Gradient

A generalization of these concepts is the gradient in vector calculus; and this article is mostly about this vector gradient. The gradient of a scalar field is a vector field Vector field

In mathematics [i] a vector field is a construction in vector calculus [i] which associates a vector [i] ...

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.

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See slope Slope

The slope or the gradient is commonly used to describe the measurement of the steepness, incline o...

for the measure of steepness of a straight line.

See grade for the grade or gradient of roads and other geographic features.

A generalization of these concepts is the gradient in vector calculus; and this article is mostly about this vector gradient. The gradient of a scalar field is a vector field Vector field

In mathematics [i] a vector field is a construction in vector calculus [i] which associates a vector [i] ...

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 , so at each point the temperature is . 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 is . The gradient of at a point is in the direction of the steepest slope Slope

The slope or the gradient is commonly used to describe the measurement of the steepness, incline o...

/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 dotted Dot product

In mathematics [i], the dot product, also known as the scalar product, is a binary operation [i] w ...

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 thus line integrals through a gradient field are path independent and can be evaluated with the gradient theorem.

Formal definition

The gradient of a scalar function with respect to a vector variable is denoted by where denotes the vector differential operator del.
These other symbols are equivalent and carry the same meaning: , .

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

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

Example

In 3 dimension Dimension

In common usage, a dimension is a parameter [i] or measurement [i] required to define the characteristi ...

s, the expression expands to in Cartesian coordinates Cartesian coordinate system

In mathematics [i], the Cartesian coordinate system is used to uniquely determine each point [i]...

.
For example, the gradient of the function

is:

Linear approximation to a function

The gradient of a function from the Euclidean space Rⁿ to R characterizes the best linear approximation Linear approximation

In mathematics [i], a linear approximation is an approximation of a general function [i] using ...

to that function at any particular point in Rⁿ.
The approximation is as follows:

for close to , where is the gradient computed at .

The gradient on manifolds

For any differentiable function f on a Riemannian manifold M, the gradient of f is the vector field Vector field

In mathematics [i] a vector field is a construction in vector calculus [i] which associates a vector [i] ...

such that for any vector ,
where denotes the inner product Inner product space

In mathematics [i], an inner product space is a vector space [i] with additional structure, an inner...

on M and
is the function that takes any point p to the directional derivative of in the direction evaluated at p. In other words, under some coordinate chart, will be:

The gradient of a function is related to the exterior derivative Exterior derivative

In mathematics [i], the exterior derivative operator of differential geometry [i] extends the concept of ...

, since . Indeed, the metric allows one to associate canonically the 1-form df to the vector field . In Rⁿ the flat metric is implicit and the gradient can be identified with the exterior derivative.