Gradient

In mathematics, the gradient is a generalization of the usual concept of derivative of a function in one dimension to a function in several dimensions. If f(x₁, ..., x_n) is a differentiable, scalar-valued function of standard Cartesian coordinates in Euclidean space, its gradient is the vector whose components are the n partial derivatives of f. It is thus a vector-valued function.

Similarly to the usual derivative, the gradient represents the slope of the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph. This characterizing property of the gradient allows it to be defined independently of a choice of coordinate system, as a vector field whose components in a coordinate system will transform when going from one coordinate system to another.

Gradient (disambiguation)

The Gradient is the rate of variation of a numerical quantity. It may refer to:

Mathematics

Biology

Fluid dynamics and earth science

See also

Four-gradient

In differential geometry, the four-gradient is the four-vector analogue of the gradient from Gibbs-Heaviside vector calculus.

Definition

The covariant components compactly written in index notation are:

The comma in the last part above implies the partial differentiation with respect to . This is not the same as a semi-colon, used for the covariant derivative.

The contravariant components are:

where g^αβ is the metric tensor, which here has been chosen for flat spacetime with the metric signature (+,−,−,−).

Alternative symbols to are and D.

Usage

The 4-Gradient is used in a number of different ways in Special Relativity:

As a 4-Divergence

The 4-Divergence of the 4-Position gives the dimensionality of spacetime:

The 4-Divergence of the 4-CurrentDensity gives a conservation law - the conservation of charge:

This means that the time rate of change of the charge density must equal the negative spatial divergence of the current density . In other words, the charge inside a box cannot just change arbitrarily, it must enter and leave the box via a current. This is a continuity equation.