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gradient
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{{about|the gradient of a multivariate function|direction and steepness of a line on a graph|Slope|the slope of a road or other physical feature|Grade (slope)|the similarly spelled unit of angle also known as gon|Gradian|other uses|Gradient (disambiguation)}}{{more footnotes|date=January 2018}}{{short description|Multi-variable generalization of the derivative of a function}}(File:Gradient2.svg|thumb|300px|In the above two images, the values of the function are represented in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows.)In vector calculus, the gradient is a multi-variable generalization of the derivative.{{harvtxt|Beauregard|Fraleigh|1973|p=84}} Whereas the ordinary derivative of a function of a single variable is a scalar-valued function, the gradient of a function of several variables is a vector-valued function. Specifically, the gradient of a differentiable function f of several variables, at a point P, is the vector whose components are the partial derivatives of f at P.{{harvtxt|Bachman|2007|p=76}}{{harvtxt|Beauregard|Fraleigh|1973|p=84}}{{harvtxt|Downing|2010|p=316}}{{harvtxt|Harper|1976|p=15}}{{harvtxt|Kreyszig|1972|p=307}}{{harvtxt|McGraw-Hill|2007|p=196}}{{harvtxt|Moise|1967|p=683}}{{harvtxt|Protter|Morrey, Jr.|1970|p=714}}{{harvtxt|Swokowski et al.|1994|p=1038}}Much as the derivative of a function of a single variable represents the slope of the tangent to the graph of the function,{{harvtxt|Protter|Morrey, Jr.|1970|pp=21,88}} if at a point P, the gradient of a function of several variables is not the zero vector, it has the direction of greatest increase of the function at P, and its magnitude is the rate of increase in that direction.{{harvtxt|Bachman|2007|p=77}}{{harvtxt|Downing|2010|pp=316–317}}{{harvtxt|Kreyszig|1972|p=309}}{{harvtxt|McGraw-Hill|2007|p=196}}{{harvtxt|Moise|1967|p=684}}{{harvtxt|Protter|Morrey, Jr.|1970|p=715}}{{harvtxt|Swokowski et al.|1994|pp=1036,1038–1039}}The magnitude and direction of the gradient vector are independent of the particular coordinate representation.{{harvtxt|Kreyszig|1972|pp=308–309}}{{harvtxt|Stoker|1969|p=292}}The Jacobian is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds.{{harvtxt|Beauregard|Fraleigh|1973|pp=87,248}}{{harvtxt|Kreyszig|1972|pp=333,353,496}} A further generalization for a function between Banach spaces is the Fréchet derivative.

Motivation

(File:Gradient of a Function.tif|thumb|350px|Gradient of the 2D function {{math|1=f(x, y) = xe−(x2 + y2)}} is plotted as blue arrows over the pseudocolor plot of the function.)Consider a room in which the temperature is given by a scalar field, {{math|T}}, so at each point {{math|(x, y, z)}} the temperature is {{math|T(x, y, z)}}. (Assume that the temperature does not change over time.) At each point in the room, the gradient of {{math|T}} at that point will show the direction in which 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 point {{math|(x, y)}} is {{math|H(x, y)}}. The gradient of {{math|H}} 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 {{math|H}} is differentiable, then the gradient of {{math|H}} dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely, when {{math|H}} is differentiable, the dot product of the gradient of {{math|H}} with a given unit vector is equal to the directional derivative of {{math|H}} in the direction of that unit vector.

Definition

File:3d-gradient-cos.svg|thumb|350px|The gradient of the function {{math|f(x,y) {{=}} −(cos2x + cos2y)2}} depicted as a projected vector fieldvector fieldThe gradient (or gradient vector field) of a scalar function {{math|f(x1, x2, x3, ..., xn)}} is denoted {{math|∇f}} or {{math|{{vec|∇}}f}} where {{math|∇}} (the nabla symbol) denotes the vector differential operator, del. The notation {{math|grad f}} is also commonly used for the gradient. The gradient of {{math|f}} is defined as the unique vector field whose dot product with any unit vector {{math|v}} at each point {{math|x}} is the directional derivative of {{math|f}} along {{math|v}}. That is,
big(nabla f(x)big)cdot mathbf{v} = D_{mathbf v}f(x).
When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only (see Spatial gradient).

