00074___b74786fc9e9a0b9195e19a8de792109d - (1) dR = dx'i, =...

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SeC. 2.14 GEOMETRIC INTERPRETATION OF TENSOR COMPONENTS 53 space any three linearly independent vectors form a basis with which any other vectors can be expanded as a lin- ear combination of these three vectors. When a rectangular Cartesian frame of reference is chosen, we can choose as base vectors the unit vectors il, i2, i3 parallel to the coordinate axes: thus, if a vector A has three components (Al, A2, A3), we can write A = Aiil + A2i2 + A&. In a curvilinear coordinate system in a Euclidean space, we shall introduce the base vectors from the following considerat ion. Fig. 2.14:l. Base vectors. Let dR denote an infinitesimal vector PQ joining a point P = (xI,x~,x~) to apoint Q = (XI +dxl,x2+dx2,~3+dxg) where XI,X~,X~ are referred to a rectangular Cartesian frame of reference. (See Figure 2.14:l.) Then, obviously,
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Unformatted text preview: (1) dR = dx'i, = dx,i' , where il, i2, is, or i', i2, i3 denote the base vectors along coordinate axes. Here, since a rectangular Cartesian coordinate system is used, we can as- sign arbitrarily an index as contravariant or covariant. In the assignment chosen above, dx' and dx, are, respectively, contravariant and covariant differentials. Now let us consider a transformation from the rectangular Cartesian coordinates xi to general coordinates Bi. According to the tensor transformation law, when dxT and dx, are regarded as tensors of order one, their components in the general coordinates become (3) (4) Now in (3), d0j can be identified as the usual differential of the variable 8j, as specified in Eq. (2); hence, the superscript is justified. But in Eq. (4), dOj, is not to be identified with the usual differential; d0j is a covariant...
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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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