M408M Learning Module Pages
Main page ## Chapter 10: Parametric Equations and Polar Coordinates## Chapter 12: Vectors and the Geometry of Space## Learning 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 Functions## Chapter 14: Partial Derivatives## Chapter 15: Multiple Integrals |
## Level surfacesNow let's step up a dimension and consider functions $w=f(x,\,y,\,z) : U \subseteq {\mathbb R}^3 \to {\mathbb R}$ of $3$ variables; one such function is $${\color{darkerblue} w \ = \ f(x,\,y,\, z) \ = \ x^2 + y^2 - z^2\,.}$$ The graph of every function $w = f(x,\,y,\, z)$ will be a surface in ${\mathbb R}^4$, though it can't be drawn directly; however, slicing horizontally by $w = c$ produces relations $c = f(x,\,y,\, z)$ in $x,\, y,$ and $z$ whose graphs will be surfaces in $3$-space which can be drawn. Formally,
From the earlier example of $ w = f(x,\,y,\, z) = x^2 + y^2 - z^2$. we obtain three particularly important surfaces as level surfaces:
by taking $c = -1,\ 0,$ and $1$. The two-sheeted hyperboloid and double cone are very important in physics, while the single sheeted hyperboloid is a favorite architectural device - cooling towers etc - as is the hyperbolic paraboloid.
Again we can investigate what happens as these surfaces are sliced by planes parallel to the coordinate planes: Recognize the curves of intersection? There must be some underlying mathematical theory! To do no more than hint at what that theory might be notice
In view of the first of these comments we make the following
Cylindrical Surfaces: sometimes the intersection of a
surface in $3$-space with horizontal planes $z = c$ is the same for
all $c$ as in the surface below to the left, or is the same for all
vertical planes, say $x = a$, as in the surface to the right.
Do you see that the |