Gradient

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See image gradient for the term used in graphics; slope for the measure of steepness of a straight line, and grade for the grade or gradient of roads and other geographic features.

In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows.

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.

1 Interpretations of the gradient
2 Formal definition
- 2.1 Example
3 Linear approximation to a function
4 The gradient on manifolds
5 See also

[edit] Interpretations of the gradient

Consider a room in which the temperature is given by a scalar field $φ$ , so at each point $(x, y, z)$ the temperature is $φ(x, y, z)$ . 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 tell how fast the temperature rises in that direction.

Consider a hill whose height at a point $(x, y)$ is $H (x, y)$ . The gradient of $H$ at a point is in the direction of the steepest slope or grade at that point. The magnitude of the gradient tells how steep the slope actually is.

The gradient can also be used to tell how things change in other directions rather than the direction of largest change. Consider again the example with the hill. One can have a road which goes right uphill where the slope is largest and then its slope is the magnitude of the gradient. Or one can have a road which goes under an angle with the uphill direction, say for example an angle of 60° when projected onto the horizontal plane. Then, if the steepest slope on the hill is 40%, the road will make a shallower slope of 20% which is 40% times the cosine of 60°.

This observation can be mathematically stated as follows. The gradient of the hill height function $H$ dotted with a unit vector gives the slope of the surface in the direction of the vector. This is called the directional derivative.

The gradient is irrotational (and vice versa) and thus line integrals through a gradient field are path independent and can be evaluated with the gradient theorem.

[edit] Formal definition

The gradient of a scalar function $f (x)$ with respect to a vector variable $x = (x_1,\dots,x_n)$ is denoted by $\nabla f$ where $\nabla$ (nabla) denotes the vector differential operator del. These other symbols are equivalent and carry the same meaning: $\nabla_x f(x)$ , $\operatorname{grad}(f)$ .

By definition, the gradient is a column vector whose components are the partial derivatives of $f$ . That is:

$\nabla f = \left(\frac{\partial f}{\partial x_1 }, \dots, \frac{\partial f}{\partial x_n } \right)^T$

Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as it should be, in view of the geometric definition.

[edit] Example

In 3 dimensions, the expression expands to $\nabla f = \begin{pmatrix} {\frac{\partial f}{\partial x}}, {\frac{\partial f}{\partial y}}, {\frac{\partial f}{\partial z}} \end{pmatrix}^T$ in Cartesian coordinates. For example, the gradient of the function

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

is:

$\nabla f= \begin{pmatrix} {\frac{\partial f}{\partial x}}, {\frac{\partial f}{\partial y}}, {\frac{\partial f}{\partial z}} \end{pmatrix}^T = \begin{pmatrix} {2}, {6y}, {-\cos(z)} \end{pmatrix}^T.$

[edit] Linear approximation to a function

The gradient of a function $f$ from the Euclidean space Rⁿ to R characterizes the best linear approximation to that function at any particular point $x 0$ in Rⁿ. The approximation is as follows:

$f(x) \approx f(x_0) + (\nabla_x f(x_0))^T (x-x_0)$

for $x$ close to $x 0$ , where $\nabla_x f(x_0)$ is the gradient computed at $x 0$ .

[edit] The gradient on manifolds

For any differentiable function f on a Riemannian manifold M, the gradient of f is the vector field such that for any vector $ξ$ ,

$\langle \nabla f(x), \xi \rangle := \xi f \,\!$

where $\langle \cdot, \cdot \rangle$ denotes the inner product on M (the metric) and $\xi\, f$ is the function that takes any point p to the directional derivative of $f$ in the direction $ξ$ evaluated at p. In other words, under some coordinate chart $\varphi$ , $\xi\, f (p)$ will be:

$\sum \xi_{x_{j}} (\partial_{j}f \mid_{p}) := \sum \xi_{x_{j}} (\frac{\partial}{\partial x_{j} }(f \circ \varphi^{-1}) \mid_{\varphi(p)}).$

The gradient of a function is related to the exterior derivative, since $\xi\, f (p) = df(\xi)$ . Indeed, the metric allows one to associate canonically the 1-form df to the vector field $\nabla f$ . In Rⁿ the flat metric is implicit and the gradient can be identified with the exterior derivative.