Compute the curl of the vector field F vector x3 y4 z2 Curl

Compute the curl of the vector field F vector = (x^3, y^4, z^2). Curl (F vector (x, y, z)) = What is the curl at the point (-5, -5, -3)? Curl (F vector (-5, -5, -3)) = Is this vector field irrotational or not?

Solution

By d_v we denot partial derivative w.r.t. v

SO,

F=F_xi+F_yj+F_zk

curl F=(d_yF_z-d_zF_y)i-(d_xF_z-d_zF_x)+k(d_xF_y-d_yF_x)=0

HEnce curl F=0

curl (F(-5,-5,-3))=0

curl is 0 hence vector field given by F is irrotational.