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In vector calculus, the gradient of a scalar field is a vector field 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 on a Euclidean space which 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.

Consider a hill whose height above sea level at a point $(x,\; y)$ is $H(x,\; y)$. The gradient of $H$ 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. Consider again the example with the hill and suppose that the steepest slope on the 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 with the uphill direction (the gradient vector), 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 $H$ is differentiable, then the gradient of $H$ dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely, when $H$ is differentiable the dot product of the gradient of $H$ with a given unit vector is equal to the directional derivative of $H$ in the direction of that unit vector.

- $nabla\; f\; =\; left(frac\{partial\; f\}\{partial\; x\_1\; \},\; dots,\; frac\{partial\; f\}\{partial\; x\_n\; \}\; right).$

In Cartesian coordinates, the above expression expands to

- $nabla\; f(x,\; y,\; z)\; =$

which is often written using the standard versors i, j, k and

- $frac\{partial\; f\}\{partial\; x\}mathbf\{i\}+$

In cylindrical coordinates, the gradient is given by :

- $nabla\; f(rho,\; theta,\; z)\; =$

where $theta$ is the azimuthal angle and $z$ is the axial coordinate and e_{ρ}, e_{θ} and e_{z} are unit vectors pointing along the coordinate directions.

- $nabla\; f(r,\; theta,\; phi)\; =$

where $phi$ is the azimuth angle and $theta$ is the zenith angle.

- $f(x,y,z)=\; 2x+3y^2-sin(z)$

- $nabla\; f=\; left($

= left(2, 6y, -cos(z)right).

- $f(x)\; approx\; f(x\_0)\; +\; (nabla\; f)\_\{x\_0\}cdot(x-x\_0)$

The best linear approximation to a function $f\; :\; mathbb\{R\}^n\; to\; mathbb\{R\}$ at a point $x$ in $mathbb\{R\}^n$ is a linear map from $mathbb\{R\}^n$ to $mathbb\{R\}$ which is often denoted by $mathrm\{d\}f\_x$ or $Df(x)$ and called the differential or (total) derivative of $f$ at $x$. The gradient is therefore related to the differential by the formula

- $(nabla\; f)\_xcdot\; v\; =\; mathrm\; d\; f\_x(v)$

If $mathbb\{R\}^n$ is viewed as the space of (length $n$) column vectors (of real numbers), then one can regard $mathrm\{d\}f$ as the row vector

- $mathrm\{d\}f\; =\; left(frac\{partial\; f\}\{partial\; x\_1\},\; dots,\; frac\{partial\; f\}\{partial\; x\_n\}right)$

- $lim\_\{hto\; 0\}\; frac\{|f(x+h)-f(x)\; -nabla\; f(x)cdot\; h|\}\{|h|\}\; =\; 0$

As a consequence, the usual properties of the derivative hold for the gradient:Linearity
The gradient is linear in the sense that if f and g are two real-valued functions differentiable at the point a∈R^{n}, and α and β are two constants, then αf+βg is differentiable at a, and moreover

- $nablaleft(alpha\; f+beta\; gright)(a)\; =\; alpha\; nabla\; f(a)\; +\; betanabla\; g\; (a).$Product rule

- $nabla\; (fg)(a)\; =\; f(a)nabla\; g(a)\; +\; g(a)nabla\; f(a)$Chain rule

- $(fcirc\; g)\text{'}(c)\; =\; nabla\; f(a)cdot\; g\text{'}(c).$

More generally, if instead I⊂R^{k}, then the following holds:

- $D(fcirc\; g)(c)\; =\; (Dg(c))^Tnabla\; f(a)$

where (Dg)^{T} denotes the transpose Jacobian matrix.

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

- $nabla\; (hcirc\; f)(a)\; =\; h\text{'}(c)nabla\; f(a).$

- $nabla\; (f(Ax))\; =\; A^Tnabla\; (f(Ax))\; =\; A^\{-1\}(nabla\; f)(Ax)$

The differential is more natural than the gradient because it is invariant under all coordinate transformations (or diffeomorphisms), whereas the gradient is only invariant under orthogonal transformations (because of the implicit use of the dot product in its definition). Because of this, it is common to blur the distinction between the two concepts using the notion of covariant and contravariant vectors. From this point of view, the components of the gradient transform covariantly under changes of coordinates, so it is called a covariant vector field, whereas the components of a vector field in the usual sense transform contravariantly. In this language the gradient is the differential, as a covariant vector field is the same thing as a differential 1-form. Unfortunately this confusing language is confused further by differing conventions. Although the components of a differential 1-form transform covariantly under coordinate transformations, differential 1-forms themselves transform contravariantly (by pullback) under diffeomorphism. For this reason differential 1-forms are sometimes said to be contravariant rather than covariant, in which case vector fields are covariant rather than contravariant.

Because the gradient is orthogonal to level sets, it can be used to construct a vector normal to a surface. Consider any manifold that is one dimension less than the space it is in (e.g., a surface in 3D, a curve in 2D, etc.). Let this manifold be defined by an equation e.g. F(x, y, z) = 0 (i.e., move everything to one side of the equation). We have now turned the manifold into a level set. To find a normal vector, we simply need to find the gradient of the function F at the desired point.

The gradient of a function is called a gradient field. A gradient field is always a conservative vector field: line integrals through a gradient field are path-independent and can be evaluated with the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a conservative vector field in a simply connected region is always the gradient of a function.

- $g(nabla\; f,\; X\; )\; =\; partial\_X\; f,\; qquad\; text\{i.e.,\}quad\; g\_x((nabla\; f)\_x,\; X\_x\; )\; =\; (partial\_X\; f)\; (x)$

- $sum\_\{j=1\}^n\; X^\{j\}\; (varphi(x))\; frac\{partial\}\{partial\; x\_\{j\}\}(f\; circ\; varphi^\{-1\})\; Big|\_\{varphi(x)\},$

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

- $nabla\; f=\; g^\{ik\}frac\{partial\; f\}\{partial\; x^\{k\}\}frac\{partial\}\{partial\; x^\{i\}\}.$

Generalizing the case M=R^{n}, the gradient of a function is related to its exterior derivative, since $(partial\_X\; f)\; (x)\; =\; df\_x(X\_x)$. More precisely, the gradient $nabla\; f$ is the vector field associated to the differential 1-form df using the musical isomorphism $sharp=sharp^gcolon\; T^*Mto\; TM$ (called "sharp") defined by the metric g. The relation between the exterior derivative and the gradient of a function on R^{n} is a special case of this in which the metric is the flat metric given by the dot product.

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Last updated on Sunday October 05, 2008 at 16:31:25 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Sunday October 05, 2008 at 16:31:25 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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