Correct Answer - Option 4 :
\(\hat i - 4\hat k\)
Concept:
\(\vec F = \left( {{x^2}y} \right)̂ i + \left( {2xz} \right)̂ j + \left( {3y{z^2}} \right)̂ k\)
\({\rm{Vorticity\;vector\;}} = {\rm{\;curl\;}}\vec F = \nabla \times \vec F\)
Calculation:
Given:
\(\vec V = 2xy\hat i - {x^2}z\hat j\)
\(\therefore {\rm{curl\;}}\vec F = \left| {\begin{array}{*{20}{c}} {̂ i}&{̂ j}&{̂ k}\\ {\frac{\partial }{{\partial x}}}&{\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial z}}}\\ {{2x}y}&{-x^2z}&{{0}} \end{array}} \right|\)
curl F = [0 - (-x2)]î - [0 - 0]ĵ + [-2xz - 2x]k̂
curl F = x2î - (2xz + 2x)k̂
At point (1, 1, 1)
curl F = î - 4k̂
