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?
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
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?