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\author{Kyle Spaans}
\date{May 4, 2009}
\title{Calculus 3 Lecture Notes}
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\maketitle
\section*{Lecture 1 -- Functions of Several Variables}
We will consider functions of the form $z = f(x,y)$ (explicitly defined) and
$f(x,y,z) = c$ (for a constant $c$, implicitly defined) or we can say
$f: D \rightarrow R$. These functions will most be ones we've already seen before
in \emph{Calculus 1} and \emph{Calculus 2}, but the algebra will be different
since we're working with multiple variables.
How can we visualize these functions? Drawing will become important later, so
we are going to learn this now. We can follow two simple steps to help us
draw a function:
\begin{enumerate}
\item Fix a single variable (usually $z$) and plot the resulting function for
a couple of values. \label{draw1}
\item Fix each of the other variables in turn (generally by setting them $= 0$)
and plot those functions. \label{draw2}
\end{enumerate}
Consider $x^2 + y^2 = z^2$. Following \ref{draw1}, we let
$z = c \Rightarrow x^2 + y^2 = c^2$. This is essentially a circle, having
different radii for different values of $c$. We can draw this in 2D, with x-
and y-axes, and different ``lines'' (called ``level curves'') for each value of
$c$. Consider that when $c = 0$, the graph collapses to a point at the origin.
Continuing with step \ref{draw2} we let $y = 0 \Rightarrow x^2 = x^2$, which
implies that $z = x$ and $z = -x$. This gives us a graph with the z- and x-axes
called the ''cross-section''. Repeating for $x$, let
$x = 0 \Rightarrow y^2 = z^2$, which in this case gives us a similar graph for
the z- and y-axes.
Combining all of this, we see a cone in our 3D plot. What if we let $y = k$?
This gives us $x^2 + k^2 = z^2$, which is the equation for a hyperbola
($\frac{x^2}{k^2} - \frac{z^2}{k^2} = 1$). For next lecture, make sure you've
refreshed your memory about hyperbolas and their equations.
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