By I. Madsen, B. Oliver
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Extra info for Algebraic Topology Aarhus 1982. Proc. conf. Aarhus, 1982
We recall that a groupoid is said to be transitive if for each ordered pair of points of the base manifold there exists at least one groupoid element with one of the points as the source and the other as the target. In the case of the material groupoid, transitivity means, therefore, that each body point is materially isomorphic to every other body point. In other words, the uniformity of a body corresponds exactly to the notion of a transitive groupoid, namely, a groupoid whereby none of the sets Pij is empty.
We then calculate the jet of this composite function and obtain, by deﬁnition, the composition of the given jets. To illustrate this operation, consider the case k = 2. , p) be chosen, respectively, around the points X ∈ M, f (X) ∈ N and g(f (X)) ∈ P. , xn ), 2 where the indices take values in the appropriate ranges. 5) 46 2 Uniformity of second-grade materials B Archetype Fig. 5. Second-grade homogeneity a total of n + mn + m2 n numbers. 7) f (X) ∂2zα ∂xj ∂xi . 8) f (X) 2 To obtain the components of the composite jet jf2(X) g ◦ jX f , we consider the α i I composite functions z (x (x )) and take its ﬁrst and second derivatives.
11) X It is this peculiar composition formula for second (and indeed higher) derivatives that conﬁrms our previous observation as to the necessity of including all lower order derivatives in the deﬁnition of jets. 12) where we have economized the indication of the explicit dependence on X since it is already included in the deﬁnition of the jet. 13) where we have used the notation: F iIJ ≡ ∂F iI ∂ 2 xi = = F iJI . 13) we are not including the explicit spatial argument xi (which is certainly included in the deﬁnition of jet) because of the requirement of translation invariance of the energy.
Algebraic Topology Aarhus 1982. Proc. conf. Aarhus, 1982 by I. Madsen, B. Oliver