60 TENSOR ANALYSIS Chap. 2 The right-hand side, the sum of the vectors gk multiplied by the scalar quantities (3 . gk) is exactly equal to the vector dgild63 itself. To see this, we note that since the set of vectors g', g2, g3 are linearly independent and form a basis of the vector space, dgild6j can be expressed as a linear combinat ion agi - = X1g' + X2g2 + X3g3 , d0j (7) where XI, X2, A3 are scalars. If we multiply this equation by gl, we obtain Similarly, X2, A3 can be evaluated. A comparison of Eqs. (6), (8), and (7) thus shows the truth of Eq. (2.15:4): Q.E.D. 2.16. PHYSICAL COMPONENTS OF A VECTOR The base vectors g, and g' are in general not unit vectors. In fact, their lgrl = 6, lgrl = m, T not summed. lengths are Let us write Eq. (2.14:12) as Then, since gr/& gT/m are unit vectors, all components
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