Lecturer: Dr. T.G. Murphy
Date: 1995-96
Groups: Optional JS and SS Mathematics, SS Two-subject Moderatorship
Prerequisites:
Duration: 21 weeks
Lectures per week: 3
Assessment:
Examinations: Examined in 3 parts, one examination covering each term's work
(Provisional)
This course is divided into 2 parts:
Tensor algebra may be considered as an extension of Linear Algebra. Linear algebra is concerned mainly with vectors, which are tensors of type (1,0), and linear maps, which are tensors of type (1,1). It also deals with other types of tensor, eg linear functionals, of type (0,1), and determinants, of type (0,n).
Tensor algebra consists of the study of tensors of all types (m,n), linked by the tensor operations of addition, tensor product and trace. Classically, a tensor of type (m,n) is represented by an array with m upper and n lower indices.
Where a quadratic (or symmetric bilinear) form is given, the distinction between upper and lower indices disappears, and we are left with the simpler concept of a cartesian system, in which there are tensors of type 0 (scalars), 1 (vectors), 2, 3, 4, etc.
Tensor calculus is the study of tensor fields on manifolds or varieties. It is the fundamental concept behind general relativity. The primary tool is that of the covariant derivative, leading (in the cartesian case) from a field f of type n to a field Df of type n+1. The torsion and curvature are tensor fields associated to the derivative.