By J. N. Reddy

This textbook on continuum mechanics displays the fashionable view that scientists and engineers could be expert to imagine and paintings in multidisciplinary environments. The e-book is perfect for complicated undergraduate and starting graduate scholars. The ebook positive factors: derivations of the fundamental equations of mechanics in invariant (vector and tensor) shape and specializations of the governing equations to varied coordinate structures; a variety of illustrative examples; chapter-end summaries; and workout difficulties to check and expand the certainty of ideas awarded.

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**Sample text**

2. If a is a constant vector, show that a · dx = 0, C a × dx = 0. 3. Transform the surface integrals I= (n × ∇) · udS, I= S (n × ∇) × udS S into volume integrals and evaluate them. 4. Obtain the differential equations for the vector function φ and the scalar ψ inside an arbitrary volume if the surface integrals n × (r φ)da = 0, S n · (rψ)da = 0, S where r = (xi xi )1/2 and r = x. 5. In the torsion of shafts, the St. Venant warping function φ satisfies the Laplace equation in a simply connected 2D domain D in the x, y plane, representing the cross-section of the shaft and the boundary conditions, n · ∇φ = k · n × r, where n is the unit normal to the boundary, r = xi + y j, and k is the unit vector perpendicular to the plane.

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. 13. If T is a two-dimensional, second-rank, unsymmetrical tensor and n is a twodimensional vector (two-vector), compare n · T · n, (T · n) · n, and n · (n · T).

3 Example: Polar Decomposition Suppose we want to factor the matrix ⎡ 17 1⎢ B = ⎣ 19 5 0 −11 23 0 ⎤ 0 ⎥ 0⎦. 136) V = BRT . The eigenvalues of C are λ1 = 9, λ2 = 16, λ3 = 36. Using the expansion λi = c0 + c1 λi + c2 λi2 , we have the simultaneous equations 3 = c0 + 9c1 + 92 c2 , 4 = c0 + 16c1 + 162 c2 , 6 = c0 + 36c1 + 362 c2 . The solutions of this system are c0 = 52/35, c1 = 23/126, c2 = −1/630. Writing C1/2 = c0 I + c1 C + c2 C2 , we get ⎡ ⎤ 5 1 0 ⎢ ⎥ U = C1/2 = ⎣ 1 5 0 ⎦ , 0 0 3 ⎡ ⎤ 4 3 0 1⎢ ⎥ R = ⎣ −3 4 0 ⎦ , 5 0 0 5 ⎡ ⎤ 101 7 0 1 ⎢ ⎥ V= 149 0 ⎦ .