M408M Learning Module Pages
Main page Chapter 10: Parametric Equations and Polar CoordinatesChapter 12: Vectors and the Geometry of SpaceLearning module LM 12.1: 3dimensional rectangular coordinates:Learning module LM 12.2: Vectors:Learning module LM 12.3: Dot products:Learning module LM 12.4: Cross products:Learning module LM 12.5: Equations of Lines and Planes:Learning module LM 12.6: Surfaces:Surfaces and tracesLevel curves Level surfaces Worked problems Chapter 13: Vector FunctionsChapter 14: Partial DerivativesChapter 15: Multiple Integrals 
Surfaces and tracesJust as having a good understanding of curves in the plane is essential to interpreting the concepts of single variable calculus, so a good understanding of surfaces in $3$space is needed when developing the fundamental concepts of multivariable calculus. We've already seen surfaces like planes, circular cylinders and spheres. The next step is to look at a surface arising as the graph of a realvalued function $z = f(x,\, y) : U \subseteq {\mathbb R}^2 \to {\mathbb R}$ of two variables  in practice, such functions occur in the theory of heat flow in a bar, or vibrating strings, or the pressure at a point on the earth. The graph of $z = f(x,\, y)$ is the surface $$\color{rgb(0, 51, 153)} {S \ = \ \big\{ (x,\, y, \, f(x,\, y)\big) : \, \ (x,\, y) \ \hbox{in} \ U\, \big\} }$$ in $3$space. Two examples that will occur repeatedly are shown below in
How do we know the surfaces look like that? The basic idea is to take crosssections of the surface by plane slices. Because a plane intersects the surface in a curve that also lies in the plane, this curve is often referred to as the trace of the surface on the plane. Identifying traces gives us one way of 'picturing' the surface; reassembling the crosssections then provides a full picture of the surface. It thus reduces the problem of describing a surface to identfying curves in the plane  the 'natural progression' idea at work! When $z = f(x,\, y)$,
