Gradient of the function

Gradient

In vector calculus, the gradient is a multi-variable generalization of the derivative. Whereas the ordinary derivative of a function of a single variable is a scalar-valued function, the gradient of a function of several variables is a vector-valued function.

Much as the derivative of a function of a single variable represents the slope of the tangent to the graph of the function, if at a point P, the gradient of a function of several variables is not the zero vector, it has the direction of greatest increase of the function at some point, and its magnitude is the rate of increase in that direction.

Motivation

Consider a room in which the temperature is given by a scalar field, T, so at each point (x, y, z) the temperature is T(x, y, z). (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 (x, y) is H(x, y). 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.

Definition

The gradient (or gradient vector field) of a scalar function $f(x_1, x_2, x_3, ..., x_n)$ is denoted $\nabla f$ where $\nabla$ (the nabla symbol) denotes the vector differential operator, del. The notation $\textrm{grad} f$ is also commonly used for the gradient.

In the three-dimensional Cartesian coordinate system, the gradient, if it exists, is given by:

$\nabla f = \frac{\delta f}{\delta x}\textrm{\textbf{i}}+\frac{\delta f}{\delta y}\textrm{\textbf{j}}+\frac{\delta f}{\delta z}\textrm{\textbf{k}}$

where i, j, k are the standard unit vectors in the directions of the x, y and z coordinates, respectively.

For example, the gradient of the function

$f(x,y,z)=2x+3y^2-sin(z)$

is

$\nabla f=2\textrm{\textbf{i}}+6y\textrm{\textbf{j}}-cos(z)\textrm{\textbf{k}}$

General properties

As a consequence, the usual properties of the derivative hold for the gradient:

Linearity

$\nabla(\alpha f+\beta g)=\alpha\nabla f + \beta\nabla g$

Product rule

$\nabla(fg)=f \nabla g + g \nabla f$

Conservative vector fields and the gradient theorem

The gradient of a function is called a gradient field. A (continuous) gradient field is always a conservative vector field: its line integral along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative vector field is always the gradient of a function.

References

Wikipedia
Wolfram Alpha