Sorry, I should have been clearer. I was trying to say that convergence in area for two-dimensional surfaces in R^3 requires convergence in normals in the same

Sorry, I should have been clearer. I was trying to say that convergence in area for two-dimensional surfaces in R^3 requires convergence in normals in the same way that convergence in length for one-dimensional curves in R^2 requires convergence in normals. For area in R^2, volume in R^3, and so on, you're definitely right.