\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

MA 227 Standard C01

Surfaces in Three-Dimensional Space
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At the end of the course, each student should be able to…

  • C01: SurfaceEQ. Identify and sketch surfaces in three-dimensional Euclidean space.

C01. Surfaces in Three-Dimensional Space

  • Simple planes parallel to coordinate axes are given by \(x=a\), \(y=b\), and \(z=c\).
  • The dot product may be used to obtain equations for arbitrary planes.
    • The equation \(\vect N\cdot(P-P_0)=0\) describes the points \(P\) on the plane passing through the point \(P_0\) and normal to the vector \(\vect N\).
    • If \(P=\<x,y,z\>\), \(P_0=\<x_0,y_0,z_0\>\), and \(\vect N=\<A,B,C\>\), then this equation may be expressed as \(A(x-x_0)+B(y-y_0)+C(z-z_0)=0\) and simplifies to \(Ax+By+Cz=D\) for some value of \(D\).
  • Quadric surfaces are another common example used in calculus.
    • These surfaces are obtained from quadratic equations (polynomials of degree \(2\)) of the variables \(x,y,z\).
    • The simplest example is that of the sphere \(\|P-P_0\|=r\) describing the points \(P\) that are distance \(r\) from the sphere’s center \(P_0\).
    • If \(P=\<x,y,z\>\), \(P_0=\<x_0,y_0,z_0\>\), and \(\vect N=\<A,B,C\>\), then this equation may be expressed as \((x-x_0)^2+(y-y_0)^2+(z-z_0)^2=r^2\).
  • The surfaces given by equations of \(x,y,z\) may be sketched by considering traces of the surface in simple planes \(x=a\), \(y=b\), or \(z=c\).

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 11.1 (exercises 1-30, 41)
    • 11.5 (exercises 21-26)
    • 11.6 (exercises 1-12, 33-44)