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72) x,j = xi g i ,j = xi ,j g i + xi g i ,j = xi ,j g i − xi Γikj g k = xi ,j −xk Γkij g i . 62) yields xi |j = xi ,j +xk Γikj , xi |j = xi ,j −xk Γkij , i, j = 1, 2, . . , n. 84) Similarly, we treat the tensor-valued function A = A θ1 , θ2 , . . , θn : A,k = Aij g i ⊗ g j ,k = Aij ,k g i ⊗ g j + Aij g i ,k ⊗g j + Aij g i ⊗ g j ,k = Aij ,k g i ⊗ g j + Aij Γlik g l ⊗ g j + Aij g i ⊗ Γljk g l = Aij ,k +Alj Γilk + Ail Γjlk g i ⊗ g j . 6 Applications in Three-Dimensional Space: Divergence and Curl 47 Thus, Aij |k = Aij ,k +Alj Γilk + Ail Γjlk , i, j, k = 1, 2, .

N. 86) i, j, k = 1, 2, . . , n. 87) By analogy, we further obtain Aij |k = Aij ,k −Alj Γlik − Ail Γljk , Ai·j |k = Ai·j ,k +Al·j Γilk − Ai·l Γljk , Similar expressions for the covariant derivative can also be formulated for tensors of higher orders. 76) coincides with the partial derivative: xi |j = xi ,j , xi |j = xi ,j , Aij |k = Aij ,k , Aij |k = Aij ,k , Ai·j |k = Ai·j ,k , i, j, k = 1, 2, . . , n. 91) where i, j, k = 1, 2, . . , n. 93) Aij |k i i = a |k bj + a bj |k for Aij i = a bj , i, j, k = 1, 2, .

125) into account we immediately come to the contradiction. 37) T A−1 = AT −1 = A−T . 126) The composition of two arbitrary invertible tensors A and B is inverted by −1 (AB) = B−1 A−1 . 127) Indeed, let y = ABx. 124) x = B−1 A−1 y, ∀x ∈ En . On the basis of transposition and inversion one defines the so-called orthogonal tensors. They do not change after consecutive transposition and inversion and form the following subset of Linn : Orthn = Q ∈ Linn : Q = Q−T . 125) QQT = QT Q = I, ∀Q ∈ Orthn . 71) is orthogonal.

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