Cartesian coordinates

In the three-dimensional Cartesian coordinate system with a Euclidean metric, the gradient, if it exists, is given by:
nabla f = frac{partial f}{partial x} mathbf{i} + frac{partial f}{partial y} mathbf{j} + frac{partial f}{partial z} mathbf{k},
where {{math|i}}, {{math|j}}, {{math|k}} are the standard unit vectors in the directions of the {{math|x}}, {{math|y}} and {{math|z}} coordinates, respectively. For example, the gradient of the function
f(x,y,z)= 2x+3y^2-sin(z)
is
nabla f = 2mathbf{i}+ 6ymathbf{j} -cos(z)mathbf{k}.
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.

Cylindrical and spherical coordinates

In cylindrical coordinates with a Euclidean metric, the gradient is given by:{{harvnb|Schey|1992|pp=139–142}}.
nabla f(rho, varphi, z) = frac{partial f}{partial rho}mathbf{e}_rho + frac{1}{rho}frac{partial f}{partial varphi}mathbf{e}_varphi + frac{partial f}{partial z}mathbf{e}_z,
where {{math|ρ}} is the axial distance, {{math|φ}} is the azimuthal or azimuth angle, {{math|z}} is the axial coordinate, and {{math|eρ}}, {{math|eφ}} and {{math|ez}} are unit vectors pointing along the coordinate directions.In spherical coordinates, the gradient is given by:
nabla f(r, theta, varphi) = frac{partial f}{partial r}mathbf{e}_r + frac{1}{r}frac{partial f}{partial theta}mathbf{e}_theta + frac{1}{r sintheta}frac{partial f}{partial varphi}mathbf{e}_varphi,
where {{math|r}} is the radial distance, {{math|φ}} is the azimuthal angle and {{math|θ}} is the polar angle, and {{math|er}}, {{math|eθ}} and {{math|eφ}} are again local unit vectors pointing in the coordinate directions (i.e. the normalized covariant basis).For the gradient in other orthogonal coordinate systems, see Orthogonal coordinates (Differential operators in three dimensions).

General coordinates

We consider general coordinates, which we write as {{math|x1, ..., x'i, ..., x'n}}, where {{mvar|n}} is the number of dimensions of the domain. Here, the upper index refers to the position in the list of the coordinate or component, so {{math|x2}} refers to the second component—not the quantity {{math|x}} squared. The index variable {{math|i}} refers to an arbitrary element {{math|xi}}. Using Einstein notation, the gradient can then be written as:
nabla f = frac{partial f}{partial x^{i}}g^{ij} mathbf{e}_j ( Note that its dual is mathrm{d}f= frac{partial f}{partial x^{i}}mathbf{e}^i ),
where mathbf{e}_i = partial mathbf{x}/partial x^i and mathbf{e}^i = mathrm{d}x^i refer to the unnormalized local covariant and contravariant bases respectively, g^{ij} is the inverse metric tensor, and the Einstein summation convention implies summation over i and j. If the coordinates are orthogonal we can easily express the gradient (and the differential) in terms of the normalized bases, which we refer to as hat{mathbf{e}}_i and hat{mathbf{e}}^i, using the scale factors (also known as Lamé coefficients) h_i= lVert mathbf{e}_i rVert = 1, / lVert mathbf{e}^i ,rVert :
nabla f = sum_{i=1}^n , frac{partial f}{partial x^{i}}frac{1}{h_i}mathbf{hat{e}}_i ( and mathrm{d}f = sum_{i=1}^n , frac{partial f}{partial x^{i}}frac{1}{h_i}mathbf{hat{e}}^i ),
where we cannot use Einstein notation, since it is impossible to avoid the repetition of more than two indices. Despite the use of upper and lower indices, mathbf{hat{e}}_i, mathbf{hat{e}}^i, and h_i are neither contravariant nor covariant.The latter expression evaluates to the expressions given above for cylindrical and spherical coordinates.

Gradient and the derivative or differential

{{Calculus |Vector}}

Linear approximation to a function

The gradient of a function {{math|f}} from the Euclidean space {{math|Rn}} to {{math|R}} at any particular point {{math|x0}} in {{math|Rn}} characterizes the best linear approximation to {{math|f}} at {{math|x0}}. The approximation is as follows:
f(x) approx f(x_0) + (nabla f)_{x_0}cdot(x-x_0)
for {{math|x}} close to {{math|x0}}, where {{math|(∇f )x0}} is the gradient of {{math|f}} computed at {{math|x0}}, and the dot denotes the dot product on {{math|Rn}}. This equation is equivalent to the first two terms in the multivariable Taylor series expansion of {{math|f}} at {{math|x0}}.

