Thread: quick question vector fields

Define the vector field on the complement of the z axis by
F(x ,y, z) = (-y i + x j)/(x^2 + y^2)

Im trying to find the curl and i know how to do it but for my k direction im confused because of this whole complement of z axis thing, do i just take x^2 + y^2 to be k direction??

The vector field is defined at all points except where x = y = 0.

This is precisely all points on the z axis.

So when you say :

Define the vector field on the complement of the z axis by
F(x ,y, z) = (-y i + x j)/(x^2 + y^2)

You are simply saying that F is defined everywhere but along the z axis.

So simply compute the curl as you normally would with 0 k component.

In General if F =f i + gj

Then curl F is[dg/dx - df/dy] k

Remember Green's Theorem in the plane ?