(Solved): Exercise 6 Calculate the \( \nabla \times \mathbf{F} \) and \( \nabla \cdot \mathbf{F} \) in cyli ...

Exercise 6 Calculate the \( \nabla \times \mathbf{F} \) and \( \nabla \cdot \mathbf{F} \) in cylindrical coordinates. \( \mathbf{F}(r, \theta, z)=\boldsymbol{r}^{2} \cos \theta \mathbf{a}_{r}+\sin \theta \mathbf{a}_{\theta} \). Exercise 7 Calculate the \( \nabla \times \mathbf{F} \) and \( \nabla \cdot \mathbf{F} \) in spherical coordinates. \( \mathbf{F}=F_{\rho} \mathbf{a}_{\rho}+F_{\phi} \mathbf{a}_{\phi} \) where \( F_{\rho}=\cos (\beta \rho) \cos \phi \) and \( F_{\phi}=\cos (\beta \rho) \sin \phi . \) Exercise 8 Calculate \( \nabla f \) in cylindrical coordinates where \( f(r, \theta, z)=\ln r \). Exercise 9 Calculate \( \nabla f \) in spherical coordinates where \[ f(\rho, \phi, \theta)=\frac{1}{\rho} \]