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\textbf{Partial Differentiation}

\textit{\textbf{Functions of more than one variable}}
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\textbf{Question}

For the given functions $f(x,y,z)$, describe the level surfaces.
\begin{description}
\item{(a)}
$f(x,y,z)=x^2+y^2+z^2$

\item{(b)}
$f(x,y,z)=x+2y+3z$

\item{(c)}
$f(x,y,z)= x^2+y^2$

\item{(d)}
$\displaystyle f(x,y,z)=\frac{x^2+y^2}{z^2}$

\item{(e)}
$f(x,y,z)=|x|+|y|+|z|$

\end{description}

\textbf{Answer}

\begin{description}

\item{(a)}
$f(x,y,z)=x^2+y^2+z^2$

The level surface $f(x,y,z)=c&gt;0$ is a sphere of radius $\sqrt{c}$
centred at the origin.


\item{(b)}
$f(x,y,z)=x+2y+3z$

The level surfaces are parallel planes with common normal vector
$\underline{i}+2 \underline{j}+ 3\underline{k}$.


\item{(c)}
$f(x,y,z)= x^2+y^2$

The level surface $f(x,y,z)=c&gt;0$ is a circular cylinder of radius
$\sqrt{c}$ with axis along the $z$-axis.


\item{(d)}
$\displaystyle f(x,y,z)=\frac{x^2+y^2}{z^2}$

The equation $f(x,y,z)=c$ can be rewritten $x^2+y^2=C^2z^2$. The level
surfaces are circular cones with vertices at the origin and axes along
the $z$-axis.


\item{(e)}
$f(x,y,z)=|x|+|y|+|z|$

The level surface $f(x,y,z)=c&gt;0$ is the surface4 of the octohedron
with vertices $(\pm c, 0,0)$, $(0, \pm c, 0)$ and $(0,0, \pm c)$. (An
octohedron is a solid with eight planar faces.)
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