Which Way Is Uphill for a Rover?

A rover can only climb so steep a slope. The directional derivative tells it how steep the ground is in whatever direction it heads, and the gradient points straight up the steepest way.

Drag on the surface to move the point, or click a red marker to jump to it.

How it works

Directional derivatives and the gradient

In one sentence

The directional derivative of $h$ in the direction of a unit vector $\mathbf u$ is $D_{\mathbf u}h = \nabla h\cdot\mathbf u$ ; it is maximized when $\mathbf u$ points along $\nabla h$ , with maximum value $\lVert\nabla h\rVert$ .

General formula

{\htmlData{sym=0}{D_{\mathbf u}h}} = {\htmlData{sym=1}{\nabla h}}\cdot{\htmlData{sym=2}{\mathbf u}} = {\htmlData{sym=4}{h_x}}\cos{\htmlData{sym=3}{\theta}} + {\htmlData{sym=5}{h_y}}\sin{\htmlData{sym=3}{\theta}}, \qquad \max_{{\htmlData{sym=2}{\mathbf u}}} {\htmlData{sym=0}{D_{\mathbf u}h}} = \lVert{\htmlData{sym=1}{\nabla h}}\rVert

The dot product keeps only the part of the gradient that points along your heading: all of it straight uphill, none of it along a contour.

What the symbols mean

$D_{\mathbf u}h$
Directional derivative
The slope of the ground in the direction the rover is heading.
$\nabla h$
Gradient
The arrow pointing straight uphill; its length is the steepest slope available.
$\mathbf u$
Heading
A unit vector (length 1) pointing where the rover drives.
$\theta$
Heading angle
The rover's direction, measured counterclockwise from east.
$h_x$
East-west slope
How fast height changes going east.
$h_y$
North-south slope
How fast height changes going north.

Pick a heading $\theta$ ; the unit vector $\mathbf u = (\cos\theta, \sin\theta)$ is that direction on the map.
$D_{\mathbf u}h = \nabla h\cdot\mathbf u$ is the slope of the ground the rover feels driving that way: the pink slice through the point.
It is largest, $\lVert\nabla h\rVert$ , when $\mathbf u$ points along $\nabla h$ , and zero when the rover drives along a contour line.
A rover that can climb a slope of at most $m$ can head in any direction with $D_{\mathbf u}h \le m$ .

Learn the math

Directional Derivatives and the GradientCalc III · Lesson 24, Partial Derivatives The Gradient, Tangent Planes and Normal LinesCalc III · Lesson 26, Applications of Partial Derivatives