By J. N. Reddy
This textbook on continuum mechanics displays the trendy view that scientists and engineers might be proficient to imagine and paintings in multidisciplinary environments. The booklet is perfect for complex undergraduate and starting graduate scholars. The publication good points: derivations of the elemental equations of mechanics in invariant (vector and tensor) shape and specializations of the governing equations to varied coordinate structures; various illustrative examples; chapter-end summaries; and workout difficulties to check and expand the knowledge of options awarded.
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Additional resources for An introduction to continuum mechanics: with applications
23) Likewise, we would find that (A × B) × C = m2 A + n2 B. 24) Thus, the parentheses cannot be interchanged or removed. It can be shown that m1 = A · C, n1 = −A · B, and hence that A × (B × C) = (A · C)B − (A · B)C. 25) The example below illustrates the use of the vector triple product. 1: Let A and B be any two vectors in space. , parallel) and perpendicular to vector B. The component of A along B is given by (A · eˆ B), where eˆ B = B/B is the unit vector in the direction of B. 7. (a) Plane area as a vector.
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. The equation would have been valid if i on the left side of the equation is replaced with j; then there will be three equations. 4. A valid expression, containing one equation: x12 + x22 + x32 = r 2 . 5. A valid expression; it contains three equations (i = 1, 2, 3): a1 b1 c1 + a1 b2 c2 + a1 b3 c3 = 3, a2 b1 c1 + a2 b2 c2 + a2 b3 c3 = 3, and a3 b1 c1 + a3 b2 c2 + a3 b3 c3 = 3.
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 = . ∂ xi r (b) 2. Similar to 1, we have ∇(r n ) = eˆ i ∂ ∂r = nr n−2 xi eˆ i = nr n−2 r. (r n ) = nr n−1 eˆ i ∂ xi ∂ xi 3. Consider the expression ∇ × (∇ F) = eˆ i ∂ ∂ xi × eˆ j ∂F ∂xj = ei jk eˆ k ∂2 F . ∂ xi ∂ x j 10:34 P1: JzG Chapter02 CUFX197-Reddy 978 0 521 87044 3 October 3, 2007 38 Vectors and Tensors Note that ∂ ∂xi ∂Fx j is symmetric in i and j.
An introduction to continuum mechanics: with applications by J. N. Reddy