Editing Gradient (section)

==Motivation==
[[File:Vector Field of a Function's Gradient imposed over a Color Plot of that Function.svg|thumb|500px|Gradient of the 2D function {{math|1=''f''(''x'', ''y'') = ''xe''<sup>−(''x''<sup>2</sup> + ''y''<sup>2</sup>)</sup>}} is plotted as arrows over the pseudocolor plot of the function.]]

Consider a room where the temperature is given by a [[scalar field]], {{math|''T''}}, so at each point {{math|(''x'', ''y'', ''z'')}} the temperature is {{math|''T''(''x'', ''y'', ''z'')}}, independent of time. At each point in the room, the gradient of {{math|''T''}} at that point will show the direction in which the temperature rises most quickly, moving away from {{math|(''x'', ''y'', ''z'')}}. The magnitude of the gradient will determine how fast the temperature rises in that direction.

Consider a surface whose height above sea level at point {{math|(''x'', ''y'')}} is {{math|''H''(''x'', ''y'')}}. The gradient of {{math|''H''}} at a point is a plane vector pointing in the direction of the steepest slope or [[Grade (slope)|grade]] at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector.

The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, by taking a [[dot product]]. Suppose that the steepest slope on a hill is 40%. A road going directly uphill has slope 40%, but a road going around the hill at an angle will have a shallower slope. For example, if the road is at a 60° angle from the uphill direction (when both directions are projected onto the horizontal plane), then the slope along the road will be the dot product between the gradient vector and a [[unit vector]] along the road, as the dot product measures how much the unit vector along the road aligns with the steepest slope,{{efn|the dot product (the slope of the road around the hill) would be 40% if the degree between the road and the steepest slope is 0°, i.e. when they are completely aligned, and flat when the degree is 90°, i.e. when the road is perpendicular to the steepest slope.}} which is 40% times the [[cosine]] of 60°, or 20%.

More generally, if the hill height function {{math|''H''}} is [[differentiable function|differentiable]], then the gradient of {{math|''H''}} [[dot product|dotted]] with a [[unit vector]] gives the slope of the hill in the direction of the vector, the [[directional derivative]] of {{math|''H''}} along the unit vector.