Φr 2 s rφ ˆ e φ s rθ r 1 r s φθ φ 1 r sin φ

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φR + 2 S ] ˆ e φ + ∂S ∂R + 1 R ∂S φθ ∂φ + 1 R sin φ ∂S θθ ∂θ + 1 R [( S φθ + S θφ ) cot φ + 2 S + S θR ] ˆ e θ . Solution: Using the del operator in the spherical coordinate system (see Table 2.4.2), the divergence of the tensor S is computed as ˆ e R ∂R + ˆ e φ 1 R ∂φ + ˆ e θ 1 R sin φ ∂θ · [ S RR ˆ e R ˆ e R + S ˆ e R ˆ e φ + S φR ˆ e φ ˆ e R + · · · ] = ∂S RR ∂R ˆ e R + ∂S ∂R ˆ e φ + ∂S ∂R ˆ e θ + 1 R S RR ˆ e R + ( S + S φR e φ + ∂S φR ∂φ ˆ e R + S ˆ e θ + ∂S φφ ∂φ ˆ e φ - S φφ ˆ e R + ∂S φθ ∂φ ˆ e θ + 1 R [ S RR ˆ e R + S ˆ e φ + ( S + S θR e θ - S θθ ˆ e R ] + cot φ R [ S φR ˆ e R + S φφ ˆ e φ + ( S φθ + S θφ e θ - S θθ ˆ e φ ] + 1 R sin φ ∂S θR ∂θ ˆ e R + ∂S θφ ∂θ ˆ e φ + ∂S θθ ∂θ ˆ e θ , where the following derivatives of the base vectors are used: ˆ e θ ∂θ = - sin φ ˆ e R - cos φ ˆ e φ , ˆ e R ∂θ = sin φ ˆ e θ , ˆ e φ ∂φ = - ˆ e R , ˆ e φ ∂θ = cos φ ˆ e θ , ˆ e R ∂φ = ˆ e φ . Collecting the coefficients of ˆ e R , ˆ e φ , and ˆ e θ , we obtain the required result. 2.52 Show that u in the spherical coordinate system is given by u = ∂u R ∂R ˆ e R ˆ e R + ∂u φ ∂R ˆ e R ˆ e φ + ∂u θ ∂R ˆ e R ˆ e θ + 1 R ∂u R ∂φ - u φ ˆ e φ ˆ e R + 1 R ∂u φ ∂φ + u R ˆ e φ ˆ e φ + 1 R ∂u θ ∂φ ˆ e φ ˆ e θ + 1 R sin φ ∂u R ∂θ - u θ sin φ ˆ e θ ˆ e R + ∂u φ ∂θ - u θ cos φ ˆ e θ ˆ e φ + ∂u θ ∂θ + u R sin φ + u φ cos φ ˆ e θ ˆ e θ . Solution: Using the del operator in the spherical coordinate system (see Table 2.4.2), the gradient of the vector u is computed as ˆ e R ∂R + ˆ e φ 1 R ∂φ + ˆ e θ 1 R sin φ ∂θ ( u R ˆ e R + u φ ˆ e φ + u θ ˆ e θ ) = ˆ e R ∂u R ∂R ˆ e R + ∂u φ ∂R ˆ e φ + ∂u θ ∂R ˆ e θ + 1 R ˆ e φ ∂u R ∂φ ˆ e R + u R ˆ e φ + ∂u φ ∂φ ˆ e φ - u φ ˆ e R + ∂u θ ∂φ ˆ e θ + 1 R sin φ ˆ e θ h ∂u R ∂θ ˆ e R + u R sin φ ˆ e φ + ∂u φ ∂θ ˆ e φ + u φ cos φ ˆ e θ + ∂u θ ∂θ ˆ e θ + u θ ( - sin φ ˆ e R - cos φ ˆ e φ ) i , where the following derivatives of the base vectors are used: ˆ e θ ∂θ = - sin φ ˆ e R - cos φ ˆ e φ , ˆ e R ∂θ = sin φ ˆ e θ , ˆ e φ ∂φ = - ˆ e R , ˆ e φ ∂θ = cos φ ˆ e θ , ˆ e R ∂φ = ˆ e φ .
CHAPTER 2: VECTORS AND TENSORS 31 Collecting the coefficients of ˆ e R ˆ e R , ˆ e R ˆ e φ , ˆ e R ˆ e θ and so on, we obtain the given expres- sion for the gradient of a vector in the spherical coordinate system. 2.53 Prove the following identities when A and B are vectors and S , R , and T are second- order tensors: (a) tr( AB ) = A · B . (b) tr( S T ) = tr S . (c) tr( R · S ) = R · · S . (d) tr( R T · S ) = R : S . (e) tr( R · S ) = tr( S · R ) . (f) tr( R · S · T ) = tr( T · R · S ) = tr( S · T · R ) .

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