7.7 Summary

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$\begin{array}{l} {((\vec{F} \cdot \vec{\nabla}) \vec{G})}_{ρ} & = & sin θ cos φ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ cos φ + \frac{\partial G_{θ}}{\partial ρ} cos θ cos φ - \frac{\partial G φ}{\partial ρ} sin φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ cos φ + G_{ρ} cos θ cos φ + \frac{\partial G_{θ}}{\partial θ} cos θ cos φ - G_{θ} sin θ cos φ \\ - \frac{\partial G_{φ}}{\partial θ} sin φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ cos φ - G_{ρ} sin θ sin φ + \\ \frac{\partial G_{θ}}{\partial φ} cos θ cos φ - G_{θ} cos θ sin φ - \frac{\partial G_{φ}}{\partial φ} sin φ - G_{φ} cos φ)) + \\ sin θ sin φ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ sin φ + \frac{\partial G_{θ}}{\partial ρ} cosθ sin φ + \frac{\partial G_{φ}}{\partial ρ} cos φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ sin φ + G_{ρ} cos θ sin φ + \frac{\partial G_{θ}}{\partial θ} cos θ sin φ \\ - G_{θ} sin θ sin φ + \frac{\partial G_{φ}}{\partial θ} cos φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ sin φ + \\ G_{ρ} sin θ cos φ + \frac{\partial G_{θ}}{\partial φ} cos θ sin φ + G_{θ} cos θ cos φ + \frac{\partial G_{φ}}{\partial φ} cos φ - G_{φ} sin φ)) + \\ cos θ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} cos θ - \frac{\partial G_{θ}}{\partial ρ} sin θ) + \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} cos θ - G_{ρ} sin θ \\ - \frac{\partial G_{θ}}{\partial θ} sin θ - G_{θ} cos θ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} cos θ - \frac{\partial G_{θ}}{\partial φ} sin θ)) \\ = & F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} {sin}^{2} θ {cos}^{2} φ + \frac{\partial G_{θ}}{\partial ρ} sin θ cos θ {cos}^{2} φ - \frac{\partial G_{φ}}{\partial ρ} sin θ sin φ cos φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} s i n^{2} θ {cos}^{2} φ + G_{ρ} sin θ cos θ {cos}^{2} φ + \frac{\partial G_{θ}}{\partial θ} sin θ cos θ {cos}^{2} φ \\ - G_{θ} {sin}^{2} θ {cos}^{2} φ - \frac{\partial G_{φ}}{\partial θ} sin θ sin φ cos φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} {sin}^{2} θ {cos}^{2} φ \\ - G_{ρ} {sin}^{2} θ sin φ cos φ + \frac{\partial G_{θ}}{\partial φ} sin θ cos θ {cos}^{2} φ - G_{θ} sin θ cos θ sin φ cos φ \\ - \frac{\partial G_{φ}}{\partial φ} sin θ sin φ cos φ - G_{φ} sin θ {cos}^{2} φ) + \\ F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} {sin}^{2} θ {cos}^{2} φ + \frac{\partial G_{θ}}{\partial ρ} sin θ cos θ {sin}^{2} φ + \frac{\partial G_{φ}}{\partial ρ} sin θ sin φ cos φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} s i n^{2} θ {sin}^{2} φ + G_{ρ} cos θ sin θ {sin}^{2} φ + \frac{\partial G_{θ}}{\partial θ} sin θ cos θ {sin}^{2} φ \\ - G_{θ} {sin}^{2} θ {sin}^{2} φ - \frac{\partial G_{φ}}{\partial θ} sin θ sin φ cos φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} {sin}^{2} θ {sin}^{2} φ + \\ G_{ρ} {sin}^{2} θ sin φ cos φ + \frac{\partial G_{θ}}{\partial φ} sin θ cos θ {sin}^{2} φ + G_{θ} sin θ cos θ sin φ cos φ + \\ \frac{\partial G_{φ}}{\partial φ} sin θ sin φ cos φ - G_{φ} sin θ {sin}^{2} φ) + \\ F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} {cos}^{2} θ - \frac{\partial G_{θ}}{\partial ρ} sin θ cos θ) + \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} {cos}^{2} θ - G_{ρ} sin θ cos φ \\ - \frac{\partial G_{θ}}{\partial θ} sin θ cos θ - G_{θ} {cos}^{2} θ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} {cos}^{2} θ - \frac{\partial G_{θ}}{\partial φ} sin θ cos θ) \\ = & F_{ρ} \frac{\partial G_{ρ}}{\partial ρ} + \frac{F_{θ}}{ρ} \frac{\partial G_{ρ}}{\partial θ} + \frac{F_{φ}}{ρ sin θ} \frac{\partial G_{ρ}}{\partial