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Calculus Small Changes, Big Ideas

Bill Shillito

Section 11.3 Slicing through dimensions

Introduction goes here.
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  • We can hold one variable constant to get a cross-section of a 3D surface. This is called a trace, and it’s the same as looking at the intersection between a surface and a plane parallel to one of the coordinate planes.
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  • Ellipsoid:
    \begin{equation*} \frac{(x-x_{0})^{2}}{a^{2}}+\frac{(y-y_{0})^{2}}{b^{2}}+\frac{(z-z_{0})^{2}}{c^{2}}=1 \end{equation*}
    Traces are all ellipses.
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  • Hyperboloid of one sheet:
    \begin{equation*} \frac{(x-x_{0})^{2}}{a^{2}}+\frac{(y-y_{0})^{2}}{b^{2}}-\frac{(z-z_{0})^{2}}{c^{2}}=1 \end{equation*}
    Traces are ellipses or hyperbolas.
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  • Hyperboloid of two sheets:
    \begin{equation*} \frac{(x-x_{0})^{2}}{a^{2}}-\frac{(y-y_{0})^{2}}{b^{2}}-\frac{(z-z_{0})^{2}}{c^{2}}=1 \end{equation*}
    Again traces are ellipses or hyperbolas, but one trace is empty, which gives two sheets.
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  • Elliptic Paraboloid:
    \begin{equation*} z=\frac{(x-x_{0})^{2}}{a^{2}}+\frac{(y-y_{0})^{2}}{b^{2}} \end{equation*}
    Traces are ellipses or parabolas.
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  • Elliptic Paraboloid:
    \begin{equation*} z=\frac{(x-x_{0})^{2}}{a^{2}}-\frac{(y-y_{0})^{2}}{b^{2}} \end{equation*}
    Traces are hyperbolas or parabolas.
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