Calculating Surface Area Perturbations in n-Sphere Theory

[andert]

Is anyone familiar with books or papers on perturbation theory for closed levels sets in which the equation for the n-sphere is perturbed? For example, the level set:

\sum_{i=1}^n x_i^2 + \epsilon p(x_i) = C

where \epsilon is a small parameter and p(x_i) is a positive polynomial such as x_i^4.

 

[andert]

In particular, say A(C,\epsilon) is the "surface area". Then we can expand it:

A(C,\epsilon) = A(C,0) + \epsilon (dA/d\epsilon)(C,0) + \dots

How do I figure out what (d^nA/d\epsilon^n)(C,0) is from the equation for the level set?