By Roger Blanford, Kip Thorne
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Extra info for Applications of Classical Physics (web draft april 2013)
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34), consider a 2-surface Σ = An with area A oriented perpendicular to some arbitrary unit vector n. e. it is a normal force with magnitude equal to the fluid pressure P times the surface area A. This is what it should be. , the stress tensor is the flux of momentum. 29). The total momentum in V3 is V3 GdV , where G is the momentum density. This changes as a result of momentum flowing into and out of V3 . The net rate at which momentum flows outward is the integral of the stress tensor over the surface ∂V3 of V3 .
12b) The orthonormality requirement for the two bases implies that δij = ei · ej = (ep¯Rp¯i ) · (eq¯Rq¯j ) = Rp¯i Rq¯j (ep¯ · eq¯) = Rp¯i Rq¯j δp¯q¯ = Rp¯i Rp¯j . This says that the transpose of [Rp¯i ] is its inverse—which we have already denoted by [Ri¯p ]; [Ri¯p ] ≡ Inverse ([Rp¯i ]) = Transpose ([Rp¯i ]) . , Goldstein, Poole and Safko (2002)]. Thus (as should be obvious and familiar), the bases associated with any two Euclidean coordinate systems are related by a reflection or rotation. Note: Eq.
4a) In Euclidean space, this is the standard inner product, familiar from elementary geometry. 4a) is a real-valued linear function of each of its vectors. Therefore, we can regard it as a tensor of rank 2. When so regarded, the inner product is denoted g( , ) and is called the metric tensor. In other words, the metric tensor g is that linear function of two vectors whose value is given by g(A, B) ≡ A · B . , one gets the same real number independently of the order in which one inserts the two vectors into the slots: g(A, B) = g(B, A) .
Applications of Classical Physics (web draft april 2013) by Roger Blanford, Kip Thorne
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