By J. N. Reddy

This textbook on continuum mechanics displays the fashionable view that scientists and engineers will be expert to imagine and paintings in multidisciplinary environments. The e-book is perfect for complex undergraduate and starting graduate scholars. The booklet good points: derivations of the elemental equations of mechanics in invariant (vector and tensor) shape and specializations of the governing equations to numerous coordinate platforms; a variety of illustrative examples; chapter-end summaries; and workout difficulties to check and expand the knowledge of ideas offered.

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3. The boundary curve C¯ would be mapped into C. 19) that satisfies the Stokes theorem, S¯ ∂ A¯ 1 ∂ A¯ 2 − dy1 dy2 = ∂ y1 ∂ y2 y2 C¯ [ A¯ 1 dy1 + A¯ 2 dy2 ]. 3. Mapping a 2D surface onto a 3D surface. 20) 46 Integral Theorems To distinguish the partial derivatives with respect to the y coordinates from those with respect to the x coordinates, we use the notation ∂α = ∂ , ∂ yα α = 1, 2; ∂i = ∂ , ∂ xi i = 1, 2, 3. 21) Next, consider a three-vector Ai (x1 , x2 , x3 ), defined on the surface S, which has unit normal n.

For a twice-differentiable, vector-valued function v, show that ∇ × (∇ × v) = ∇(∇ · v) − (∇ · ∇)v. 10. 16 25 (c) ∇ · ∇r n = n(n + 1)r (n−2) . (d) If F is any differentiable function, show that ∇ × [F (r )r] = 0. 11. Show that the differential equation (∇ 2 + k2 )ψ = 0, in three dimensions, admits the solution ψ= e±ikr . 12. If A and B are tensors of rank m and n, respectively, in N dimensions, show that (a) the rank of A · B is m + n − 2, (b) the rank of A × B is m + n − 1, and (c) the rank of A B is m + n.

86) Using index notation, we have Thus the stress components do transform in accordance with the transformation law for Cartesian tensors of rank 2. Using tensor notation, we have σ = σi j ei e j , σ = σi j ei e j , σ = Q · σ · QT . 6 Tensors 17 When the tensors involved are of the second rank or lower, we use matrix notation and write the preceding relation as σ = Qσ QT . 89) The following results are useful in dealing with tensor equations: 1. If all the components of a tensor vanish in one coordinate system, they vanish in all coordinate systems.