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4.5
The authors make a very strong, and successful, attempt to motivate the key tensor calculus concepts, in particular Christoffel symbols, the Riemann curvature tensor and scalar densities.The first 238 pages of "Tensors, differential forms, and variational principles", by David Lovelock and Hanno Rund, are metric-free. This book is very heavily into tensor subscripts and superscripts. If you don't like "coordinates", you won't like this book. Here's a round-up of the chapters.Chapter 1 (17 pages) has some interesting examples which demonstrate how tensors arise naturally, namely the (symmetric) stress tensor in elasticity, the (antisymmetric) inertia tensor for rigid bodies, and cross-product vectors (which arise in electromagnetism). Also discussed are vector components and the properties of determinants.Chapter 2 (36 pages) presents "affine tensor algebra in Euclidean geometry", which means basic tensor algebra in flat Euclidean spaces, including non-linear coordinate transformations. There's a very interesting explanation of how a metric tensor and Christoffel symbols naturally arise in flat space when parallel vector fields are subjected to non-linear transformations.Chapter 3 (47 pages) introduces manifolds (using an atlas of charts), including tensor algebra on manifolds and the derivatives of tensor fields, where once again the Christoffel symbol is introduced to make the derivatives tensorial, thereby motivating Christoffel symbols. Then there's more on absolute differentials (i.e. covariant derivatives) of tensor fields, the effects of multiple covariant differentiation (which motivates the definition of Riemannian curvature), parallelism on manifolds, and properties of the Riemannian curvature tensor.Chapter 4 (29 pages) has some miscellaneous tensor calculus topics, namely scalar densities (with transformation-invariant integrals), normal coordinates, and the Lie derivative.Chapter 5 (51 pages) is about differential forms, including exterior products, the exterior derivative, Poincaré's lemma, systems of total differential equations, the Stokes theorem, and curvature forms.Chapter 6 (58 pages) is concerned with "invariant problems in the calculus of variations".Chapter 7 (59 pages) introduces Riemannian geometry. This includes Finsler spaces and Riemannian and pseudo-Riemannian spaces. Topics include geodesics, Riemannian curvature tensor properties in the presence of a metric, and a divergence theorem for Riemannian manifolds.Chapter 8 (33 pages) is titled "invariant variational principles and physical field theories". This includes Lagrangians, vector field theory, metric field theory, and Einstein's equations.The authors have made great efforts to explain and motivate everything.
