The discussion above defined the surface integral by using a parametrization of the surface "S". A given surface might have several parametrizations. For example, when the locations of the North Pole and South Pole are moved on a sphere, the latitude and longitude change for all the points on the sphere. A natural question is then whether the definition of the surface integral depends on the chosen parametrization. For integrals of scalar fields, the answer to this question is simple: the value of the surface integral will be the same no matter what parametrization one uses.
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