φ} - \frac{F_{θ} G_{θ} + F_{φ} G_{φ}}{ρ} \end{array}$ $\begin{array}{l} {((\vec{F} \cdot \vec{\nabla}) \vec{G})}_{ρ} & = & sin θ cos φ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ cos φ + \frac{\partial G_{θ}}{\partial ρ} cos θ cos φ - \frac{\partial G φ}{\partial ρ} sin φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ cos φ + G_{ρ} cos θ cos φ + \frac{\partial G_{θ}}{\partial θ} cos θ cos φ - G_{θ} sin θ cos φ \\ - \frac{\partial G_{φ}}{\partial θ} sin φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ cos φ - G_{ρ} sin θ sin φ + \\ \frac{\partial G_{θ}}{\partial φ} cos θ cos φ - G_{θ} cos θ sin φ - \frac{\partial G_{φ}}{\partial φ} sin φ - G_{φ} cos φ)) + \\ sin θ sin φ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ sin φ + \frac{\partial G_{θ}}{\partial ρ} cosθ sin φ + \frac{\partial G_{φ}}{\partial ρ} cos φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ sin φ + G_{ρ} cos θ sin φ + \frac{\partial G_{θ}}{\partial θ} cos θ sin φ \\ - G_{θ} sin θ sin φ + \frac{\partial G_{φ}}{\partial θ} cos φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ sin φ + \\ G_{ρ} sin θ cos φ + \frac{\partial G_{θ}}{\partial φ} cos θ sin φ + G_{θ} cos θ cos φ + \frac{\partial G_{φ}}{\partial φ} cos φ - G_{φ} sin φ)) + \\ cos θ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} cos θ - \frac{\partial G_{θ}}{\partial ρ} sin θ) + \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} cos θ - G_{ρ} sin θ \\ - \frac{\partial G_{θ}}{\partial θ} sin θ - G_{θ} cos θ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} cos θ - \frac{\partial G_{θ}}{\partial φ} sin θ)) \\ = & F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} {sin}^{2} θ {cos}^{2} φ + \frac{\partial G_{θ}}{\partial ρ} sin θ cos θ {cos}^{2} φ - \frac{\partial G_{φ}}{\partial ρ} sin θ sin φ cos φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} s i n^{2} θ {cos}^{2} φ + G_{ρ} sin θ cos θ {cos}^{2} φ + \frac{\partial G_{θ}}{\partial θ} sin θ cos θ {cos}^{2} φ \\ - G_{θ} {sin}^{2} θ {cos}^{2} φ - \frac{\partial G_{φ}}{\partial θ} sin θ sin φ cos φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} {sin}^{2} θ {cos}^{2} φ \\ - G_{ρ} {sin}^{2} θ sin φ cos φ + \frac{\partial G_{θ}}{\partial φ} sin θ cos θ {cos}^{2} φ - G_{θ} sin θ cos θ sin φ cos φ \\ - \frac{\partial G_{φ}}{\partial φ} sin θ sin φ cos φ - G_{φ} sin θ {cos}^{2} φ) + \\ F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} {sin}^{2} θ {cos}^{2} φ + \frac{\partial G_{θ}}{\partial ρ} sin θ cos θ {sin}^{2} φ + \frac{\partial G_{φ}}{\partial ρ} sin θ sin φ cos φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} s i n^{2} θ {sin}^{2} φ + G_{ρ} cos θ sin θ {sin}^{2} φ + \frac{\partial G_{θ}}{\partial θ} sin θ cos θ {sin}^{2} φ \\ - G_{θ} {sin}^{2} θ {sin}^{2} φ - \frac{\partial G_{φ}}{\partial θ} sin θ sin φ cos φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} {sin}^{2} θ {sin}^{2} φ + \\ G_{ρ} {sin}^{2} θ sin φ cos φ + \frac{\partial G_{θ}}{\partial φ} sin θ cos θ {sin}^{2} φ + G_{θ} sin θ cos θ sin φ cos φ + \\ \frac{\partial G_{φ}}{\partial φ} sin θ sin φ cos φ - G_{φ} sin θ {sin}^{2} φ) + \\ F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} {cos}^{2} θ - \frac{\partial G_{θ}}{\partial ρ} sin θ cos θ) + \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} {cos}^{2} θ - G_{ρ} sin θ cos φ \\ - \frac{\partial G_{θ}}{\partial θ} sin θ cos θ - G_{θ} {cos}^{2} θ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} {cos}^{2} θ - \frac{\partial G_{θ}}{\partial φ} sin θ cos θ) \\ = & F_{ρ} \frac{\partial G_{ρ}}{\partial ρ} + \frac{F_{θ}}{ρ} \frac{\partial G_{ρ}}{\partial θ} + \frac{F_{φ}}{ρ sin θ} \frac{\partial G_{ρ}}{\partial φ} - \frac{F_{θ} G_{θ} + F_{φ} G_{φ}}{ρ} \end{array}$

