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Gradient is commonly used to describe the measure of the ''' Slope ''' (also called ''steepness'', ''fall'' or ''incline'') of a Straight Line . In the UK it is the usual term for the ''inclination'' of a Surface along a given direction, which is usually called the '' Grade '' in the U.S.A . Given a surface, the grade (inclination) of the surface in a particular direction given a unit vector is the Dot Product of the vector gradient with that vector.

A generalization of these concepts is the gradient in Vector Calculus ; and this article will be mostly about this vector gradient. 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 \phi, so at each point (x,y,z) the temperature is \phi(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 it gets hot most quickly. The magnitude of the gradient will tell how fast it gets hot 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 / 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 .


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
abla f where
abla ( Nabla ) denotes the vector Differential Operator Del .
These other symbols are equivalent and carry the same meaning:
abla_x f(x), grad(f).


By definition, the gradient is a column Vector whose components are the Partial Derivatives of f. That is:

:
abla f = \left( rac{\partial f}{\partial x_1 }, \dots, rac{\partial f}{\partial x_n } ight)

Although expressed in terms of coordinates, the result is invariant under Orthogonal Transformation s, as it should, in view of the geometric definition.


Example


In 3 Dimension s, the expression expands to
abla f = \begin{pmatrix}
{ rac{\partial f}{\partial x}},
{ rac{\partial f}{\partial y}},
{ rac{\partial f}{\partial z}}
\end{pmatrix} in Cartesian Coordinates .
For example, the gradient of the function
: f(x,y,z)=2x+3y^2-\sin(z)
is:
:
abla f= \begin{pmatrix}
{ rac{\partial f}{\partial x}},
{ rac{\partial f}{\partial y}},
{ rac{\partial f}{\partial z}}
\end{pmatrix} =
\begin{pmatrix}
{2},
{6y},
{-\cos(z)}
\end{pmatrix}.


LINEAR APPROXIMATION TO A FUNCTION


The gradient of a Function f from the Euclidean Space R''n'' to R charaterizes the best Linear Approximation to that function at any particular point x_0 in R''n''.
The approximation is as follows:
: g(x) = f(x_0) + (
abla_x f(x_0))^T (x-x_0)
where
abla_x f(x_0) is the gradient computed at x_0.


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 \xi,
:\langle
abla f(x), \xi angle := \xi f
where \langle \cdot, \cdot angle 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 \xi evaluated at ''p''. In other words, under some Coordinate Chart arphi, \xi f (p) will be:

:\sum \xi_{x_{j}} (\partial_{j}f \mid_{p}) := \sum \xi_{x_{j}} ( rac{\partial}{\partial x_{j} }(f \circ arphi^{-1}) \mid_{ arphi(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
abla f. In R''n'' the flat metric is implicit and the gradient can be identified with the exterior derivative.


SEE ALSO