Differential or (exterior) derivative

The best linear approximation to a differentiable function
f colon mathbf{R}^n to mathbf{R}
at a point {{math|x}} in {{math|Rn}} is a linear map from {{math|Rn}} to {{math|R}} which is often denoted by {{math|dfx}} or {{math|Df(x)}} and called the differential or (total) derivative of {{math|f}} at {{math|x}}. The gradient is therefore related to the differential by the formula
(nabla f)_xcdot v = df_x(v)
for any {{math|v ∈ Rn}}. The function {{math|df}}, which maps {{math|x}} to {{math|dfx}}, is called the differential or exterior derivative of {{math|f}} and is an example of a differential 1-form.If {{math|Rn}} is viewed as the space of (dimension {{math|n}}) column vectors (of real numbers), then one can regard {{math|df}} as the row vector with components
left( frac{partial f}{partial x_1}, dots, frac{partial f}{partial x_n}right),
so that {{math|df'x(v)}} is given by matrix multiplication. Assuming the standard Euclidean metric on {{math|R'n''}}, the gradient is then the corresponding column vector, i.e.,
(nabla f)_i = df^mathsf{T}_i.

Gradient as a derivative

Let {{math|U}} be an open set in {{math|Rn}}. If the function {{math|f : U → R}} is differentiable, then the differential of {{math|f}} is the (Fréchet) derivative of {{math|f}}. Thus {{math|∇f}} is a function from {{math|U}} to the space {{math|Rn}} such that
lim_{hto 0} frac{|f(x+h)-f(x) -nabla f(x)cdot h|}{|h|} = 0,
where · is the dot product.As a consequence, the usual properties of the derivative hold for the gradient:

Linearity

The gradient is linear in the sense that if {{math|f}} and {{math|g}} are two real-valued functions differentiable at the point {{math|a ∈ Rn}}, and {{mvar|α}} and {{mvar|β}} are two constants, then {{math|αf + βg}} is differentiable at {{math|a}}, and moreover
nablaleft(alpha f+beta gright)(a) = alpha nabla f(a) + betanabla g (a).

Product rule

If {{math|f}} and {{math|g}} are real-valued functions differentiable at a point {{math|a ∈ Rn}}, then the product rule asserts that the product {{math|fg}} is differentiable at {{math|a}}, and
nabla (fg)(a) = f(a)nabla g(a) + g(a)nabla f(a).

Chain rule

Suppose that {{math|f : A → R}} is a real-valued function defined on a subset {{math|A}} of {{math|Rn}}, and that {{math|f}} is differentiable at a point {{math|a}}. There are two forms of the chain rule applying to the gradient. First, suppose that the function {{math|g}} is a parametric curve; that is, a function {{math|g : I → Rn}} maps a subset {{math|I ⊂ R}} into {{math|Rn}}. If {{math|g}} is differentiable at a point {{math|c ∈ I}} such that {{math|g(c) {{=}} a}}, then
(fcirc g)'(c) = nabla f(a)cdot g'(c),
where ∘ is the composition operator: {{math|( f ∘ g)(x) {{=}} f(g(x))}}.More generally, if instead {{math|I ⊂ Rk}}, then the following holds:
nabla (fcirc g)(c) = big(Dg(c)big)^mathsf{T} big(nabla f(a)big),
where {{math|(Dg)}}T denotes the transpose Jacobian matrix.For the second form of the chain rule, suppose that {{math|h : I → R}} is a real valued function on a subset {{math|I}} of {{math|R}}, and that {{math|h}} is differentiable at the point {{math|f(a) ∈ I}}. Then
nabla (hcirc f)(a) = h'big(f(a)big)nabla f(a).

Further properties and applications

Level sets

{{see also|Level set#Level sets versus the gradient}}A level surface, or isosurface, is the set of all points where some function has a given value.If {{math|f}} is differentiable, then the dot product {{math|(∇f )x â‹… v}} of the gradient at a point {{math|x}} with a vector {{math|v}} gives the directional derivative of {{math|f}} at {{math|x}} in the direction {{math|v}}. It follows that in this case the gradient of {{math|f}} is orthogonal to the level sets of {{math|f}}. For example, a level surface in three-dimensional space is defined by an equation of the form {{math|1=F(x, y, z) = c}}. The gradient of {{math|F}} 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 {{math|1=F(P) = 0}} such that {{math|dF}} is nowhere zero. The gradient of {{math|F}} is then normal to the hypersurface.Similarly, an affine algebraic hypersurface may be defined by an equation {{math|1=F(x1, ..., xn) = 0}}, where {{math|F}} is a polynomial. The gradient of {{math|F}} is zero at a singular point of the hypersurface (this is the definition of a singular point). At a non-singular point, it is a nonzero normal vector.