si410_e (Eq. 7.140)

Proceeding likewise for the θ -component we find

$\begin{array}{l} {((\vec{F} \cdot \vec{\nabla}) \vec{G})}_{θ} & = & cos θ cos φ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ cos φ + \frac{\partial G_{θ}}{\partial ρ} cos θ cos φ - \frac{\partial G_{φ}}{\partial ρ} sin φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ cos φ + G_{ρ} cos θ cos φ + \frac{\partial G_{θ}}{\partial θ} cos θ cos φ - G_{θ} sin θ cos φ \\ - \frac{\partial G_{φ}}{\partial θ} sin φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ cos φ - G_{ρ} sin θ sin φ + \\ \frac{\partial G_{θ}}{\partial φ} cos θ cos φ - G_{θ} cos θ sin φ - \frac{\partial G_{φ}}{\partial φ} sin φ - G_{φ} cos φ)) + \\ sin φ cos θ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ sin φ + \frac{\partial G_{θ}}{\partial ρ} cosθ sin φ + \frac{\partial G_{φ}}{\partial ρ} cos φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ sin φ + G_{ρ} cos θ sin φ + \frac{\partial G_{θ}}{\partial θ} cos θ sin φ - G_{θ} sin θ sin φ + \\ \frac{\partial G_{φ}}{\partial θ} cos φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ sin φ + G_{ρ} sin θ cos φ + \\ \frac{\partial G_{θ}}{\partial φ} cos θ sin φ + G_{θ} cos θ cos φ + \frac{\partial G_{φ}}{\partial φ} cos φ - G_{φ} sin φ)) \\ - sin θ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} cos θ - \frac{\partial G_{θ}}{\partial ρ} sin θ) + \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} cos θ - G_{ρ} sin θ \\ - \frac{\partial G_{θ}}{\partial θ} sin θ - G_{θ} cos θ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} cos θ - \frac{\partial G_{θ}}{\partial φ} sin θ)) \\ = & F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ cos θ {cos}^{2} φ + \frac{\partial G_{θ}}{\partial ρ} {cos}^{2} θ {cos}^{2} φ - \frac{\partial G_{φ}}{\partial ρ} cos θ sin φ cos φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ cos θ {cos}^{2} φ + G_{ρ} {cos}^{2} θ {cos}^{2} φ + \frac{\partial G_{θ}}{\partial θ} {cos}^{2} θ {cos}^{2} φ \\ - G_{θ} sin θ cos θ {cos}^{2} φ - \frac{\partial G_{φ}}{\partial θ} cos θ sin φ cos φ) + \\ \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ cos θ {cos}^{2} φ - G_{ρ} sin θ cos θ sin φ cos φ + \\ \frac{\partial G_{θ}}{\partial φ} {cos}^{2} θ {cos}^{2} φ - G_{θ} {cos}^{2} θ sin φ cos φ - \frac{\partial G_{φ}}{\partial φ} cos θ sin φ cos φ \\ - G_{φ} cos θ {cos}^{2} φ) + \\ F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ cos θ {sin}^{2} φ + \frac{\partial