00053___d43b9593d6b45a31b5255725bcdf41b0 - nected by the...

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32 TENSOR ANALYSIS Chap. 2 all a12 a13 a21 a22 a23 a31 a32 a33 alla22a33 + a21a32a13 + a31a12a23 - a11a32a23 - a12a21a33 - a13a22a31 . = If we denote the general term in the determinant by aij and write the determinant as laijl, then the equation above can be written as where erst, the permutation symbol, is defined by the equations In other words, eijk vanishes whenever the values of any two indices coincide; eijk = 1 when the subscripts permute like 1,2,3; and eijk = -1 otherwise. The Kronecker delta and the permutation symbol are very important quantities which will appear again and again in this book. They are con-
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Unformatted text preview: nected by the identity This e-6 identity is used sufficiently frequently to warrant special attention here. It can be verified by actual trial. Finally, we shall extend the summation convention to differentiation formulas. Let f (xl, x2,. . . , P) be a function of n variables x1 , x2,. . . , xn. Then its differential shall be written as The Kronecker delta and the permutation symbol play important roles in vector or tensor operations in rectangular Cartesian coordinate system. The dot product of two vectors x = ( ~ 1 ~ x 2 , x3) and y = (yl, y2, y3) is a scalar c = 2 . y = ZlYl + x2y2 + x3y3,...
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