This chapter is meant to be a student’s first introduction to tensors. Self-contained and complete, the student learns how tensors are defined, written, and used. The scalar and vector products are defined along with the physical meaning of the divergence and curl differential operations that act on tensors of any order. The integro-differential theorems are introduced in three dimensions, which include the fundamental theorem of calculus in three dimensions, Stokes’ theorem and the Reynolds’ transport theorem. The student learns how to derive a long list of tensor-calculus product rules that are valid in any coordinate system. The Taylor series in three-dimensional space is derived, which involves tensors of all orders. Functions of second-order tensors are defined. Isotropic tensors of all tensorial orders are obtained and used in proving Curie’s principle for the constitutive laws in an isotropic material. Tensor calculus in orthogonal curvilinear coordinates is developed. Finally, the Dirac delta function is introduced along with its integral and differential properties and uses.

Wepresent the theory as a number of postulates about a mathematical model for spacetime.

In §3.1 we introduce the mathematical model and in §3.2 the first two postulates, local causality and local energy conservation. These postulates are common to both special and general relativity, and thus may be regarded as tested by the many experiments that have been performed to check the former. In §3.3 we derive the equations of the matter fields and obtain the energy–momentum tensor from a Lagrangian.

The third postulate, the field equations, is given in §3.4. This is not so well established experimentally as the first two postulates, but we shall see that any alternative equations would seem to have one or more undesirable properties, or else require the existence of extra fields which have not yet been detected experimentally.

The mathematical model we shall use for spacetime, i.e. the collection of all events, is a pair (ℳ, g) where ℳ is a connected four-dimensional Hausdorff C∞ manifold and g is a Lorentz metric (i.e. a metric of signature + 2) on ℳ.

We discuss the action of a generic group on a complex vector space, representing the elements of the group as matrices.

A very short and simple introduction to vectors and tensors and some related quantities.

Tensor notation, the mechanics of particles and fields in the Lagrangian formulation.

Chapter 2 contains the problem statements of the 150 problems in general relativity theory. The chapter is divided into 12 sections with problems organized by different topics defined by the keywords in the section headings.

This last part of the book introduces the Einstein equation – the basic equation of general relativity, in much the same way that Maxwell’s equations are the basic equations of electromagnetism. Geometries such as the Schwarzschild geometry, or those of the FRW cosmological models, are particular solutions of the Einstein equation. Just three new mathematical ideas are needed to give an efficient and standard discussion of the Einstein equation: a more precise definition of vectors in terms of directional derivatives; the notion of dual vectors as a linear map from vectors to real numbers; and the covariant derivative of a vector field in curved spacetime. These mathematical concepts are introduced in this chapter.

Vector and matrix calculus provides a powerful set of tools for analying and manipulating scalars, vectors, and tensors in continuum mechanics.This includes transformations between coordinate systems and provides the foundation for optimization methods.

Glaciers are classified according to size, shape, and temperature. Temperate glaciers are at the pressure melting point throughout, whereas polar glaciers are below the pressure melting point except, in some places, at the bed. Ice is basically incompressible, which simplifies many analyses. Stresses, strains, and strain rates are second rank tensor quantities, so nine quantities are needed to describe them. Ice flows in response to stresses in excess of hydrostatic, or deviatoric stresses. To a good approximation, the strain rate, “ε”̇, can be described by “ε” ̇= A“σ” ^n in which A is a temperature-dependent rate factor, “σ”isthe stress, and n is a constant, usually taken to be 3.

In the previous chapter, it was shown that an aligned composite is usually stiff along the fibre axis, but much more compliant in the transverse directions. Sometimes, this is all that is required. For example, in a slender beam, such as a fishing rod, the loading is often predominantly axial and transverse or shear stiffness are not important. However, there are many applications in which loading is distributed within a plane: these range from panels of various types to cylindrical pressure vessels. Equal stiffness in all directions within a plane can be produced using a planar random assembly of fibres. This is the basis of chopped-strand mat. However, demanding applications require material with higher fibre volume fractions than can readily be achieved in a planar random (or woven) array. The approach adopted is to stack and bond together a sequence of thin ‘plies’ or ‘laminae’, each composed of long fibres aligned in a single direction, into a laminate. It is important to be able to predict how such a construction responds to an applied load. In this chapter, attention is concentrated on the stress distributions that are created and the elastic deformations that result. This involves consideration of how a single lamina deforms on loading at an arbitrary angle to the fibre direction. A summary is given first of some matrix algebra and analysis tools used in elasticity theory.