Gradients and Directional Derivatives

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Gradients on a Contour Map
Steepestdescent.gif
Field: Calculus
Image Created By: Golden Software
Website: [1]

Gradients on a Contour Map

This image shows gradient vectors at different points on a contour map. These vectors show the paths of steepest descent at different points on the landscape.


Basic Description

Graph and contour map of a function: Click to enlarge

A contour map utilizes the concept of level sets. A level set is the set of all points generated when a function is made equal to a constant. For example, one level set of the function  z = x^2 -y^2 is  3=x^2-y^2 . Setting a 3-d function equal to a constant in this way yields a contour curve. These curves are curves with constant z-component, and when representing a map, they show constant height.

A contour map is simply a collection of contour curves, each with the given function set equal to a different constant, meaning each curve represents a different constant height.

Now suppose instead of seeking points of constant height, we wish to find points along which height changes most rapidly. Intuitively, we travel perpendicular to contour curves, since even partially traveling along contour curves would involve traveling along a level set. This page's main image shows a number of vectors perpendicular to contours, meaning they represent the most rapid change of height (in this case, in the descending direction) from a point.

A More Mathematical Explanation

The gradient is a useful idea for finding the path of steepest descent or ascent. For a scalar [...]

The gradient is a useful idea for finding the path of steepest descent or ascent. For a scalar function f in three variables, the gradient is defined as  \nabla f = (\partial{f}/\partial{x} , \partial{f}/\partial{y} ,\partial{f}/\partial{z}) . Intuitively, this definition means that if our function has a large rate of change in a certain direction, the gradient vector would have a large component in this direction.

The gradient thus has a domain of scalars, such as height, and has a codomain of vectors. Each vector points in the direction of steepest ascent from the point the vector originates, with the vector's magnitude corresponding to the rate of ascent one would experience if one followed the vector.




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  [[Description::This image shows gradient vectors at different points on a contour map.  These vectors show the paths of steepest descent at different points on the landscape.|]]