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One of the fundamental concepts in vector analysis and the theory of non-linear mappings.

The gradient of a scalar function $f$ of a vector argument $t = ( t ^ {1} \dots t ^ {n} )$ from a Euclidean space $E ^ {n}$ is the derivative of $f$ with respect to the vector argument $t$, i.e. the $n$- dimensional vector with components $\partial f / \partial t ^ {i}$, $1 \leq i \leq n$. The following notations exist for the gradient of $f$ at $t _ {0}$:

$$\mathop{\rm grad} f ( t _ {0} ),\ \ \nabla f ( t _ {0} ),\ \ \frac{\partial f ( t _ {0} ) }{\partial t } ,\ \ f ^ { \prime } ( t _ {0} ) ,\ \ \left . \frac{\partial f }{\partial t } \right | _ {t _ {0} } .$$

The gradient is a covariant vector: the components of the gradient, computed in two different coordinate systems $t = ( t ^ {1} \dots t ^ {n} )$ and $\tau = ( \tau ^ {1} \dots \tau ^ {n} )$, are connected by the relations:

$$\frac{\partial f }{\partial t ^ {i} } ( \tau ( t)) = \ \sum _ {j = 1 } ^ { n } \frac{\partial f ( \tau ) }{\partial \tau ^ {j} } \ \frac{\partial \tau ^ {j} }{\partial t ^ {i} } .$$

The vector $f ^ { \prime } ( t _ {0} )$, with its origin at $t _ {0}$, points to the direction of fastest increase of $f$, and is orthogonal to the level lines or surfaces of $f$ passing through $t _ {0}$.

The derivative of the function at $t _ {0}$ in the direction of an arbitrary unit vector $\mathbf N = ( N ^ {1} \dots N ^ {n} )$ is equal to the projection of the gradient function onto this direction:

$$\tag{1 } \frac{\partial f ( t _ {0} ) }{\partial \mathbf N } = \ ( f ^ { \prime } ( t _ {0} ), \mathbf N ) \equiv \ \sum _ {j = 1 } ^ { n } \frac{\partial f ( t _ {0} ) }{\partial t ^ {j} } N ^ {j} = | f ^ { \prime } ( t _ {0} ) | \cos \phi ,$$

where $\phi$ is the angle between $\mathbf N$ and $f ^ { \prime } ( t _ {0} )$. The maximal directional derivative is attained if $\phi = 0$, i.e. in the direction of the gradient, and that maximum is equal to the length of the gradient.

The concept of a gradient is closely connected with the concept of the differential of a function. If $f$ is differentiable at $t _ {0}$, then, in a neighbourhood of that point,

$$\tag{2 } f ( t) = f ( t _ {0} ) + ( f ^ { \prime } ( t _ {0} ),\ t - t _ {0} ) + o ( | t - t _ {0} | ),$$

i.e. $df = ( f ^ { \prime } ( t _ {0} ), dt)$. The existence of the gradient of $f$ at $t _ {0}$ is not sufficient for formula (2) to be valid.

A point $t _ {0}$ at which $f ^ { \prime } ( t _ {0} ) = 0$ is called a stationary (critical or extremal) point of $f$. An example of such a point is a local extremal point of $f$, and the system $\partial f ( t _ {0} ) / \partial t ^ {i} = 0$, $1 \leq i \leq n$, is employed to find an extremal point $t _ {0}$.

The following formulas can be used to compute the value of the gradient:

$$\mathop{\rm grad} ( \lambda f ) = \ \lambda \mathop{\rm grad} f,\ \ \lambda = \textrm{ const } ,$$

$$\mathop{\rm grad} ( f + g) = \mathop{\rm grad} f + \mathop{\rm grad} g,$$

$$\mathop{\rm grad} ( fg) = g \mathop{\rm grad} f + f \mathop{\rm grad} g,$$

$$\mathop{\rm grad} \left ( { \frac{f}{g} } \right ) = \frac{1}{g ^ {2} } ( g \mathop{\rm grad} f - f \mathop{\rm grad} g).$$

