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Analytical Elements of Mechanics by Thomas R. Kane

By Thomas R. Kane

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8, [a, b, c] can be expressed in the following determinantal form: [a, b, c] = «1 0,2 6i b, as b3 Ci Ci ca Proof: Resolve each of the vectors a, b and c into three mutually perpendicular components, and carry out the operations indicated by a-(b X c). Expand the determinant given above. Compare the results. Problem: Referring to Fig. 6b, evaluate [F, ni, n 2 ]. 17 The vector triple product of three vectors a , b, c: a X (bXe) The expression a X (b X c) denotes the cross product of the vectors a and b X c.

T

From Eqs. (1), (2) and (3), I pdr P* =V- (4) The integral j dr gives the total length, area, or volume of F, that is, JFdr = r (5) (This follows from Eq. (3) and the fact that the total length, area, )r volume of F is the sum of the lengths, areas, or volumes of the elements r», i = 1, . . ) Substitute from Eq. (5) into Eq. (4): P' = -ipdr (6) Eq. (6) is of no use unless one knows how to evaluate the integral in the right-hand member. 2 and Eq. (2) to reduce the integral to a sum involving integrals of scalar functions, the theory of integration of scalar functions being presumed known.

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