Question

Consider the vector field f(x,y,z) = (5yz, 2xz, 5xy). find the divergence and curl of f.

Answer 1

Recall the formula of the curl :
$Curl(f)=(R_y-Q_z)i+(P_z-R_x)j+(Q_x-P_y)k\\\text{Wherein }P=5yz,Q=2xz,R=5xy\\\text{Compute the partial derivative like this:}\\R_y=5x,Q_z=2x,P_z=5y,R_x=5y,Q_x=2z,P_y=5z\\\text{Apply the formula like this:}\\Curl(f)=(5x-2x)i+(5y-5y)j+(2z-5z)k\\=3xi-3zk\\\text{We can write in the following form:}\\Curl(f)=(3x,0,-3z)\\\textbf{Divergence:}\\\nabla(f)=P_x+Q_y+R_z=0+0+0=0$