G_{θ}}{\partial ρ} {cos}^{2} θ {sin}^{2} φ + \frac{\partial G_{φ}}{\partial ρ} cos θ sin φ cos φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ cos θ {sin}^{2} φ + G_{ρ} {cos}^{2} θ {sin}^{2} φ + \frac{\partial G_{θ}}{\partial θ} {cos}^{2} θ {sin}^{2} φ \\ - G_{θ} sin θ cos θ {sin}^{2} φ + \frac{\partial G_{φ}}{\partial θ} cos θ sin φ cos φ) + \\ \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ cos θ {sin}^{2} φ + G_{ρ} sin θ cos θ sin φ cos φ + \\ \frac{\partial G_{θ}}{\partial φ} {cos}^{2} θ {sin}^{2} φ + G_{θ} {cos}^{2} θ sin φ cos φ + \frac{\partial G_{φ}}{\partial φ} cos θ sin φ cos φ \\ - G_{φ} cos θ {sin}^{2} φ) \\ - F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sinθ cos θ - \frac{\partial G_{θ}}{\partial ρ} {sin}^{2} θ) \\ - \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sinθ cos θ - G_{ρ} {sin}^{2} θ - \frac{\partial G_{θ}}{\partial θ} {sin}^{2} θ - G_{θ} sinθ cos θ) \\ - \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sinθ cos θ - \frac{\partial G_{θ}}{\partial φ} {sin}^{2} θ) \\ = & F_{ρ} \frac{\partial G_{θ}}{\partial ρ} + \frac{F_{θ}}{ρ} \frac{\partial G_{θ}}{\partial θ} + \frac{F_{φ}}{ρ sin θ} \frac{\partial G_{θ}}{\partial φ} \frac{F_{θ} G_{ρ}}{ρ} - \frac{F_{φ} G_{φ}}{ρ} \frac{cos θ}{sin θ} \end{array}$ $\begin{array}{l} {((\vec{F} \cdot \vec{\nabla}) \vec{G})}_{θ} & = & cos θ cos φ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ cos φ + \frac{\partial G_{θ}}{\partial ρ} cos θ cos φ - \frac{\partial G_{φ}}{\partial ρ} sin φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ cos φ + G_{ρ} cos θ cos φ + \frac{\partial G_{θ}}{\partial θ} cos θ cos φ - G_{θ} sin θ cos φ \\ - \frac{\partial G_{φ}}{\partial θ} sin φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ cos φ - G_{ρ} sin θ sin φ + \\ \frac{\partial G_{θ}}{\partial φ} cos θ cos φ - G_{θ} cos θ sin φ - \frac{\partial G_{φ}}{\partial φ} sin φ - G_{φ} cos φ)) + \\ sin φ cos θ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ sin φ + \frac{\partial G_{θ}}{\partial ρ} cosθ sin φ + \frac{\partial G_{φ}}{\partial ρ} cos φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ sin φ + G_{ρ} cos θ sin φ + \frac{\partial G_{θ}}{\partial θ} cos θ sin φ - G_{θ} sin θ sin φ + \\ \frac{\partial G_{φ}}{\partial θ} cos φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ sin φ + G_{ρ} sin θ cos φ + \\ \frac{\partial G_{θ}}{\partial φ} cos θ sin φ + G_{θ} cos θ cos φ + \frac{\partial G_{φ}}{\partial φ} cos φ - G_{φ} sin