Directional Derivatives

The Grids | Calculate | Calculus directional derivatives provide you with information about the slope, or rate of change of slope, of the gridded surface in a specified direction. Because this takes a specified direction into account, this slope, or rate of change in slope, might not be the steepest slope at a given point. For example, if the specified direction is due East, but the gradient is due North, the directional derivative slope is zero at that point. In the specified direction, there is no slope at that point, although there is a slope to the North.

Directional derivatives are different from the values given in Terrain Modeling where the direction of the slope is defined as the gradient, or the direction of steepest ascent at a given point (i.e., straight uphill at that point). In the above example, the terrain model would report the slope in the north direction at that point. The two methods would report different values at that particular point on the grid.

Let Z be defined as a function of X and Y in a domain that includes point P. At what rate does Z change if we move from point P in a specified direction? In the X axis direction, the rate of change for Z is $image\img00151.png$ , and the rate of change in the Y axis direction for Z is $image\img00152.png$ .

Definition - The Directional Derivative (Schwartz, 1974)

Let f(x,

$image\img00154.png$ ,

such that the line segment PQ lies in D, and let

$image\img00155.png$ .

Then

$image\img00156.png$

is the directional derivative of f(x,y) at P in the direction of

There are three directional derivative options available: First Derivative, Second Derivative, and Curvature. You can specify the angular direction for the operation.

First Derivative

Second Derivative

Curvature

Directional Derivative Example

This example is based on the Demogrid.grd file included with Surfer, although the limits of the grid have been extended to 10 units in both the X and Y dimensions. The map shows the contours and the direction chosen for the Grids | Calculate | Calculus directional derivative operations. A profile line is also included on the map. The graphs included are based on Grids | Calculate | Slice command data collected along the profile lines. Each operation was performed, and the Slice command was applied to the resultant grid files. The results are shown in the graphs that were produced in Golden Software, LLC's Grapher™ . A comparison of the map and the graphs gives a good visual analysis of the results of the directional derivative operations.

This map shows the profile line for the directional derivative and directional curvature calculations.

These are cross sectional views of Curvature, Second, and First Derivatives, and the original grid surface.

Directional Derivatives

Directional Derivative Example

See Also