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 … 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 fluid or gas (the center of the ball being fixed at a certain point). If the ball has a rough … 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 … See more

• Helmholtz decomposition
• Hiptmair–Xu preconditioner
• 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 be applied using some set of curvilinear coordinates, for which simpler representations have been derived.
The notation ∇ × F … 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 … 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 magnetic field, characterized by zero divergence, can be … See more