Calculate the div and rot:
Correct?
The divergence is correct
Not sure what you have in mind for rotF
but it is 5z^2 i +3xz^2 j +4 y^4x k
The middle row of the determinant are the operators d/dx d/dy and d/dz
and the last row are the x , y , and z components of F not the parial derivatives
is an operator and means d/dx i +d/dy j +d/dz k where d/dx d/dy d/dz ar the partial derivative operators
f is the gradient where f is a scalar valued function
X F is the curl or rotation where F is a vector field
*F is the divergence where F is a vector field * means dot product
From the previous identity * ( x F) = 0
r= xi +yj +zk
||r|| = (x^2 +y^2 +z^2)^(1/2)
Compute ||r|| and the result follows directly