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00068___3ba57f2c3369aaabcf129bea8f949711 - derivative terms...

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Sec. 2.11 PARTIAL DERIVATIVES IN CARTESIAN COORDINATES 47 this, let us consider two Cartesian coordinates (xi, ~2~x3) and (Zi, Z2,33) related by zi = a..x, + bi, (1) 23 3 where aij and bi are constants. From Eq. (l), we have (3) (4) Now, if ti(xl, ~2~x3) is a contravariant tensor, so that Then, on differentiating both sides of the equation, one obtains (5) When xi and zi are Cartesian coordinates, the last term vanishes according to Eq. (3). Hence, a? - ap axp a3.i (6) a~~ axg azj ax, . Thus, the set of partial derivatives a<"/axg follows the transformation law for a mixed tensor of rank two under a transformation from Cartesian
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Unformatted text preview: derivative terms in Eq. (5) which does not vanish in curvilinear coordinates shows that atcY/axp are not really the components of a tensor field in general coordinates. A similar situation holds obviously also for tensor fields of higher ranks. See Sec. 2.12 below. When Cartesian coordinates are used, we shall use a comma to denote partial differentiation. Thus, when we restrict ourselves to Cartesian coordinates, +,i, &,j, f f i j , k are ten- sors of rank one, two, three, respectively, provided that 4, &, aij are tensors. Warning: Cartesian tensor equations derived through the use of differ- entiation are in general not valid in curvilinear coordinates. This important point is discussed in the following five sections....
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