(Solved): Using index notation, demonstrate mathematically the following equations: 1- \( \underline{\mathbf ...

Using index notation, demonstrate mathematically the following equations: 1- \( \underline{\mathbf{u}} \times \underline{\mathbf{v}}=-\underline{\mathbf{v}} \times \underline{\mathbf{u}} \) 2- \( \nabla \cdot(\nabla a)=\nabla^{2} a \) 3- \( \nabla^{2}(a b)=a \nabla^{2} b+2 \nabla a \cdot \nabla b+b \nabla^{2} a \) 4- \( \nabla \cdot(a \mathbf{b})=\nabla a \cdot \underline{\mathbf{b}}+a \nabla \cdot \underline{\mathbf{b}} \) 5- \( \nabla \times \nabla \phi=0 \) 6- \( \nabla \cdot(\nabla \times \underline{\mathbf{V}})=0 \) 7- \( \nabla \cdot \underline{\mathbf{v}} \neq \underline{\mathbf{v}} \cdot \nabla \) (Express each term in index notation and expand in Cartesian coordinates to prove the inequality) 8- \( (\underline{\mathbf{a}} \times \underline{\mathbf{b}}) \cdot(\underline{\mathbf{c}} \times \underline{\mathbf{d}})=(\underline{\mathbf{a}} \cdot \underline{\mathbf{c}})(\underline{\mathbf{b}} \cdot \underline{\mathbf{d}})-(\underline{\mathbf{a}} \cdot \mathbf{\mathbf { d }})(\underline{\mathbf{b}} \cdot \underline{\mathbf{c}}) \)