3.4: Grad, curl and div
( \newcommand{\kernel}{\mathrm{null}\,}\)
Gradient. For a function f(x,y), the gradient is defined as gradf=∇f=(fx,fy). A vector field F which is the gradient of some function is called a gradient vector field.
Curl. For a vector in the plane F(x,y)=(M(x,y),N(x,y)) we define
curlF = Nx−My.
Note. The curl is a scalar. In general, the curl of a vector field is another vector field. However, for vectors fields in the plane the curl is always in the ˆk direction, so we have simply dropped the ˆk and made curl a scalar.
Divergence. The divergence of the vector field F =(M,N) is
divF = Mx+Ny.