The curl of a vector field F, denoted by curl F, or $${\displaystyle \nabla \times \mathbf {F} }$$, or rot F, is an operator that maps C functions in R to C functions in R , and in particular, it maps continuously differentiable … See more

Example 1
Suppose the vector field describes the velocity field of a fluid flow (such as a large tank of liquid or gas) and a small ball is located within the … See more

The vector calculus operations of grad, curl, and div are most easily generalized in the context of differential forms, which involves a number of steps. In short, they correspond to the … See more

• Helmholtz decomposition
• Del in cylindrical and spherical coordinates
• Vorticity See more

In practice, the two coordinate-free definitions described above are rarely used because in virtually all cases, the curl operator can … See more

In general curvilinear coordinates (not only in Cartesian coordinates), the curl of a cross product of vector fields v and F can be shown to be
Interchanging the vector field v and ∇ operator, we arrive at the cross product of a vector field with … See more

In the case where the divergence of a vector field V is zero, a vector field W exists such that V = curl(W). This is why the See more