φ)) \\ - sin θ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} cos θ - \frac{\partial G_{θ}}{\partial ρ} sin θ) + \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} cos θ - G_{ρ} sin θ \\ - \frac{\partial G_{θ}}{\partial θ} sin θ - G_{θ} cos θ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} cos θ - \frac{\partial G_{θ}}{\partial φ} sin θ)) \\ = & F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ cos θ {cos}^{2} φ + \frac{\partial G_{θ}}{\partial ρ} {cos}^{2} θ {cos}^{2} φ - \frac{\partial G_{φ}}{\partial ρ} cos θ sin φ cos φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ cos θ {cos}^{2} φ + G_{ρ} {cos}^{2} θ {cos}^{2} φ + \frac{\partial G_{θ}}{\partial θ} {cos}^{2} θ {cos}^{2} φ \\ - G_{θ} sin θ cos θ {cos}^{2} φ - \frac{\partial G_{φ}}{\partial θ} cos θ sin φ cos φ) + \\ \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ cos θ {cos}^{2} φ - G_{ρ} sin θ cos θ sin φ cos φ + \\ \frac{\partial G_{θ}}{\partial φ} {cos}^{2} θ {cos}^{2} φ - G_{θ} {cos}^{2} θ sin φ cos φ - \frac{\partial G_{φ}}{\partial φ} cos θ sin φ cos φ \\ - G_{φ} cos θ {cos}^{2} φ) + \\ F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ cos θ {sin}^{2} φ + \frac{\partial G_{θ}}{\partial ρ} {cos}^{2} θ {sin}^{2} φ + \frac{\partial G_{φ}}{\partial ρ} cos θ sin φ cos φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ cos θ {sin}^{2} φ + G_{ρ} {cos}^{2} θ {sin}^{2} φ + \frac{\partial G_{θ}}{\partial θ} {cos}^{2} θ {sin}^{2} φ \\ - G_{θ} sin θ cos θ {sin}^{2} φ + \frac{\partial G_{φ}}{\partial θ} cos θ sin φ cos φ) + \\ \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ cos θ {sin}^{2} φ + G_{ρ} sin θ cos θ sin φ cos φ + \\ \frac{\partial G_{θ}}{\partial φ} {cos}^{2} θ {sin}^{2} φ + G_{θ} {cos}^{2} θ sin φ cos φ + \frac{\partial G_{φ}}{\partial φ} cos θ sin φ cos φ \\ - G_{φ} cos θ {sin}^{2} φ) \\ - F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sinθ cos θ - \frac{\partial G_{θ}}{\partial ρ} {sin}^{2} θ) \\ - \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sinθ cos θ - G_{ρ} {sin}^{2} θ - \frac{\partial G_{θ}}{\partial θ} {sin}^{2} θ - G_{θ} sinθ cos θ) \\ - \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sinθ cos θ - \frac{\partial G_{θ}}{\partial φ} {sin}^{2} θ) \\ = & F_{ρ} \frac{\partial G_{θ}}{\partial ρ} + \frac{F_{θ}}{ρ} \frac{\partial G_{θ}}{\partial θ} + \frac{F_{φ}}{ρ sin θ} \frac{\partial G_{θ}}{\partial φ} \frac{F_{θ} G_{ρ}}{ρ} - \frac{F_{φ} G_{φ}}{ρ} \frac{cos θ}{sin θ} \end{array}$

