By Paul Renteln

Delivering a succinct but complete therapy of the necessities of recent differential geometry and topology, this book's transparent prose and casual type make it obtainable to complex undergraduate and graduate scholars in arithmetic and the actual sciences. The textual content covers the fundamentals of multilinear algebra, differentiation and integration on manifolds, Lie teams and Lie algebras, homotopy and de Rham cohomology, homology, vector bundles, Riemannian and pseudo-Riemannian geometry, and measure concept. It additionally gains over 250 specified workouts, and various purposes revealing primary connections to classical mechanics, electromagnetism (including circuit theory), basic relativity and gauge conception. options to the issues can be found for teachers at www.cambridge.org/9781107042193

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**Extra resources for Manifolds, Tensors, and Forms: An Introduction for Mathematicians and Physicists**

**Example text**

Amm Bn1 A1m B12 A1m B22 . . A1m Bn2 . . . Amm B12 Amm B22 .. Amm Bn2 ··· ··· .. ··· ··· ··· .. ··· A1m B1n ⎞ A1m B2n ⎟ ⎟ . ⎟ . ⎟ . ⎟ A1m Bnn ⎟ ⎟ ⎟ . ⎟ .. ⎟ ⎟. ⎟ . ⎟ . ⎟ Amm B1n ⎟ ⎟ Amm B2n ⎟ ⎟ ⎟ .. ⎠ . Amm Bnn Now let A : V → V and B : W → W be linear maps and let v ∈ V and w ∈ W . The tensor product map A ⊗ B : V ⊗ W → V ⊗ W is naturally defined by (A ⊗ B)(v ⊗ w) = Av ⊗ Bw, extended by linearity. Pick a basis {e1 , . . , em } for V and { f 1 , . . , f n } for W .

The same idea works in general. For example, the components of u ⊗ v ⊗ w are just u i v j wk and so on. It is perhaps worth observing that a tensor of the form v ⊗ w for some vectors v and w is not the most general order-2 tensor. The reason is that the most general order-2 tensor has nine algebraically independent components, whereas vi w j has only six algebraically independent components (three from each vector). For example, v2 w1 = (v1 w1 )(v2 w2 )/v1 w2 . 1 Some examples of tensors you may have already encountered include the inertia tensor and the electromagnetic field strength tensor.

62) becomes σ = cg(e J , e J )σ. It follows that c = g(e J , e J ), so we may write e I = g(e J , e J )e J . 63) A similar argument shows that e J = bg(e I , e I )e I for some constant b, determined as follows. We have (−1) p(n− p) σ = (−1) p(n− p) e I ∧ e J = e J ∧ e I = g( e J , e I )σ = g(e I , e J )σ, so that (−1) p(n− p) = g(e I , e J ) = b[g(e I , e I )]2 = b and thus e J = (−1) p(n− p) g(e I , e I )e I . 55). 61). 4 Let V = M4 and let (0, 0, 1, 0), and e3 = (0, 0, 0, 1). Thus ⎛ −1 ⎜ 0 (gi j ) = ⎜ ⎝ 0 0 e0 = (1, 0, 0, 0), e1 = (0, 1, 0, 0), e2 = 0 1 0 0 0 0 1 0 ⎞ 0 0⎟ ⎟.