Area Element of Elliptic Cylinder Coordinates

[jameson2]

Homework Statement

Compute the area element for elliptic cylinder coordinates

Homework Equations

The coordinates are defined as follows:
x=a*cosh(u)*cos(v)
y=a*sinh(u)*sin(v)

The Attempt at a Solution

Starting from the assumption that the area element dA=dx*dy, I found dx and dy:
dx=a*du*sinh(u)*cos(v) - a*cosh(u)*dv*sin(v)
dy=a*du*cosh(u)*sin(v) - a*sinh(u)*dv*cos(v)

Then multiplying these together, to get dA:
dA=[(a^2)*sinh(u)*cosh(u)*sin(v)*cos(v)*(du^2 - dv^2)] +
[(a^2)*du*dv*((sinh(u))^2)*((cos(v))^2) - (cosh(u))^2)*((sin(v))^2)]

I don't like this answer for a couple of reasons. It seems like there should be a tidier, more compact expression than what I have. Compared to surface elements in other coordinate systems, this is frankly a mess. Also, I don't think I've seen a "du^2" in any area element formulae either, which I'm not sure makes it wrong, but I feel a little uneasy about it anyway.

 

[HallsofIvy]

When you say "multiplying these together", what kind of product did you use? If a surface is given by parametric equations, x= x(u,v), y= y(u,v), z= z(u,v), then the "position vector" is $\vec{r}= x(u,v)\vec{i}+ y(u,v)\vec{j}+ z(u,v)\vec{k}$, the derivatives in the directions of the u, v axes are $\vec{r}_u= x_u \vec{i}+ y_u\vec{j}+ z_u\vec{k}$ and $\vec{r}_v= x_v \vec{i}+ y_v\vec{j}+ z_v\vec{k}$. The differential of surface area is the length of the cross product of those two vectors times dudv.

 

[jameson2]

So in vector form: dr/du=(a*sinh(u)*cos(v) , a*cosh(u)*sin(v) , 0)
and dr/dv=(-a*cosh(u)*sin(v) , a*sinh(u)*cos(v) , 0)

Multiplying across by du and dv respectively:
dr=(a*sinh(u)*cos(v)du , a*cosh(u)*sin(v)du , 0) with dr w.r.t "u"
and dr=(-a*cosh(u)*sin(v)dv , a*sinh(u)*cos(v)dv , 0) with dr w.r.t "v"

Then cross product of these vectors gives:
dA=length of the vector: [0,0,(a^2*du*dv*(sinh(u)^2*cos(v)^2 + cosh(u)^2*sin(v)^2))]

And the length of this vector is just the third component, so is this the answer for dA?
It seems a better answer than my original one, and the reasoning behind this method makes perfect sense to me.

dA= a^2*du*dv*(sinh(u)^2*cos(v)^2 + cosh(u)^2*sin(v)^2)