Elliptical coordinates

[izi96]

Hi

Im having problems finding a way to solve this question. We got it on a test and i couldnt do it then and im trying to see if i can now. The question is as follows (its not in english so im trying my best to translate)

The elliptical coordinates (u,v) are related to cartesian coordinates through position vector
~r = a cosh u cos v i + a sinh u sin v j

a is a constant and we can assume the value is 1. u is in the interval of (0,1) and v is in (0,2pi)

Find the unit vectors e_u and e_v and also the scaling factors to the elliptical coordinates. Are the vectors orthogonal?
Draw a sketch of the coordinate curves in the cartesian coordinate system and also give the curves a constant value of either u or v.

Any ideas would help me. I need to understand this before my finals and im really struggling

 

[LCKurtz]

You have [MATH]\vec r = \langle \cosh u \cos v, \sinh u \sin v\rangle[/MATH]. The partials [MATH]\vec r_u[/MATH] and [MATH]\vec r_v[/MATH] will give you tangent vectors in the coordinate directions. Make unit vectors out of them by dividing them by their lengths. Take their dot product to see if they are orthogonal. Use your definition of scaling factor to calculate it.
To see what your coordinate system looks like, take a couple of values of [MATH]u[/MATH] and let [MATH]v[/MATH] vary and plot the parametric curve. Do the same with the other variable constant. You should get something like: