\noindent
\begin{figure}[htbp]
\centering
% \begin{subfigure}[t]{0.3\textwidth}
% \centering
% \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/gradient_surface4}}}
\caption{Gradient vectors point in the direction of greatest increase of the function $f$.}
% \caption{$\frac{\partial f}{\partial x} = 0$ and $\frac{\partial f}{\partial y} > 0$}
\label{fig:partialy_gradient}
\end{subfigure}
\hspace{.5cm}
\begin{subfigure}[t]{0.3\textwidth}
\centering
\fbox{\resizebox{2in}{!}{\includegraphics{../images/instructors_guide_illustrations/gradient_surface3}}}
\caption{$\left|\vec{\nabla} f\right|$ is the rate of change in the indicated direction.}
% \caption{$\frac{\partial f}{\partial x} < 0$ and $\frac{\partial f}{\partial y} = 0$}
\label{fig:partialx_gradient}
\end{subfigure}
\hspace{.5cm}
\begin{subfigure}[t]{0.3\textwidth}
\centering
\fbox{\resizebox{2in}{!}{\includegraphics{../images/instructors_guide_illustrations/gradient_surface2}}}
\caption{The gradient lives in the domain. $\vec{\nabla} f$ is perpendicular to level curves and $\left| \vec{\nabla} f\right|$ varies inversely to the spacing of the level curves.}
% \caption{$\frac{\partial f}{\partial x} < 0$ and $\frac{\partial f}{\partial y} < 0$}
\label{fig:partialxandy_gradient}
\end{subfigure}
\caption[Gradient, level curves, and the function's graph]{The geometry of gradient vector in relation to level curves and the spacing of level curves.}
\label{fig:gradient_vectors}
\end{figure}
\noindent
\begin{center}
\begin{figure}[tbp]
\fbox{\resizebox{5.2in}{!}{\includegraphics{../images/instructors_guide_illustrations/level_curves_gradient_vectors}}}
\caption[Gradient vector field on level curves and scale]{The gradient vector field overlaying the level curves. Notice the three scales on one plot: the domain ($m$), the level curves ($L/m^2$), and the gradient vectors ($L/m^3$).}
\label{fig:gradient_level_curves}
\end{figure}
\end{center}
The value of a multivariable function $f$ will depend upon the value of multiple inputs.
Hence, the rate of change of $f$ will depend upon how much of each of these inputs in changed.
At a given point $P$, the {\it direction providing the greatest rate of change} in $f$, and the {\it amount of that change}, is encoded into a vector called the {\bf gradient vector} $\vec{\nabla} f$.
The gradient vector $\vec{\nabla} f$ exists in the domain, and is {\it flat}, as illustrated in Figure \ref{fig:gradient_vectors}. Further, $\left| \vec{\nabla} f \right|$ is inversely proportional to the spacing of the local level curves, as shown in Figure \ref{fig:gradient_level_curves}.
Since $f$ increases most quickly in the direction of the gradient, then it will decrease most rapidly in the direction of $-\vec{\nabla} f$.
%At a point $P$, the function will increase most quickly in the direction of $\vec{\nabla} f$ and decrease by the same rate in the opposite direction, namely $-\vec{\nabla} f$.
%The direction of greatest increase in $f$ must be opposite the direction of greatest decrease;
By symmetry, the function will have a zero rate of change in directions orthogonal to $\vec{\nabla} f$ and $-\vec{\nabla} f$.
Similar to partial derivatives, the units of $\vec{\nabla} f$ will be the units of change in $f$ per unit of change in the domain. For example, the units of $\vec{\nabla} f$ in Figure \ref{fig:gradient_level_curves} will be $\left(\frac{L}{m^2}\right) / m$ or $\frac{L}{m^3}$. More importantly, note that Figure \ref{fig:gradient_level_curves} contains three units of measurement: (a) units of the domain ($m$), (b) units of the function on the level curves ($L/m^2$), and (c) units for the gradient vectors ($L/m^3$)
The gradient can also be expressed using coordinates, with the amount of each coordinate direction in proportion to the corresponding partial derivatives. That is, a function $f$ increasing $5$ times more quickly in the $x$ direction than the $y$ direction will have a gradient vector $\vec{\nabla} f$ pointing $5$ times further in the $x$ direction than the $y$ direction. Indeed, in rectangular coordinates,
\[ \vec{\nabla} f = \frac{\partial f}{\partial x} \hat{\imath} + \frac{\partial f}{\partial y} \hat{\jmath} \]
Thus, if $\frac{\partial f}{\partial x} < 0$ at a point $P$, then the gradient vector will point in the negative $x$ direction (and similarly for the $y$ direction).
The maximum rate of change of $f$ at the point $P$ is given by
\[ \left| \vec{\nabla} f \right| = \sqrt{\vec{\nabla} f \cdot \vec{\nabla} f } = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2} \]
\noindent
In polar coordinates, these expressions are
\[
\vec{\nabla} f = \frac{\partial f}{\partial r} \hat{r} + \frac{1}{r} \frac{\partial f}{\partial \varphi} \hat{\varphi}
\textrm{\hspace{.5cm} and \hspace{.5cm}}
\left| \vec{\nabla} f \right| = \sqrt{\vec{\nabla} f \cdot \vec{\nabla} f} = \sqrt{\left(\frac{\partial f}{\partial r}\right)^2 + \left(\frac{1}{r} \frac{\partial f}{\partial \varphi}\right)^2}
\]
\noindent
Notice the factor of $\frac{1}{r}$ the expression above so that $\frac{1}{r} \frac{\partial f}{\partial r}$ is a ratio of small measurable changes.
\noindent
The gradient vectors for cylindrical and spherical coordinates are:
\[
\begin{array}{ll}
\textrm{cylindrical coordinates:} & \vec{\nabla} f = \frac{\partial f}{\partial r}\; \hat{r} + \frac{1}{r} \frac{\partial f}{\partial \varphi} \; \hat{\varphi} + \frac{\partial f}{\partial z} \; \hat{z} \\[8pt]
\textrm{spherical coordinates:} & \vec{\nabla} f = \frac{\partial f}{\partial r}\; \hat{r} + \frac{1}{r} \frac{\partial f}{\partial \theta} \; \hat{\theta} + \frac{1}{r\sin \theta}\frac{\partial f}{\partial \varphi } \; \hat{\varphi} \\[8pt]
\end{array}
\]
% \begin{enumerate}
% \item Combination of units
% \item Lives in Domain
% \item Rectangular coordinates
% \item Polar coordinates (thought experiment)
% \end{enumerate}