Gradient

Wikipedia:

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is that rate of increase.

A generalization of the gradient for functions on a Euclidean space that have values in another Euclidean space is the Jacobian. A further generalization for a function from one Banach space to another is the Fréchet derivative.

Interpretations

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 over time.) At each point in the room, the gradient of at that point will show the direction the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature rises in that direction.

Consider a surface whose height above sea level at a point is . The gradient of at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector.

The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, by taking a dot product. Suppose that the steepest slope on a hill is 40%. If a road goes directly up the hill, then the steepest slope on the road will also be 40%. If, instead, the road goes around the hill at an angle, then it will have a shallower slope. For example, if the angle between the road and the uphill direction, projected onto the horizontal plane, is 60°, then the steepest slope along the road will be 20%, which is 40% times the cosine of 60°.

This observation can be mathematically stated as follows. If the hill height function is differentiable, then the gradient of dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely, when is differentiable, the dot product of the gradient of with a given unit vector is equal to the directional derivative of in the direction of that unit vector.

Definition

The gradient (or gradient vector field) of a scalar function is denoted or where (the nabla symbol) denotes the vector differential operator, del. The notation is also commonly used for the gradient. The gradient of is defined as the unique vector field whose dot product with any unit vector at each point is the directional derivative of along . That is,

In a rectangular coordinate system, the gradient is the vector field whose components are the partial derivatives of :

where the are the orthogonal unit vectors pointing in the coordinate directions. When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only.

In the three-dimensional Cartesian coordinate system, this is given by

where are the standard unit vectors. For example, the gradient of the function

is:

In some applications it is customary to represent the gradient as a row vector or column vector of its components in a rectangular coordinate system.

Gradient and the derivative or differential

Linear approximation to a function

The gradient of a function from the Euclidean space to at any particular point in characterizes the best linear approximation to at . The approximation is as follows: for close to , where is the gradient of computed at , and the dot denotes the dot product on . This equation is equivalent to the first two terms in the multi-variable Taylor Series expansion of at .

Differential or (exterior) derivative

The best linear approximation to a function at a point in is a linear map from to which is often denoted by or and called the differential or (total) derivative of at . The gradient is therefore related to the differential by the formula for any . The function , which maps to , is called the differential or exterior derivative of and is an example of a differential 1-form.

If is viewed as the space of (length ) column vectors (of real numbers), then one can regard as the row vector

so that is given by matrix multiplication. The gradient is then the corresponding column vector, i.e., .

Gradient as a derivative

Let be an open set in . If the function : →  is differentiable, then the differential of is the (Fréchet) derivative of . Thus is a function from to the space such that

where • is the dot product.

As a consequence, the usual properties of the derivative hold for the gradient:

The gradient is linear in the sense that if and are two real-valued functions differentiable at the point ∈, and α and β are two constants, then α+β is differentiable at , and moreover

If and are real-valued functions differentiable at a point ∈, then the product rule asserts that the product (fg)() = ()() of the functions and is differentiable at , and

Suppose that :→ is a real-valued function defined on a subset of , and that is differentiable at a point . There are two forms of the chain rule applying to the gradient. First, suppose that the function is a parametric curve; that is, a function  : → maps a subset ⊂ into . If is differentiable at a point ∈ such that () = , then

where is the composition operator. More generally, if instead ⊂, then the following holds:

where (Dg) denotes the transpose Jacobian matrix.

For the second form of the chain rule, suppose that  : → is a real valued function on a subset of , and that is differentiable at the point = () ∈ . Then

Further properties and applications

Level sets

If the partial derivatives of are continuous, then the dot product of the gradient at a point with a vector gives the directional derivative of at in the direction . It follows that in this case the gradient of is orthogonal to the level sets of . For example, a level surface in three-dimensional space is defined by an equation of the form (, , ) = . The gradient of is then normal to the surface.

More generally, any embedded hypersurface in a Riemannian manifold can be cut out by an equation of the form () = 0 such that dF is nowhere zero. The gradient of is then normal to the hypersurface.

Let us consider a function at a point P. If we draw a surface through this point P and the function has the same value at all points on this surface,then this surface is called a 'level surface'.

Conservative vector fields and the gradient theorem

The gradient of a function is called a gradient field. A (continuous) gradient field is always a conservative vector field: its line integral along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative vector field is always the gradient of a function.

Riemannian manifolds

For any smooth function f on a Riemannian manifold (,), the gradient of is the vector field such that for any vector field ,

where denotes the inner product of tangent vectors at defined by the metric and (sometimes denoted ()) is the function that takes any point ∈ to the directional derivative of in the direction , evaluated at . In other words, in a coordinate chart from an open subset of to an open subset of , is given by:

where denotes the th component of in this coordinate chart.

So, the local form of the gradient takes the form:

Generalizing the case =, the gradient of a function is related to its exterior derivative, since . More precisely, the gradient is the vector field associated to the differential 1-form d using the musical isomorphism (called "sharp") defined by the metric . The relation between the exterior derivative and the gradient of a function on is a special case of this in which the metric is the flat metric given by the dot product.

Cylindrical and spherical coordinates

In cylindrical coordinates, the gradient is given by (Schey 1992, pp. 139–142):

where is the azimuthal angle, is the axial coordinate, and ρ, φ and are unit vectors pointing along the coordinate directions.

In spherical coordinates (Schey 1992, pp. 139–142):

where is the azimuth angle and is the zenith angle.

For the gradient in other orthogonal coordinate systems, see Orthogonal coordinates#Differential operators in three dimensions.

Gradient of a vector

In rectangular coordinates, the gradient of a vector is defined by

or the Jacobian matrix .

In curvilinear coordinates, the gradient involves Christoffel symbols.