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A surface at each point of which a tangent plane can be defined. At each point of S, one can then define an orthonormal set consisting of two unit tangent vectors, t1 and t2 , lying on the tangent plane, and a unit normal vector, n, which is perpendicular to the tangent plane: n · n = t 1 · t1 = t2 · t2 = 1 and n · t1 = t 1 · t2 = t 2 · n = 0 . Obviously, there are two choices for n; the first is the vector t1 × t2 , |t1 × t2 | and the second one is just its opposite. Once one of these two vectors is chosen as the unit normal vector n, the surface is said to be oriented; n is then called the orientation of the surface.

77) ei · (ej ek ) ≡ (ei · ej ) ek = δij ek . 78) or Obviously, this operation is not commutative. Operations involving tensors are easily performed by expanding the tensors into components with respect to a given basis and using the elementary unit dyad operations defined in Eqs. 78). The most important operations involving tensors are summarized below. The single-dot product of two tensors If σ and τ are tensors, then  σ ·τ 3 =  3  σij ei ej  · i=1 j=1 3 3 3 3 τkl ek el k=1 l=1 3 3 σij τkl (ei ej ) · (ek el ) = i=1 j=1 k=1 l=1 3 3 3 3 = σij τkl δjk ei el i=1 j=1 k=1 l=1 3 3 3 = σij τjl ei el =⇒ i=1 j=1 l=1 3 3 σ ·τ = i=1 l=1   3  σij τjl  ei el .

75) This operation is not commutative. © 2000 by CRC Press LLC (ii) The double-dot product (or scalar product or inner product) of two unit dyads is a scalar defined by (ei ej ) : (ek el ) ≡ (ei · el ) (ej · ek ) = δil δjk . 76) It is easily seen that this operation is commutative. 77) ei · (ej ek ) ≡ (ei · ej ) ek = δij ek . 78) or Obviously, this operation is not commutative. Operations involving tensors are easily performed by expanding the tensors into components with respect to a given basis and using the elementary unit dyad operations defined in Eqs.

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