Applet: Directional derivative on a mountain shown as level curves
The height of a mountain ranged described by a function $f(x,y)$ is shown as a level curve plot. A point $\vc{a}$ (in dark red) can be moved with the mouse. The height $f(\vc{a})$ is shown on the bottom cyan slider labeled by “f”. The direction vector $\vc{u}$ (the light green vector) points at an angle $\theta$ from east, which can be changed by dragging the red point on the top slider. The value of the directional derivative $D_{\vc{u}}f(\vc{a})$ is shown by the middle (light green) slider labeled by “Duf”.
If you set $\vc{u}$ to point straight east ($\theta=0$ in the applet), then $\vc{u}$ points in the positive $x$ direction ($\vc{u}=(1,0)$) so that $\displaystyle D_{\vc{u}}f(\vc{a}) = \pdiff{f}{x}(\vc{a})$. Similarly, when $\vc{u}$ points straight north ($\theta=\pi/2$), then $\vc{u}$ points in the positive $y$ direction ($\vc{u}=(0,1)$) so that $\displaystyle D_{\vc{u}}f(\vc{a}) = \pdiff{f}{y}(\vc{a})$.
f you make $\vc{u}$ point in a direction parallel to the level curve, what happens to $D_{\vc{u}} f(\vc{a})$? (Since the height is constant along a level curve, you should be able to infer what the slope in that direction should be.) What happens to $D_{\vc{u}}f(\vc{a})$ when you turn $\vc{u}$ to point in the opposite direction (i.e., add or subtract $\pi$ from $\theta$)?
Applet file: mountcondirect.m
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