(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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?

2. Relevant equations

3. 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?

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# Homework Help: Differential form

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