnon-parametric equations of curves and surfaces., 1.2 , 1.3 Vector Differentiation:Differentiation of vector valued functions with respect to a scalar. Geometry of curves. Scalar and vector fields. Gradient of a scalarfield, and divergence and curl of a vector field. Sum and product rules for these differentiation operators. Second order vector operators. Directional derivatives. Normal and tangent plane to a surface. Solenoidal and irrotational fields., 1.4 , 1.5 Vector Integration:Curvilinear line integrals. Surface integrals. The divergence theorem, Green’s theorem and Stoke’s theorem., 1.6 , 1.7 Curvilinear Coordinate Systems:Coordinate free vector derivatives. Vector derivatives in curvilinear coordinates. Spherical, polar and cylindrical coordinates, 1.8 , 1.9 Potential Theory:Gradient fields. Rotation fields. Harmonic functions. Helmholtz’s fundamental theorem of vector calculus.
By the end of the course, students should be able to do the following:
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SCQF Level: 9
Credits: 15