Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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)

share|cite|improve this question

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 $\ldots$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.