Express the uniform vector field \(\mathbf{F}=5 \mathbf{a}_{x}\) in \((a)\) cylindrical components; (b) spherical components.

The given vector field \(\mathbf{F}\) is already expressed in Cartesian components: $$ \mathbf{F}=5 \mathbf{a}_{x}. $$

To convert into cylindrical components, we'll use the following transformation equations for the \(x\) component: $$\mathbf{a}_x = \cos{\theta} \, \mathbf{a}_\rho - \sin{\theta} \, \mathbf{a}_\theta$$ Thus, the vector field in cylindrical components can be expressed as: $$ \mathbf{F'} = 5 \mathbf{a}_x = 5(\cos{\theta} \, \mathbf{a}_\rho - \sin{\theta} \, \mathbf{a}_\theta) $$

To convert into spherical components, we'll use the following transformation equations for the \(x\) component: $$\mathbf{a}_x = \sin{\phi} \cos{\theta} \, \mathbf{a}_r + \cos{\phi} \cos{\theta}\, \mathbf{a}_\phi - \sin{\theta}\, \mathbf{a}_\theta$$ Then, the vector field in spherical components can be expressed as: $$ \mathbf{F''} = 5 \mathbf{a}_x = 5(\sin{\phi} \cos{\theta} \, \mathbf{a}_r + \cos{\phi} \cos{\theta}\, \mathbf{a}_\phi - \sin{\theta}\, \mathbf{a}_\theta) $$ So, the vector field \(\mathbf{F}\) is expressed in (a) cylindrical components as: $$ \mathbf{F'} = 5(\cos{\theta} \, \mathbf{a}_{\rho} - \sin{\theta} \, \mathbf{a}_{\theta}) $$ and (b) spherical components as: $$ \mathbf{F''} = 5(\sin{\phi} \cos{\theta} \, \mathbf{a}_{r} + \cos{\phi} \cos{\theta}\, \mathbf{a}_{\phi} - \sin{\theta}\, \mathbf{a}_{\theta}) $$