Distance element in the elliptical coordinates

As we found earlier,

We have further

The derivatives are

Please substitute. Calculation of the tangential unit vector . Let us consider two close points on the ellipse: and . The vector connecting these two points is a tangential vector, that is, . However, we have to take into account that and are not independent, since they have to be calculated with r=\mbox{const}, that is,

Using this we find

Please substitute. After is found, the normal unit vector can be found as .