00052___991d30f68d3ff865e407fcc0373c18ca - (ad2(a2I2(w3)2 =...

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Sec. 2.1 NOTATION AND SUMMATION CONVENTION 31 to the dummy variable in an integral b s,” f(z)dx = s, f(Y)dY. The use of index and summation convention may be illustrated by other examples. Consider a unit vector u in a three-dimensional Euclidean space with rectangular Cartesian coordinates x, y, and z. Let the direction cosines ai be defined as a1 = cos(u, x) , a2 = cos(u, y) , a3 = cos(u, z) , where (v,x) denotes the angle between u and the x-axis, etc. The set of numbers ai(i = 1,2,3) represents the projections of the unit vector on the coordinate axes. The fact that the length of the vector is unity is expressed
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Unformatted text preview: (ad2 + (a2I2 + (w3)2 = 1, or simply (4) aiai = 1. As a further illustration, consider a line element (dx, dy, d z ) in a three- dimensional Euclidean space with rectangular Cartesian coordinates x, y, z. The square of the length of the line element is (5) ds2 = dx2 + dy2 + d z 2 . If we define (6) dx’ = d x , dx2 = dy , dx3 = d z , and Then (5) may be written as (8) ds2 = bijdxidx’, with the understanding that the range of the indices i and j is 1 to 3. Note that there are two summations in the expression above, one over i and one over j. The symbol 6ij as defined in Eq. (7) is called the Kroneclcer delta....
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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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