This is about change of parameter from (u,v) plane to (x,y) plane. If you read the begining, it said σ is a smooth parametric surface on a region R. It went on and talk about continuous and all in region R shown in Fig 15.4.10 (a). But that is not correct. If you read on, this is about mapping of (u,v) plane of Fig (a) to (x,y) plane in Fig (b)!!! It is obvious that σ is a region in (x,y) plane shown in Fig (b). Seems like the book is wrong. It should be ##\vec r=\vec r(x(u,v), y(u,v))=\hat x x(u,v)+\hat y y(u,v)+\hat z z(u,v)## which is a vector value function in (x,y) plane shown in Fig (b). Attached is the scan of the book, please ignore all my scribbles, just read the book. [PLAIN]http://i46.tinypic.com/2byywh.jpg[/PLAIN] Am I missing something?