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 
Level curvesFor a general function $z = f(x,\,y)$, slicing horizontally is a particularly important idea:
Notice the critical difference between a level curve $C$ of value $c$ and the trace on the plane $z = c$: a level curve $C$ always lies in the $xy$plane, and is the set $C$ of points in the $xy$plane on which $f(x,\,y) = c$, whereas the trace lies in the plane $z = c$, and is the set of points $(x,\,y,\, c)$ with $(x,\,y)$ in $C$. By combining the level curves $f(x,\,y) = c$ for equally spaced values of $c$ into one figure, say $c = 1, \,0,\, 1,\, 2,\, \ldots \,,$ in the $xy$plane, we obtain a contour map of the graph of $z=f(x,\,y)$. Thus the graph of $z = f(x,\,y)$ can be visualized in two ways,
