When $z = x^2 + y^2$, the trace on $y = b$ is the Parabolas or hyperbolas, and the surface is called a I.e., hyperbolas opening in the $x$-direction if $c > 0$ and On the other hand, slicing horizontally by $z = c$ gives I.e., parabolas opening up and down respectively. Vertically by $y = b$ means fixing $y = b$ and graphing $$z \ = \į(x,\, b) \ = \ x^2 - b^2\,$$ while slicing vertically by the plane The next step is to look at a surface arising as the graph ofĪ real-valued function $z = f(x,\, y) : U \subseteq $$ in the plane $z = c$. We've already seen surfaces like planes, circular cylinders and spheres. Just as having a good understanding of curves in the plane isĮssential to interpreting the concepts of single variable calculus, soĪ good understanding of surfaces in $3$-space is needed whenĭeveloping the fundamental concepts of multi-variable calculus. Surfaces and traces M408M Learning Module PagesĪnd Polar Coordinates Chapter 12: Vectors and the Geometry of Spaceģ-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 traces
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