si411_e (Eq. 7.141)

Finally we find for the φ -component

$\begin{array}{l} {((\vec{F} \cdot \vec{\nabla}) \vec{G})}_{θ} & = & - sin φ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ cos φ + \frac{\partial G_{θ}}{\partial ρ} cos θ cos φ - \frac{\partial G_{φ}}{\partial ρ} sin φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ cos φ + G_{ρ} cos θ cos φ + \frac{\partial G_{θ}}{\partial θ} cos θ cos φ \\ - G_{θ} sin θ cos φ - \frac{\partial G_{φ}}{\partial θ} sin φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ cos φ \\ - G_{ρ} sin θ sin φ + \frac{\partial G_{θ}}{\partial φ} cos θ cos φ - G_{θ} cos θ sin φ - \frac{\partial G_{φ}}{\partial φ} sin φ - G_{φ} cos φ)) + \\ cos φ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ sin φ + \frac{\partial G_{θ}}{\partial ρ} cosθ sin φ + \frac{\partial G_{φ}}{\partial ρ} cos φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ sin φ + G_{ρ} cos θ sin φ + \frac{\partial G_{θ}}{\partial θ} cos θ sin φ \\ - G_{θ} sin θ sin φ + \frac{\partial G_{φ}}{\partial θ} cos φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ sin φ + \\ G_{ρ} sin θ cos φ + \frac{\partial G_{θ}}{\partial φ} cos θ sin φ + G_{θ} cos θ cos φ + \frac{\partial G_{φ}}{\partial φ} cos φ - G_{φ} sin φ)) \\ = & - F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ sin φ cos φ + \frac{\partial G_{θ}}{\partial ρ} cos θ sin φ cos φ - \frac{\partial G_{φ}}{\partial ρ} {sin}^{2} φ) \\ - \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ sin φ cos φ + G_{ρ} cos θ sin φ cos φ + \\ \frac{\partial G_{θ}}{\partial θ} cos θ sin φ cos φ - G_{θ} sin θ sin φ cos φ - \frac{\partial G_{φ}}{\partial θ} {sin}^{2} φ) \\ - \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ sin φ cos φ - G_{ρ} sin θ {sin}^{2} φ + \\ \frac{\partial G_{θ}}{\partial φ} cos θ sin φ cos φ - G_{θ} cos θ {sin}^{2} φ - \frac{\partial G_{φ}}{\partial φ} {sin}^{2} φ - G_{φ} sin φ cos φ) + \\ F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ sin φ cos φ + \frac{\partial G_{θ}}{\partial ρ} cos θ sin φ cos φ + \frac{\partial G_{φ}}{\partial ρ} {cos}^{2} φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ sin φ cos φ + G_{ρ} cos θ sin φ cos φ + \\ \frac{\partial G_{θ}}{\partial θ} cos θ sin φ cos φ - G_{θ} sin θ cos φ cos φ + \frac{\partial G_{φ}}{\partial θ} {cos}^{2} φ) + \\ \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ sin φ cos φ + G_{ρ} sin θ {cos}^{2} φ + \\ \frac{\partial G_{θ}}{\partial φ} cos θ sin φ cos φ + G_{θ} cos θ {cos}^{2} φ + \frac{\partial G_{φ}}{\partial φ} {cos}^{2} φ - G_{φ} sin φ cos φ) \\ = & F_{ρ} \frac{\partial G_{φ}}{\partial ρ} + \frac{F_{θ}}{ρ} \frac{\partial G_{φ}}{\partial θ} + \frac{F_{φ}}{ρ sin θ} \frac{\partial G_{φ}}{\partial φ} + \frac{F_{φ} G_{ρ}}{ρ} + \frac{F_{φ} G_{θ}}{ρ} \frac{cos θ}{sin