vector calculus parametric equations

Hi.

I wanted to ask a question about the attached exercise 10.10(a).
I'm confident using the method explained to calculate the surface area. However, in questions of this type that I usually see the vector has two parameters, say, u and v for example. Then you proceed by finding the partial derivatives of r with respect to u and v etc.

In this question though, there appears to be three parameters; u, φ and Ω.The authors take the partial derivatives of r with respect to u and φ (there is a typo I believe), but what is their reasoning behind this? why do they chose these two parameters and not u and Ω, or φ and Ω?

Is Ω not a true parameter or is the choice irrelevant and any two of the three could be used?

Thanks

Omega is defined as a fixed angle (not a parameter)
A surface parameterization need only 2 parameters