Rate of change requires two important pieces of information: \begin{enumerate} \item A specified direction from a point \item The ratio of small change in the function to small change in the specified direction \end{enumerate} \noindent These quantities can be combined to form the direction and magnitude of a vector, which we call {\bf Directional Derivative Vectors}. Directional derivative vectors have three important geometric features. First, as shown in the second image in Figure \ref{fig:dir_deriv_projection}, the set of these vectors form two circles of radius $|\vec{\nabla} f|$, one where the rate of change in any direction is positive and one where the rate of change in any direction is negative. If the magnitude is allowed to be negative, these two circles form a {\it Figure Eight} and each circle is tangent to the level curve through the point. Second, the rate of change in any given direction is proportional to the length of the chord extending from the base point out to the edge of the circle in the given direction. Third, the length of this chord is the perpendicular projection from the gradient vector onto the unit vector extending in the given direction, as shown in the first image of Figure \ref{fig:dir_deriv_projection}. Let {\bf Directional Derivative Sphere} refer to the set of directional derivative vectors. Then, as shown in Figure \ref{fig:dir_deriv_generalization}, the diameter of the directional derivative sphere is equal to $\left| \vec{\nabla} f \right|$ and inversely proportional to the space between level curves. The geometric properties of the directional derivative vectors extend to higher dimensions, providing a pair of $n-1$ dimensional spheres for a function of $n$ variables which have diameter $|\vec{\nabla}|$. From the diagram of these spheres, it is evident that if $m$ is a rate of change with $-\left| \vec{\nabla} f \right| < m < \left| \vec{\nabla} f \right|$, then is an $n-2$ sphere of direction vectors in which the directional derivative of $f$ is $m$. For a function of $n$ dimensions, the gradient vector can be recovered using any set of $n$ linearly independent directional derivative vectors. If the vectors are pairwise orthogonal, then their signed sum\footnote{Directional derivative vectors from the negatively-oriented circle must be subtracted.} is the gradient vector. For the case in which the vectors are not pairwise orthogonal, notice that the perpendicular bisector of any chord of any given sphere must pass through the center of that sphere. Hence, the $n-1$ directional derivative sphere can also be determined from $n$ directional derivative vectors for a function of $n$ variables. Stated another way, the tangent plane to a multivariable function is determined by the value of $n$ linearly independent tangent vectors. \noindent \begin{center} \begin{figure}[tbp] \fbox{\resizebox{2.6in}{!}{\includegraphics{../images/instructors_guide_illustrations/gradient_directional_derivative_projected_onto_unit_vector_with_circle}}} \hspace{.5cm} \fbox{\resizebox{2.6in}{!}{\includegraphics{../images/instructors_guide_illustrations/gradient_projected_onto_different_unit_directions}}} \caption[Directional Derivative Vectors]{The rate of change in any direction $\hat{u}$ plotted as a vector, or {\it directional derivative vector}. These vectors form a {\it Figure Eight} of vectors, called a {\it Directional Derivative Sphere}. Vectors from the sphere have magnitude $\vec{\nabla} f \cdot \hat{u}$.} \label{fig:dir_deriv_projection} \end{figure} \end{center} %The rate of change in a direction $\hat{u}$ is the projection of the vector $\vec{\nabla} f$ onto the given direction. This rate of change and direction can be represented with a vector. The result gives two circles of vectors of diameter $\left|\vec{\nabla} f\right|$ which are perpendicular to the level curve through a point. \noindent \begin{figure}[tbp] \centering \begin{subfigure}[t]{0.42\textwidth} \centering \fbox{\resizebox{2.6in}{!}{\includegraphics{../images/instructors_guide_illustrations/double_circles_on_level_curves}}} \caption{The diameter of a directional derivative vector sphere is inversely related to the space between the level curves.} \label{fig:dir_deriv_generalization_level_curves} \end{subfigure} \hspace{.5cm} \begin{subfigure}[t]{0.42\textwidth} \centering \fbox{\resizebox{2.6in}{!}{\includegraphics[clip=true, trim=.5in 1.33in .5in 1.33in]{../images/instructors_guide_illustrations/higher_dimensions}}} \caption{The directional derivative vector spheres naturally extend to higher dimensional settings. Direction vectors through the blue ring have the same rate of change.} \label{fig:dir_deriv_generalization_higher_dimensions} \end{subfigure} \caption[Directional derivative vector circles]{The diameter of the directional derivative circles are inversely proportional to space between the level curves.} \label{fig:dir_deriv_generalization} \end{figure}