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Alternatively, if we take V = W = 0, Eq. 3a) R Similar results are obtained for U = W = 0 and U = V = 0. 3a) is called Gauss’s theorem. 3b) V ∂xi S where Fi = Fi (x1 , x2 , x3 ), V denotes volume, S denotes surface of volume V with unit normal vector n : (n1 , n2 , n3 ), and x1 ≡ x, x2 ≡ y, and x3 ≡ z. 4) or div a = φ div A + (grad φ) · A Accordingly, Eq. 5) R If, furthermore, the vector A is representable as the gradient of a scalar function ψ(A = grad ψ), then by Eq. 5), An = dψ/dn and div A = ∂ 2ψ ∂ 2ψ ∂ 2ψ + + 2 = ∇ 2ψ 2 2 ∂x ∂y ∂z Hence, for A = grad ψ, Eq.

For this reason, they are called orthogonality relations. 1 may be obtained as follows. Noting that the direction cosines of a unit vector with respect to (x, y, z) axes are identical to the projections of the unit vector on the coordinate axes, we regard the direction cosines (a11 , a12 , a13 ) as the components on (x, y, z) axes of a unit vector in the X direction. Similarly, (a21 , a22 , a23 ) and (a31 , a32 , a33 ) represent unit vectors in the Y direction and the Z direction, respectively.

2)]. 1) [or Eq. 3)] is called the quadratic form associated with the array [aij ]. Without loss of generality, the array may be assumed symmetrical; that is, we may set aij = aji . Then Eq. 4) where we have simply replaced the notation (a12 + a21 ) in Eq. 3) by 2a12 in Eq. 4), and so on.

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