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## Definition

File:3d-gradient-cos.svg|thumb|350px|The gradient of the function {{math|f(x,y) {{=}} âˆ’(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|âˆ‡f}} or {{math|{{vec|âˆ‡}}f}} where {{math|âˆ‡}} (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=139â€“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|Ï}} is the axial distance, {{math|Ï†}} is the azimuthal or azimuth angle, {{math|z}} is the axial coordinate, and {{math|eÏ}}, {{math|eÏ†}} 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|Ï†}} is the azimuthal angle and {{math|Î¸}} is the polar angle, and {{math|er}}, {{math|eÎ¸}} and {{math|eÏ†}} 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 componentâ€”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 LamÃ© 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|(âˆ‡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 âˆˆ 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.

Let {{math|U}} be an open set in {{math|Rn}}. If the function {{math|f : U â†’ R}} is differentiable, then the differential of {{math|f}} is the (FrÃ©chet) derivative of {{math|f}}. Thus {{math|âˆ‡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 Â· 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 âˆˆ Rn}}, and {{mvar|Î±}} and {{mvar|Î²}} are two constants, then {{math|Î±f + Î²g}} 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 âˆˆ 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 â†’ 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 â†’ Rn}} maps a subset {{math|I âŠ‚ R}} into {{math|Rn}}. If {{math|g}} is differentiable at a point {{math|c âˆˆ I}} such that {{math|g(c) {{=}} a}}, then
(fcirc g)'(c) = nabla f(a)cdot g'(c),
where âˆ˜ is the composition operator: {{math|( fâ€‰âˆ˜â€‰g)(x) {{=}} f(g(x))}}.More generally, if instead {{math|I âŠ‚ 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 â†’ 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) âˆˆ 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|(âˆ‡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

{{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=348â€“349}}.
nabla^a f^b = g^{ac} nabla_c f^b ,
where {{math|âˆ‡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|âˆ‡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|âˆ‚Xf}} is the function that takes any point {{math|x âˆˆ 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|Ï†}} from an open subset of {{math|M}} to an open subset of {{math|R'n}}, {{math|(âˆ‚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|âˆ‡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.

{{reflist}}

## References

• BOOK

, B. A.
, Dubrovin
, A. T.
, Fomenko
, S. P.
, Novikov
, Modern Geometryâ€”Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields
, 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
,
• }}

• BOOK

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

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• WEB

,
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| id = G/g044680
| last = Kuptsov
| first = L.P.
}}.
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