θ} \end{array}$ $\begin{array}{l} {((\vec{F} \cdot \vec{\nabla}) \vec{G})}_{θ} & = & - sin φ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ cos φ + \frac{\partial G_{θ}}{\partial ρ} cos θ cos φ - \frac{\partial G_{φ}}{\partial ρ} sin φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ cos φ + G_{ρ} cos θ cos φ + \frac{\partial G_{θ}}{\partial θ} cos θ cos φ \\ - G_{θ} sin θ cos φ - \frac{\partial G_{φ}}{\partial θ} sin φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ cos φ \\ - G_{ρ} sin θ sin φ + \frac{\partial G_{θ}}{\partial φ} cos θ cos φ - G_{θ} cos θ sin φ - \frac{\partial G_{φ}}{\partial φ} sin φ - G_{φ} cos φ)) + \\ cos φ (F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ sin φ + \frac{\partial G_{θ}}{\partial ρ} cosθ sin φ + \frac{\partial G_{φ}}{\partial ρ} cos φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ sin φ + G_{ρ} cos θ sin φ + \frac{\partial G_{θ}}{\partial θ} cos θ sin φ \\ - G_{θ} sin θ sin φ + \frac{\partial G_{φ}}{\partial θ} cos φ) + \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ sin φ + \\ G_{ρ} sin θ cos φ + \frac{\partial G_{θ}}{\partial φ} cos θ sin φ + G_{θ} cos θ cos φ + \frac{\partial G_{φ}}{\partial φ} cos φ - G_{φ} sin φ)) \\ = & - F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ sin φ cos φ + \frac{\partial G_{θ}}{\partial ρ} cos θ sin φ cos φ - \frac{\partial G_{φ}}{\partial ρ} {sin}^{2} φ) \\ - \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ sin φ cos φ + G_{ρ} cos θ sin φ cos φ + \\ \frac{\partial G_{θ}}{\partial θ} cos θ sin φ cos φ - G_{θ} sin θ sin φ cos φ - \frac{\partial G_{φ}}{\partial θ} {sin}^{2} φ) \\ - \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ sin φ cos φ - G_{ρ} sin θ {sin}^{2} φ + \\ \frac{\partial G_{θ}}{\partial φ} cos θ sin φ cos φ - G_{θ} cos θ {sin}^{2} φ - \frac{\partial G_{φ}}{\partial φ} {sin}^{2} φ - G_{φ} sin φ cos φ) + \\ F_{ρ} (\frac{\partial G_{ρ}}{\partial ρ} sin θ sin φ cos φ + \frac{\partial G_{θ}}{\partial ρ} cos θ sin φ cos φ + \frac{\partial G_{φ}}{\partial ρ} {cos}^{2} φ) + \\ \frac{F_{θ}}{ρ} (\frac{\partial G_{ρ}}{\partial θ} sin θ sin φ cos φ + G_{ρ} cos θ sin φ cos φ + \\ \frac{\partial G_{θ}}{\partial θ} cos θ sin φ cos φ - G_{θ} sin θ cos φ cos φ + \frac{\partial G_{φ}}{\partial θ} {cos}^{2} φ) + \\ \frac{F_{φ}}{sin θ ρ} (\frac{\partial G_{ρ}}{\partial φ} sin θ sin φ cos φ + G_{ρ} sin θ {cos}^{2} φ + \\ \frac{\partial G_{θ}}{\partial φ} cos θ sin φ cos φ + G_{θ} cos θ {cos}^{2} φ + \frac{\partial G_{φ}}{\partial φ} {cos}^{2} φ - G_{φ} sin φ cos φ) \\ = & F_{ρ} \frac{\partial G_{φ}}{\partial ρ} + \frac{F_{θ}}{ρ} \frac{\partial G_{φ}}{\partial θ} + \frac{F_{φ}}{ρ sin θ} \frac{\partial G_{φ}}{\partial φ} + \frac{F_{φ} G_{ρ}}{ρ} + \frac{F_{φ} G_{θ}}{ρ} \frac{cos θ}{sin θ} \end{array}$

si412_e (Eq. 7.142)

