By Roger Blanford, Kip Thorne
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Extra info for Applications of Classical Physics (web draft april 2013)
2 Foundational Concepts . . . . . . . . . . . . . . . 1 Inertial frames, inertial coordinates, events, vectors, and spacetime diagrams . . . . . . . . . . . . . . . . . 2 The Principle of Relativity and Constancy of Light Speed . . . 3 The Interval and its Invariance . . . . . . . . . . . 3 Tensor Algebra Without a Coordinate System . . . . . . . . . 4 Particle Kinetics and Lorentz Force Without a Reference Frame .
Iv) Compatibility with the metric (the fact that orthonormal parallelopipeds have volume ±1) means that ǫ is fully determined by its antisymmetry and a single sign: the choice of which parallelopipeds have positive volume and which negative. 8 **Example and Practice: Vectorial Identities for the Cross Product and Curl Here is an example of how to use index notation to derive a vector identity for the double cross product A ×(B ×C): In index notation this quantity is ǫijk Aj (ǫklm Bl Cm ). By permuting the indices on the second ǫ and then invoking Eq.
4a); tensor product (outer product), Eqs. 5); contraction, Eqs. 6) • Component representation of tensor algebra: Sec. 5 • Slot-naming index notation: Sec. 1 • Orthogonal transformations: Sec. 6 • Differentiation of tensors: Sec. 7 • Levi-Civita tensor: Sec. 7 – Relation to volume: Eqs. 24a) in 2 dimensions – Vector cross product and curl defined using Levi-Civita tensor: Eqs. 22) – Contraction of Levi-Civita tensor with itself: Eqs. 24b) in 2 dimensions • Integration of vectors; Gauss and Stokes theorems: Sec.