(Solved): Use indicial notation to verify the following equations. Start by transcribing the left-hand side ...

Use indicial notation to verify the following equations. Start by transcribing the left-hand side of the equations, then using the indicated operations, show that the result is the right-hand side. \( \bar{u} \) and \( \bar{v} \) are vectors and \( \lambda \) and \( \phi \) are scalars. a) \( \bar{v} \times(\nabla \times \bar{v})=\frac{1}{2} \nabla(\bar{v} \cdot \bar{v})-(\bar{v} \cdot \nabla) \bar{v} \) b) \( \nabla \times(\nabla \times \bar{v})=\nabla(\nabla \cdot \bar{v})-\nabla^{2} \bar{v} \) c) \( \nabla \cdot(\lambda \nabla \phi)=\lambda \nabla^{2} \phi+\nabla \lambda \cdot \nabla \phi \) d) \( \operatorname{div}(\bar{u} \times \bar{v})=\bar{v} \cdot \operatorname{curl} \bar{u}-\bar{u} \cdot \operatorname{curl} \bar{v} \) e) \( \operatorname{curl}(\operatorname{curl} \bar{u})=\operatorname{grad}(\operatorname{div} \bar{u})-\nabla^{2} \bar{u} \)