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6.3 Fields Conservative in Some but Not All Regions

Consider its gradient, which is a vector field:

This vector field has cross partials equal at all points except at x = y =0, where these derivatives don't exist. It is in general not the gradient of a function. It is the gradient of which is not a function. Howeveris a well defined function in any simply connected region of 3-space which does not meet the line r = 0.
Thus is conservative in any such region. The seemingly confusing fact that vector fields can be conservative in one region and not so in another simply reflects the properties of objects like the angle which is a true single valued function only when restricted to be so; confining its range is one way to keep it single valued.