By direct differentiation using Cartesian coordinates verify the following relations.

(a) The divergence of the curl of any vector is zero.

\[abla \cdot(abla \times \boldsymbol{A})=0\]

(b) The curl of the gradient of any scalar is zero

\[abla \times(abla \phi)=0\]

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 \(3 \mathrm{D}\) flow

(b) the curl is zero, the flow is \(3 \mathrm{D}\)

(c) the divergence is zero, the flow is \(2 \mathrm{D}\)

(d) both divergence and curl are zero, the flow is 3D

(e) both divergence and curl are zero, the flow is \(2 \mathrm{D}\)