\noindent

\begin{center}

\begin{figure}[htbp]

\fbox{\resizebox{5.2in}{!}{\includegraphics{../images/instructors_guide_illustrations/level_curves_partial_derivatives}}}

\caption[Partial derivatives using contour lines]{Measuring changes in the domain and contour lines to approximate partial derivatives.}

\label{fig:dir_partial_deriv}

\end{figure}

\end{center}

\noindent

\begin{figure}[btp]

\centering

% \begin{subfigure}[t]{0.3\textwidth}

% \fbox{\resizebox{2in}{!}{\includegraphics{../images/instructors_guide_illustrations/partial_derivative_surface1}}}

% \caption{$\frac{\partial f}{\partial x} = 0$ and $\frac{\partial f}{\partial y} < 0$}

% \label{fig:partialy_lessthan_zero}

% \end{subfigure}

% \hspace{.5cm}

\begin{subfigure}[t]{0.3\textwidth}

\centering

\fbox{\resizebox{2in}{!}{\includegraphics{../images/instructors_guide_illustrations/partial_derivative_surface4}}}

\caption{$\vec{\imath}$ crosses 0 level curves and $\frac{\partial f}{\partial x} = 0$}

% \caption{$\frac{\partial f}{\partial x} = 0$ and $\frac{\partial f}{\partial y} > 0$}

\label{fig:partialy_morethan_zero}

\end{subfigure}

\hspace{.5cm}

\begin{subfigure}[t]{0.3\textwidth}

\centering

\fbox{\resizebox{2in}{!}{\includegraphics{../images/instructors_guide_illustrations/partial_derivative_surface3}}}

\caption{$\vec{\imath}$ crosses a few level curves decreasing in value; $\frac{\partial f}{\partial x} < 0$}

% \caption{$\frac{\partial f}{\partial x} < 0$ and $\frac{\partial f}{\partial y} = 0$}

\label{fig:partialx_lessthan_zero}

\end{subfigure}

\hspace{.5cm}

\begin{subfigure}[t]{0.3\textwidth}

\centering

\fbox{\resizebox{2in}{!}{\includegraphics{../images/instructors_guide_illustrations/partial_derivative_surface2}}}

\caption{$\vec{\imath}$ crosses more level curves than $\hat{\jmath}$ and $\frac{\partial f}{\partial x} < \frac{\partial f}{\partial y} < 0$}

% \caption{$\frac{\partial f}{\partial x} < 0$ and $\frac{\partial f}{\partial y} < 0$}

\label{fig:partialxandy_lessthan_zero}

\end{subfigure}

\caption[Partial derivatives on the surface and unit vectors with level curves]{The spacing of level curves in the coordinate directions influences the function's graph and its partial derivatives.\footnote{This is a special case; $\frac{\partial f}{r\partial \varphi}$ in polar coordinates is associated with the spacing of level curves crossed not by $\hat{\varphi}$ but by $r\hat{\varphi}$.}}

\label{fig:partial_tangent_plane}

\end{figure}

Partial derivatives describe how a function $f$ changes in a coordinate direction; $\frac{\partial f}{\partial x}$ is the ratio of infinitesimal changes in $f$ to an infinitesimal change in $x$ in the domain. This quantity can be estimated using a line segment or vector on the contour plot, as shown in Figure \ref{fig:dir_partial_deriv}. Although unnecessary, arrows can help coordinate these changes.

{\it Zooming in} on the function provides a more accurate estimate of the partial derivative by using more uniformly spaced level curves, as shown in Figure \ref{fig:partial_tangent_plane}.

%This results in more uniformly spaced level curves near

%When the level curves are uniformly spaced, the surface has essentially been replaced locally by a plane, as shown in Figure \ref{fig:partial_tangent_plane}.

%Hence, measurements made on the surface and in the domain are equivalent.

Also notice that the {\it tilt} of the surface is associated with the values of the partial derivatives (and hence spacing of level curves) in the coordinate directions.

This is a natural extension of {\it slope} from single variable calculus.

The units of partial derivatives are the units of change for the output per units of change for the domain.

For the example shown in Figure \ref{fig:dir_partial_deriv}, both $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ have units of $L/m^3$ since $f$ measures density ($L/m^2$) and a length in the domain is measured in meters.

If the domain is described with polar coordinates, then the measured changes in the radial or angular directions must have units of meters.

This causes an apparent problem in the angular direction, since $\varphi$ is unit-less.

However, the length measured is $r\triangle \varphi$, not just $\triangle \varphi$.

%Note that the measurement in the angular direction is not $\triangle \varphi$, since the angle $\varphi$ is unitless.

Unlike the values of $r$ and $\triangle \varphi$, the measured length does not depend upon the location of the origin in polar coordinates.

%Instead, the length measured ($r\triangle \varphi$) depends on both the angle and distance $r$ of this measurement from the origin.

In fact, the value of $\triangle \varphi$ is tied to the value of $r$ for a given measured length, as shown in Figure \ref{fig:dir_partial_deriv}.

Hence, $\frac{\partial f}{r \partial \varphi}$ is the measurable partial derivative in the angular direction in polar coordinates.

The measurable partial derivatives in polar, cylindrical, and spherical coordinates are thus:

\noindent

\[

\begin{array}{l|lc}

\multirow{2}{*}{\textrm{Polar Coordinates}}& \textrm{Partial derivative along radial line: } & \frac{\partial f}{\partial r}\\

& \textrm{Partial derivative in angular direction: } & \frac{1}{r} \; \frac{\partial f}{\partial \varphi} \\

\hline

\multirow{3}{*}{\textrm{Cylindrical Coordinates}}& \textrm{Partial derivative along radial line: } & \frac{\partial f}{\partial r}\\

& \textrm{Partial derivative in angular direction: } & \frac{1}{r} \; \frac{\partial f}{ \partial \varphi} \\

& \textrm{Partial derivative in vertical direction: } & \frac{\partial f}{\partial z} \\

\hline

\multirow{3}{*}{\textrm{Spherical Coordinates}} & \textrm{Partial derivative along radial line: } & \frac{\partial f}{\partial r}\\

& \textrm{Partial derivative along line of longitude: } & \frac{1}{r} \; \frac{\partial f}{ \partial \theta} \\

& \textrm{Partial derivative along line of latitude: } & \frac{1}{r \sin \theta}\; \frac{\partial f}{\partial \varphi} \\

\end{array}

\]

%Derivatives are the ratio of small quantities.

%

% \begin{enumerate}

% \item Ratio of small (measurable) change in quantities (vs. operator)

% \item {\it Tilt}

% \item rectangular coordinates

% \item polar coordinates ( $\frac{\partial f}{\partial \varphi}$ vs. $\frac{\partial f}{r\partial \varphi}$). Picture of change in angle depends upon distance away from origin - necessitating need for dependence on $r$.

% \end{enumerate}