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.

Generalizations

Gradient of a vector

{{see also|Covariant derivative}}Since the total derivative of a vector field is a linear mapping from vectors to vectors, it is a tensor quantity.In rectangular coordinates, the gradient of a vector field {{math|1=f = ( f{{i sup|1}}, f{{i sup|2}}, f{{i sup|3}})}} is defined by:
nabla mathbf{f}=g^{jk}frac{partial f^i}{partial x^j} mathbf{e}_i otimes mathbf{e}_k,
(where the Einstein summation notation is used and the tensor product of the vectors {{math|ei}} and {{math|ek}} is a dyadic tensor of type (2,0)). Overall, this expression equals the transpose of the Jacobian matrix:
frac{partial f^i}{partial x^j} = frac{partial (f^1,f^2,f^3)}{partial (x^1,x^2,x^3)}.
In curvilinear coordinates, or more generally on a curved manifold, the gradient involves Christoffel symbols:
nabla mathbf{f}=g^{jk}left(frac{partial f^i}{partial x^j}+{Gamma^i}_{jl}f^lright) mathbf{e}_i otimes mathbf{e}_k,
where {{math|g{{i sup|jk}}}} are the components of the inverse metric tensor and the {{math|ei}} are the coordinate basis vectors.Expressed more invariantly, the gradient of a vector field {{math|f}} can be defined by the Levi-Civita connection and metric tensor:{{harvnb|Dubrovin|Fomenko|Novikov|1991|pages=348–349}}.
nabla^a f^b = g^{ac} nabla_c f^b ,
where {{math|∇c}} is the connection.

Riemannian manifolds

For any smooth function {{mvar|f}} on a Riemannian manifold {{math|(M, g)}}, the gradient of {{math|f}} is the vector field {{math|∇f}} such that for any vector field {{math|X}},
g(nabla f, X) = partial_X f,
i.e.,
g_xbig((nabla f)_x, X_x big) = (partial_X f) (x),
where {{math|g'x( , )}} denotes the inner product of tangent vectors at {{math|x}} defined by the metric {{math|g}} and {{math|∂Xf}} is the function that takes any point {{math|x ∈ M}} to the directional derivative of {{math|f}} in the direction {{math|X}}, evaluated at {{math|x}}. In other words, in a coordinate chart {{math|φ}} from an open subset of {{math|M}} to an open subset of {{math|R'n}}, {{math|(∂Xf )(x'')}} is given by:
sum_{j=1}^n X^{j} big(varphi(x)big) frac{partial}{partial x_{j}}(f circ varphi^{-1}) Bigg|_{varphi(x)},
where {{math|X{{isup|j}}}} denotes the {{math|j}}th component of {{math|X}} in this coordinate chart.So, the local form of the gradient takes the form:
nabla f = g^{ik} frac{partial f}{partial x^k} {textbf e}_i .
Generalizing the case {{math|1=M = Rn}}, the gradient of a function is related to its exterior derivative, since
(partial_X f) (x) = (df)_x(X_x) .
More precisely, the gradient {{math|∇f}} is the vector field associated to the differential 1-form {{math|df}} using the musical isomorphism
sharp=sharp^gcolon T^*Mto TM
(called "sharp") defined by the metric {{math|g}}. The relation between the exterior derivative and the gradient of a function on {{math|Rn}} is a special case of this in which the metric is the flat metric given by the dot product.

See also

Notes

{{reflist}}

References

  • BOOK


, B. A.
, Dubrovin
, A. T.
, Fomenko
, S. P.
, Novikov
, Modern Geometry—Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields
, Graduate Texts in Mathematics
, Springer
, 2nd
, 1991
, 978-0-387-97663-1
, harv
,
  • }}
  • BOOK


, H. M.
, Schey
, Div, Grad, Curl, and All That
, W. W. Norton
, 2nd
, 1992
, 0-393-96251-2
, 25048561
, harv
,
  • }}

Further reading

  • BOOK


, Theresa M.
, Korn
, Granino Arthur
, Korn
, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review
, Dover Publications
, 2000
, 157–160
, 0-486-41147-8
, 43864234
, harv
,

External links

{{wiktionary}}
  • WEB


,weblink
, Gradient
, Khan Academy
,
  • {{springer


| title = Gradient
| id = G/g044680
| last = Kuptsov
| first = L.P.
}}.
  • {hide}MathWorld


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| urlname = Gradient
{edih}

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