00073_a41d2b7d8c7e - xi are rectangular coordinates g = 1 Hence Eq(5 shows that d m =(axi/d8jj the Jacobian of the transformation Thus the volume

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52 TENSOR ANALYSIS Chap. 2 If p(8) in the last term is defined as the density distribution in the 0- coordinates, then it is a relative scalar of weight one. On the other hand, po[x(9)] = po(x) is an absolute scalar which defines the (physical) density of the medium. As another example, consider the determinant g of the Euclidean metric tensor gij whose transformation law is Let 3 = (gap(, the determinant of gap. By a double use of the formula for the product of two determinants when applied to Eq. (6), it is easy to prove that (7) which shows that g is a relative scalar of weight two. It follows that fi is a relative scalar of weight one. We note that if
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Unformatted text preview: xi are rectangular coordinates, g = 1. Hence Eq. (5) shows that d m = (axi/d8jj, the Jacobian of the transformation. Thus the volume enclosed by a closed surface can be written as The last integral shows the importance of fi in mechanics. The method of tensor equations does not apply to relative tensors. Therefore it is important to properly define all quantities involved in an equation to be tensors. 2.14. GEOMETRIC INTERPRETATION OF TENSOR COMPONENTS Before we conclude this chapter we shall consider briefly the geometric interpretation of tensor components. For this purpose we must use the concept of base vectors. We know that in a three-dimensional Euclidean...
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This note was uploaded on 01/04/2012 for the course ENG 501 taught by Professor Thomson during the Fall '05 term at MIT.

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