Q: Representing area as an integral independent of the coordinate systems 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)

A: Arc length, area and volume are geometrical concepts that make sense in a coordinate-free world. Note that Archimedes computed many interesting areas and volumes without using coordinates!  But now we have this edifice called "calculus" with its primitives, Fubini theorem, change-of-variables formula, etc. In order to make this edifice useful for computing lengths and areas we have to coordinatize our curves and surfaces. We then have to ${\it prove}$ that our intuitive idea of arc length as being the $\sup$ of the lengths of inscribed polygons, leads to the formula $L(\gamma)=\int_a^b |\dot{\bf x}(t)|dt$. This is hard work! What we have gained in this way is access to a machinery that allows to compute an unlimited number of lengths and areas in a more or less automatic way. But, alas, we shall not obtain a simple formula for the circumference of an ellipse $\ldots$
