Find the curl of the vector and state its nature at (1,1,-0.2) F = 30 i + 2xy j + 5xz2 k

To find the curl of the vector field ( mathbf{F} = 30 mathbf{i} + 2xy mathbf{j} + 5xz^2 mathbf{k} ), we will use the formula for curl in three-dimensional Cartesian coordinates:

[

nabla times mathbf{F} = begin{vmatrix}

mathbf{i} & mathbf{j} & mathbf{k} \

frac{partial}{partial x} & frac{partial}{partial y} & frac{partial}{partial z} \

30 & 2xy & 5xz^2

end{vmatrix}

]

Calculating the determinant gives us:

[

nabla times mathbf{F} = mathbf{i} left( frac{partial}{partial y}(5xz^2) – frac{partial}{partial z}(2xy) right) – mathbf{j} left( frac{partial}{partial x}(5xz^2) – frac{partial}{partial z}(30) right) + mathbf{k} left( frac{partial}{partial x}(2xy) – frac{partial}{partial y}(30) right)

]

Calculating each component:

1. The ( mathbf{i} ) component:

[

frac{partial}{partial