By J. N. Reddy
This textbook on continuum mechanics displays the fashionable view that scientists and engineers might be informed to imagine and paintings in multidisciplinary environments. The publication is perfect for complex undergraduate and starting graduate scholars. The ebook beneficial properties: derivations of the elemental equations of mechanics in invariant (vector and tensor) shape and specializations of the governing equations to varied coordinate platforms; quite a few illustrative examples; chapter-end summaries; and workout difficulties to check and expand the certainty of innovations offered.
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Extra resources for An introduction to continuum mechanics: with applications
28) Next, we present several examples to illustrate the use of index notation to prove certain identities involving vector calculus. Establish the following identities using the index notation: ∇(r ) = rr . ∇(r n ) = nr n−2 r. ∇ × (∇ F) = 0. ∇ · (∇ F × ∇G) = 0. ∇ × (∇ × v) = ∇(∇ · v) − ∇ 2 v. div (A × B) = ∇ × A · B − ∇ × B · A. 1: 1. 2. 3. 4. 5. 6. SOLUTION: 1. Consider ∇(r ) = eˆ i = eˆ i 1 ∂r ∂ = eˆ i (x j x j ) 2 ∂ xi ∂ xi 1 1 1 r x (x j x j ) 2 −1 2xi = eˆ i xi (x j x j )− 2 = = , 2 r r (a) from which we note the identity ∂r xi = .
2. am bs = cm (ds − fs ). 2 Vector Algebra 21 3. ai = b j ci di . 4. xi xi = r 2 . 5. ai b j c j = 3. SOLUTION: 1. Not a valid expression because the free indices r and s do not match. 2. Valid; both m and s are free indices. There are nine equations (m, s = 1, 2, 3). 3. Not a valid expression because the free index j is not matched on both sides of the equality, and index i is a dummy index in one expression and a free index in the other; i cannot be used both as a free and dummy index in the same equation.
Thus, the expression in Eq. 34) and so on. As a rule, no index must appear more than twice in an expression. For example, Ai Bi Ci is not a valid expression because the index i appears more than twice. Other examples of dummy indices are Fi = Ai Bj C j , Gk = Hk (2 − 3Ai Bi ) + Pj Q j Fk . 2 Vector Algebra 19 The first equation above expresses three equations when the range of i and j is 1 to 3. We have F1 = A1 (B1 C1 + B2 C2 + B3 C3 ), F2 = A2 (B1 C1 + B2 C2 + B3 C3 ), F3 = A3 (B1 C1 + B2 C2 + B3 C3 ).
An introduction to continuum mechanics: with applications by J. N. Reddy