The curl of the divergence of a vector is always zero. Mathematically, it can be expressed as:
∇ × (∇ ⋅ V) = 0
where ∇ is the del operator, × denotes the cross product, ⋅ represents the dot product, and V is a vector field.
This result can be derived using vector calculus identities. The divergence of a curl of any vector field is always zero (∇ ⋅ (∇ × V) = 0), and conversely, the curl of the gradient of any scalar field is also zero (∇ × (∇f) = 0).
Applying this concept to the divergence of a vector field V, we have:
∇ × (∇ ⋅ V) = ∇ × 0 = 0
Therefore, the curl of the divergence of a vector field is identically zero in any coordinate system.