Reference:-Introduction to Electrodynamics-David J. Griffith 
My Question
I have come across surface integral and line integral over a vector field. usually, the integrand is a scalar function, after dot product. What about the equation (2.5). Can you suggest some textbook of Mathematics for this kind of integral? Here integrand is purely vector field. In lower class, I integrated just component-wise. I really want to know, How is it working? Please help me.
You should read "Div, Grad, Curl, and All That"
https://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161
It is an exceptional text to pair with any vector calculus text. My personal favorite is
https://www.amazon.com/Calculus-III-Undergraduate-Texts-Mathematics/dp/0387909850/ref=sr_1_1?s=books&ie=UTF8&qid=1518622074&sr=1-1&keywords=marsden+calculus+3
which is very thorough, and very inexpensive if purchases used.