By direct differentiation using Cartesian coordinates verify the following relations

By direct differentiation using Cartesian coordinates verify the following relations. (a) The divergence of the curl of any vector is zero. ∇ · (∇ × A) = 0 (b) The curl of the gradient of any scalar is zero ∇ × (∇φ) = 0 Use these properties in the Helmholtz decomposition to confirm that any vector can be decomposed into two vectors, where one vector is divergence-free while the other is curl-free. Discuss the use of these in simplifying the description of the velocity vector for some limiting situations. In particular, indicate the proper representation for the following cases: (a) the divergence is zero, the flow is a general 3D flow (b) the curl is zero, the flow is 3D (c) the divergence is zero, the flow is 2D (d) both divergence and curl are zero, the flow is 3D (e) both divergence and curl are zero, the flow is 2D