Path-Independent Integrals on a Limited Subspace: Vector Fields with Non-Vanishing Curl Except on a Particular Surface

For a last example, suppose the curl vanishes on the cylindrical surface defined above:
Suppose we can find a function that has a non-vanishing curl on this surface

We want to find a function which is generally non-curl free, but for which the curl vanishes on a surface. Let's pick the cylinder as our surface.



If  a function can be found, that has the following curl, then we will have constructed such a function,



It is easy to see that this is the curl of Stooge, where we construct Stooge by integrating.



In fact, we could add to Stooge, any vector function that has vanishing curl--there are an infinite number of these



Its integral doesn't care which path around the cylinder it takes, the integrand doesn't depend on P(t)



This is the value for *any* path on the cylinder that is closed.



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