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## Homework Statement

Let w be the form w= xdydz in R^3. Let S^2 be the unit sphere in R^3.

If we restrict w on S^2, is w exact?

## Homework Equations

## The Attempt at a Solution

My guess is w is not exact on S^2.

Suppose w is exact on S^2. Then w=da for some 1-form a=fdx+gdy+hdz.

Then by definition of exterior derivative, we get

w=(-df/dy+dg/dx)(dx^dy)+(-df/dz+dh/dx)(dx^dz)+(-dg/dz+dh/dy)(dy^dz)

So we get the conditions:

df/dy=dg/dx, df/dz=dh/dx, x=-dg/dz+dh/dy.

I think I should use a fact that I am working on a unit sphere. Could anybody help me?