The gradient $f ^ { \prime } ( t _ {0} )$ is the derivative at $t _ {0}$ with respect to volume of the vector function given by

$$\Phi ( E) = \ \int\limits _ {t \in \partial E } f ( t) \mathbf M ds,$$

where $E$ is a domain with boundary $\partial E$, $t _ {0} \in E$, $ds$ is the area element of $\partial E$, and $\mathbf M$ is the unit vector of the outward normal to $\partial E$. In other words,

$$f ^ { \prime } ( t _ {0} ) = \ \lim\limits \frac{\Phi ( E) }{ \mathop{\rm vol} E } \ \textrm{ as } \ \ E \rightarrow t _ {0} .$$

Formulas (1), (2) and the properties of the gradient listed above indicate that the concept of a gradient is invariant with respect to the choice of a coordinate system.

In a curvilinear coordinate system $x = ( x ^ {1} \dots x ^ {n} )$, in which the square of the linear element is

$$ds ^ {2} = \ \sum _ {i, j = 1 } ^ { n } g _ {ij} ( x) dx ^ {i} dx ^ {j} ,$$

the components of the gradient of $f$ with respect to the unit vectors tangent to coordinate lines at $x$ are

$$\sum _ {j = 1 } ^ { n } g ^ {ij} ( x) \frac{\partial f }{\partial x ^ {j} } ,\ \ 1 \leq i \leq n,$$

where the matrix $\| g ^ {ij} \|$ is the inverse of the matrix $\| g _ {ij} \|$.

The concept of a gradient for more general vector functions of a vector argument is introduced by means of equation (2). Thus, the gradient is a linear operator the effect of which on the increment $t - t _ {0}$ of the argument is to yield the principal linear part of the increment $f( t) - f( t _ {0} )$ of the vector function $f$. E.g., if $f = ( f ^ { 1 } \dots f ^ { m } )$ is an $m$- dimensional vector function of the argument $t = ( t ^ {1} \dots t ^ {n} )$, then its gradient at a point $t _ {0}$ is the Jacobi matrix $J = J ( t _ {0} )$ with components $( \partial f ^ { i } / \partial t ^ {j} ) ( t _ {0} )$, $1 \leq i \leq m$, $1 \leq j \leq n$, and

$$f ( t) = f ( t _ {0} ) + J ( t - t _ {0} ) + o ( t - t _ {0} ),$$

where $o ( t - t _ {0} )$ is an $m$- dimensional vector of length $o ( | t - t _ {0} | )$. The matrix $J$ is defined by the limit transition

$$\tag{3 } \lim\limits _ {\rho \rightarrow 0 } \ \frac{f ( t _ {0} + \rho \tau ) - f ( t _ {0} ) } \rho = J \tau ,$$

for any fixed $n$- dimensional vector $\tau$.

In an infinite-dimensional Hilbert space definition (3) is equivalent to the definition of differentiability according to Fréchet, the gradient then being identical with the Fréchet derivative.

If the values of $f$ lie in an infinite-dimensional vector space, various types of limit transitions in (3) are possible (see, for example, Gâteaux derivative).

In the theory of tensor fields on a domain of an $n$- dimensional affine space with a connection, the gradient serves to describe the principal linear part of increment of the tensor components under parallel displacement corresponding to the connection. The gradient of a tensor field

$$f ( t) = \ \{ { f _ {j _ {1} \dots j _ {q} } ^ { i _ {1} \dots i _ {p} } ( t) } : { 1 \leq i _ \alpha , j _ \beta \leq n } \}$$

of type $( p, q)$ is the tensor of type $( p, q + 1 )$ with components

$$\{ { \nabla _ {k} f _ {j _ {1} \dots j _ {q} } ^ { i _ {1} \dots i _ {p} } ( t) } : { 1 \leq k, i _ \alpha , j _ \beta \leq n } \} ,$$

where $\nabla _ {k}$ is the operator of absolute (covariant) differentiation (cf. Covariant differentiation).

The concept of a gradient is widely employed in many problems in mathematics, mechanics and physics. Many physical fields can be regarded as gradient fields (cf. Potential field).

#### References

 [1] N.E. Kochin, "Vector calculus and fundamentals of tensor calculus" , Moscow (1965) (In Russian) [2] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)