A vector function F is not irrotational.Show that it is always possible to find a scalar function f so that the function fF is irrotational.
A vector field V is not irrotational.Show that it is always possible to find f such that fV is irrotational.
ÑX[fV]=fÑxV-VxÑf
V is not irrotational means:
curl(V)=U
U non equal to zero.
f V irrotational means:
curl(fV)=0
But: curl(fV)=grad(f)xV+fcurl(V)=grad(f)xV+f U
So you get:
grad(f)xV+f U=0
grad(f)/f xV+ U=0
grad(ln(f))xV=-U
Then,what should I do?
I did! The curl of the zero vector is a zero vector. Right?
So given any irrotational vector field.
If you multiply it by f(x,y,z)=0 then it because a zero vector. Whose circulation is a zero vector. Hence the scalar function is always zero.
My construction is trivial. But there is nothing in your question which forbids me from using that.