Gradient
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Template:EnWiktionary In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.
A generalization of the gradient, for functions which have vectorial values, is the Jacobian.
Interpretations of the gradient
Consider a room in which the temperature is given by a scalar field [math]\displaystyle{ \phi }[/math], so at each point [math]\displaystyle{ (x,y,z) }[/math] the temperature is [math]\displaystyle{ \phi(x,y,z) }[/math]. We will assume that the temperature does not change in time. Then, at each point in the room, the gradient at that point will show the direction in which the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature rises in that direction.
Consider a hill whose height above sea level at a point [math]\displaystyle{ (x, y) }[/math] is [math]\displaystyle{ H(x, y) }[/math]. The gradient of [math]\displaystyle{ H }[/math] at a point is a vector pointing in the direction of the steepest slope or 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. Consider again the example with the hill and suppose that the steepest slope on the hill is 40%. If a road goes directly up the hill, then the steepest slope on the road will also be 40%. If instead, the road goes around the hill at an angle with the uphill direction (the gradient vector), then it will have a shallower slope. For example, if the angle between the road and the uphill direction, projected onto the horizontal plane, is 60°, then the steepest slope along the road will be 20% which is 40% times the cosine of 60°.