The area/arc-length is given by an integral or the integral defines the area/arc-length is one the first things we learn in Calculus, but that is done in the Cartesian coordinates, next one moves to polar coordinates and the area is transformed/redefined by a new integral using the Jacobian. My question is : Since area and arc length are invariant of coordinate system, is there a way to define them other than in Cartesian coordinates and transform them from coordinate system to coordinate system with the help of Jacobian? In other words is there a way that area/arc-length is defined independet of coordinate system and then according to the structure of coordinate system it's integral form is reached, without tranforming between coordinate systems, but for a given coordinate system it is derived. (and without the use of Jacobian to move between coordinate systems)
Representing area as an integral independent of the coordinate systems
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calculus
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6http://en.wikipedia.org/wiki/Volume_form – 2011-01-29