on the tangent space of an n−surface

we are supposed to characterize the tangent space of an n−surface s = f⁻¹(c) for some f : u ® r, u open in rⁿ⁺¹, and f is a smooth function with the property that ñf(p) ¹ 0 for all p ϵ s, in the case that the whole space admits the general form of the inner product. finally we introduce a vector field x with integral curve α through p such that the covariant derivative of f with respect to α´(0) at p, i.e., ñ_α˙(0)f has maximum value.