Gradient (Ofer Abarbanel online library)

The gradient is closely related to the derivative, but it is not itself a derivative: the value of the gradient at a point is a tangent vector – a vector at each point; while the value of the derivative at a point is a cotangent vector – a function of vectors at each point.[c]

They are related in that the dot product of the gradient of f at a point p with another tangent vector v equals the directional derivative of f at p of the function along v. See § Definition and relationship with the derivative. The nabla symbol, a character that looks like an upside down triangle, shown above is called Del, the vector differential operator.

The gradient can be interpreted as the “direction and rate of fastest increase”. If at a point p, the gradient of a function of several variables is not the zero vector, the direction of the gradient is the direction of fastest increase of the function at p, and its magnitude is the rate of increase in that direction.[10][11][12][13][14][15][16] Conversely, the gradient at a point is the zero vector if and only if the derivative vanishes at that point (a stationary point). The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent.

The gradient admits multiple generalizations to more general functions on manifolds; see § Generalizations.

Motivation

Consider a room where the temperature is given by a scalar field, T, so at each point (xyz) the temperature is T(xyz). (Assume that the temperature does not change over time.) At each point in the room, the gradient of T 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 surface whose height above sea level at point (xy) is H(xy). The gradient of H 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. Suppose that the steepest slope on a 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, 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°.

This observation can be mathematically stated as follows. If the hill height function H is differentiable, then the gradient of H dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely, when H is differentiable, the dot product of the gradient of H with a given unit vector is equal to the directional derivative of H in the direction of that unit vector.

 

Ofer Abarbanel online library