By Ralph Abraham

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Within the Spring of 1966, I gave a sequence of lectures within the Princeton college division of Physics, geared toward contemporary mathematical ends up in mechanics, particularly the paintings of Kolmogorov, Arnold, and Moser and its software to Laplace's query of balance of the sunlight process. Mr. Marsden's notes of the lectures, with a few revision and growth via either one of us, grew to become this booklet.

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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⎟ ⎟.