and in a number of equivalent forms. The area of any portion of the plane is represented by the double integral
∫∫ J−1 dξ dη,
where J denotes the above Jacobian, and the integration is taken through a suitable domain. When the boundary consists of portions of curves for which ξ = const., or η = const., the above is generally the simplest way of evaluating it.
(vi.) The problem of “rectifying” a plane curve, or finding its length, is solved by evaluating the integral
| ∫ { 1 + ( | dy | )2 }1/2 dx, |
| dx |
or, in polar coordinates, by evaluating the integral
| ∫ { r2 + ( | dr | )2 }1/2 dθ. |
| dθ |
In both cases the integrals are line integrals taken along the curve.
(vii.) When we use curvilinear coordinates ξ, η as in (v.) above, the length of any portion of a curve ξ = const. is given by the integral
∫ J−1/2 dη