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By Arthur S. Lodge

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Extra info for Body Tensor Fields in Continuum Mechanics. With Applications to Polymer Rheology

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The unit tensors δ and S do not depend on the choice of B in the definition. Moreover, δ and δ are tensors at P whose components δ) have the same values at every particle P. The zero tensors have the same feature. In general tensor analysis, there is no covariant or contravariant unit tensor. -ψ = δ{. , the matrix inverses) of the transformation laws in (8), it is evident that the contravariant law goes over into the covariant law, and vice versa, but the mixed laws go over into themselves. This enables us to define the reciprocal of a covariant tensor as being that contravariant tensor whose representative matrix in any coordinate system B is the recip­ rocal of the representative matrix of the given tensor; the reciprocal tensor so defined is independent of the choice of B.

In a general Riemannian manifold, a Cartesian coordinate system does 50 2 THE BODY METRIC TENSOR FIELD not exist. 7(5); such a manifold is called Euclidean. The space manifold (in the nonrelativistic approximation) is assumed to be Euclidean. We shall show that it follows that the body manifold is also Euclidean; but it is clear that a body coordinate system that is rectangular Cartesian at one instant will not be rectangular Cartesian at other instants unless the body is moving rigidly so that the γ^(ξ, t) are independent of t.

Where σ(ζ) and σ(ξ) are the component functions representing the scalar field in any two body coordinate systems B and B, respectively. (12) Problem Prove that, for any given body scalar field s(P), there exists a covariant body vector field (called grad s) whose representative matrix in an arbitrary body coordinate system B is [οσ/οξ''], where σ = σ{ξ) is the com­ ponent function (assumed differentiable) of s(P) in B. We are here using the shortened notation ^(P) for a scalar field instead of the full notation {s(P) | P G ^ } .

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