M408M Learning Module Pages
Main page Chapter 10: Parametric Equations and Polar CoordinatesChapter 12: Vectors and the Geometry of SpaceLearning module LM 12.1: 3-dimensional 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,…, 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,
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