Thread: Vector field question

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?

Originally Posted by kolahalb

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.

Chose f(x,y,z)=0
What is the problem?

The problem is to find f(x,y,z) to prove that ÑX[fV] can be made equal to zero even if ÑX[V] is not zero.

Originally Posted by kolahalb

The problem is to find f(x,y,z) to prove that ÑX[fV] can be made equal to zero even if ÑX[V] is not zero.

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.

The examiner will not be pleased with this trivial answer.So,please try seriously.

Originally Posted by kolahalb

The examiner will not be pleased with this trivial answer.So,please try seriously.

I will try tomorrow. Now I have to leave. But again there is nothing in the question which forbids that.

That is what I hate about engineering classes. They never, ever pay to attention to divison by zero and trivial cases. That is their problem.