Stepping stone from vectors to surfaces in multivariable/vector calculus I'm trying to study Vector Calculus, and think that the book by Michael Corral, is a wonderful book. I went through the first few chapters with ease, but I came across something entirely not included in any of my math studies up until now - surfaces. The chapter that follows is marked curvilinear coordinates.  
There's not much description or proofs about how the equations of surfaces came to be, just equation representations of 3-dimensional surfaces. I also have a secondary book (Thomas' calculus), which pretty much does the same thing. By now, I am completely lost how to continue my studies on this? What would be a stepping stone for my level of knowledge of the usual vectors to these completely new subject matters?
 A: Most students don't go from calculus to vector calculus; you need to look into linear algebra.  Take a gander here.
A: http://tutorial.math.lamar.edu/
Try these notes. I don't know how much you know but all the material is really good here and it's free. There are very lengthy introductions about how to parametrise surfaces, what surfaces are, and all you need for a vector calculus course. Stuff in Vector Calculus will be at the end of Calculus 2 and in Calculus 3.
There are also sections covering the topics in Linear Algebra that you need to know. Otherwise, if you really need more algebra you can go on youtube and watch MIT Linear Algebra by Strang
A: Looking at Section 1.6 of Corral on surfaces, especially quadric surfaces, the stepping stone you might need is to go over conic sections (i.e. circles, ellipses, parabolas and hyperbolas) again. You can temporarily set the material on vectors aside - it will return soon enough in Section 1.8 on vector-valued functions. Start with the circle and see whether the equation of a circle of radius $r$ centred on the origin makes sense:
$$x^2 + y^2 = r^2$$
It is based on Pythagoras' Theorem and captures the idea of the set of all points $(x,y)$ of distance $r>0$ from the origin $(0,0)$, measured using the Euclidean distance in the plane. If you prefer, you could write it as
$$(x-0)^2 + (y-0)^2 = r^2$$
or
$$\sqrt{(x-0)^2 + (y-0)^2} = r$$
Similarly, the equation of a circle of radius $r$ centred on $(x_0,y_0)$ is
$$(x-x_0)^2 + (y-y_0)^2 = r^2$$
Going up one dimension, we have the equation of a sphere of radius $r$ centred on $(x_0,y_0,z_0)$, measured using the Euclidean distance in three dimensions:
$$(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 = r^2$$
From here, you can investigate the other conic sections and then the quadric surfaces, perhaps with the aid of Math Insight's quadrics page.