Reassembling Eq. 7.140, Eq. 7.141, and Eq. 7.142 yields the material derivative in cylindrical coordinates given by

$\begin{array}{l} (\vec{F} \cdot \vec{\nabla}) \vec{G} & = & (F_{ρ} \frac{\partial G_{ρ}}{\partial ρ} + \frac{F_{θ}}{ρ} \frac{\partial G_{ρ}}{\partial θ} + \frac{F_{φ}}{ρ sin θ} \frac{\partial G_{ρ}}{\partial φ} - \frac{F_{θ} G_{θ} + F_{φ} G_{φ}}{ρ}) {\vec{e}}_{ρ} + \\ (F_{ρ} \frac{\partial G_{θ}}{\partial ρ} + \frac{F_{θ}}{ρ} \frac{\partial G_{θ}}{\partial θ} + \frac{F_{φ}}{ρ sin θ} \frac{\partial G_{θ}}{\partial φ} + \frac{F_{θ} G_{ρ}}{ρ} - \frac{F_{φ} G_{φ}}{ρ} \frac{cos θ}{sin θ}) {\vec{e}}_{θ} + \\ (F_{ρ} \frac{\partial G_{φ}}{\partial ρ} + \frac{F_{θ}}{ρ} \frac{\partial G_{φ}}{\partial θ} + \frac{F_{φ}}{ρ sin θ} \frac{\partial G_{φ}}{\partial φ} + \frac{F_{φ} G_{ρ}}{ρ} + \frac{F_{φ} G_{θ}}{ρ} \frac{cos θ}{sin θ}) {\vec{e}}_{φ} \end{array}$ $\begin{array}{l} (\vec{F} \cdot \vec{\nabla}) \vec{G} & = & (F_{ρ} \frac{\partial G_{ρ}}{\partial ρ} + \frac{F_{θ}}{ρ} \frac{\partial G_{ρ}}{\partial θ} + \frac{F_{φ}}{ρ sin θ} \frac{\partial G_{ρ}}{\partial φ} - \frac{F_{θ} G_{θ} + F_{φ} G_{φ}}{ρ}) {\vec{e}}_{ρ} + \\ (F_{ρ} \frac{\partial G_{θ}}{\partial ρ} + \frac{F_{θ}}{ρ} \frac{\partial G_{θ}}{\partial θ} + \frac{F_{φ}}{ρ sin θ} \frac{\partial G_{θ}}{\partial φ} + \frac{F_{θ} G_{ρ}}{ρ} - \frac{F_{φ} G_{φ}}{ρ} \frac{cos θ}{sin θ}) {\vec{e}}_{θ} + \\ (F_{ρ} \frac{\partial G_{φ}}{\partial ρ} + \frac{F_{θ}}{ρ} \frac{\partial G_{φ}}{\partial θ} + \frac{F_{φ}}{ρ sin θ} \frac{\partial G_{φ}}{\partial φ} + \frac{F_{φ} G_{ρ}}{ρ} + \frac{F_{φ} G_{θ}}{ρ} \frac{cos θ}{sin θ}) {\vec{e}}_{φ} \end{array}$

si413_e (Eq. 7.143)

7.7 Summary

In this section we have studied the fundamental concepts of vector calculus and introduced the most important operators. We also introduced the four important theorems of vector calculus, i.e., the theorems of Gauß, Stokes, and Green, as well as Reynolds’ transport theorem. In addition, we studied the four di˙erent coordinate systems that we require for solving fluid mechanical problems in di˙erent geometries. As we will see, the equations often become significantly simpler when transferred into the right coordinate system. Besides position, velocity, and acceleration vectors, we have also derived the most important operators in the respective coordinate systems. Once we have established the fundamental equations of fluid mechanics, it will be very easy to convert them to the appropriate coordinate system using the equations we set up in this section.

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7.7 Summary

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7.7 Summary