Gradient

Template:Short description Template:About Template:More citations needed

The gradient, represented by the blue arrows, denotes the direction of greatest change of a scalar function. The values of the function are represented in greyscale and increase in value from white (low) to dark (high).

In vector calculus, the gradient of a scalar-valued differentiable function <math>f</math> of several variables is the vector field (or vector-valued function) <math>\nabla f</math> whose value at a point <math>p</math> gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of <math>f</math>. If the gradient of a function is non-zero at a point <math>p</math>, the direction of the gradient is the direction in which the function increases most quickly from <math>p</math>, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative.<ref>

Template:Harvtxt
Template:Harvtxt
Template:Harvtxt
Template:Harvtxt
Template:Harvtxt
Template:Harvtxt
Template:Harvtxt</ref> Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function <math>f(\mathbf{r})</math> may be defined by:

<math display="block">df=\nabla f \cdot d\mathbf{r}</math>

where <math>df</math> is the total infinitesimal change in <math>f</math> for an infinitesimal displacement <math>d\mathbf{r}</math>, and is seen to be maximal when <math>d\mathbf{r}</math> is in the direction of the gradient <math>\nabla f</math>. The nabla symbol <math>\nabla</math>, written as an upside-down triangle and pronounced "del", denotes the vector differential operator.

When a coordinate system is used in which the basis vectors are not functions of position, the gradient is given by the vector Template:Efn whose components are the partial derivatives of <math>f</math> at <math>p</math>.<ref>

Template:Harvtxt
Template:Harvtxt
Template:Harvtxt
Template:Harvtxt
Template:Harvtxt
Template:Harvtxt
Template:Harvtxt
Template:Harvtxt
Template:Harvtxt</ref> That is, for <math>f \colon \R^n \to \R</math>, its gradient <math>\nabla f \colon \R^n \to \R^n</math> is defined at the point <math>p = (x_1,\ldots,x_n)</math> in n-dimensional space as the vectorTemplate:Efn

<math display="block">\nabla f(p) = \begin{bmatrix}

\frac{\partial f}{\partial x_1}(p) \\
\vdots \\
\frac{\partial f}{\partial x_n}(p)

\end{bmatrix}.</math>

Note that the above definition for gradient is defined for the function <math>f</math> only if <math>f</math> is differentiable at <math>p</math>. There can be functions for which partial derivatives exist in every direction but fail to be differentiable. Furthermore, this definition as the vector of partial derivatives is only valid when the basis of the coordinate system is orthonormal. For any other basis, the metric tensor at that point needs to be taken into account.

For example, the function <math>f(x,y)=\frac {x^2 y}{x^2+y^2}</math> unless at origin where <math>f(0,0)=0</math>, is not differentiable at the origin as it does not have a well defined tangent plane despite having well defined partial derivatives in every direction at the origin.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> In this particular example, under rotation of x-y coordinate system, the above formula for gradient fails to transform like a vector (gradient becomes dependent on choice of basis for coordinate system) and also fails to point towards the 'steepest ascent' in some orientations. For differentiable functions where the formula for gradient holds, it can be shown to always transform as a vector under transformation of the basis so as to always point towards the fastest increase.

The gradient is dual to the total derivative <math>df</math>: the value of the gradient at a point is a tangent vector – a vector at each point; while the value of the derivative at a point is a cotangent vector – a linear functional on vectors.Template:Efn They are related in that the dot product of the gradient of <math>f</math> at a point <math>p</math> with another tangent vector <math>\mathbf{v}</math> equals the directional derivative of <math>f</math> at <math>p</math> of the function along <math>\mathbf{v}</math>; that is, <math display="inline">\nabla f(p) \cdot \mathbf v = \frac{\partial f}{\partial\mathbf{v}}(p) = df_{p}(\mathbf{v}) </math>. The gradient admits multiple generalizations to more general functions on manifolds; see Template:Slink.

MotivationEdit

File:Vector Field of a Function's Gradient imposed over a Color Plot of that Function.svg

Gradient of the 2D function Template:Math is plotted as arrows over the pseudocolor plot of the function.

Consider a room where the temperature is given by a scalar field, Template:Math, so at each point Template:Math the temperature is Template:Math, independent of time. At each point in the room, the gradient of Template:Math at that point will show the direction in which the temperature rises most quickly, moving away from Template:Math. 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 Template:Math is Template:Math. The gradient of Template:Math at a point is a plane 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%. A road going directly uphill has slope 40%, but a road going around the hill at an angle will have a shallower slope. For example, if the road is at a 60° angle from the uphill direction (when both directions are projected onto the horizontal plane), then the slope along the road will be the dot product between the gradient vector and a unit vector along the road, as the dot product measures how much the unit vector along the road aligns with the steepest slope,Template:Efn which is 40% times the cosine of 60°, or 20%.

More generally, if the hill height function Template:Math is differentiable, then the gradient of Template:Math dotted with a unit vector gives the slope of the hill in the direction of the vector, the directional derivative of Template:Math along the unit vector.

NotationEdit

The gradient of a function <math>f</math> at point <math>a</math> is usually written as <math>\nabla f (a)</math>. It may also be denoted by any of the following:

<math>\vec{\nabla} f (a)</math> : to emphasize the vector nature of the result.
<math>\operatorname{grad} f</math>
<math>\partial_i f</math> and <math>f_{i}</math> : Written with Einstein notation, where repeated indices (Template:Math) are summed over.

DefinitionEdit

File:3d-gradient-cos.svg

The gradient of the function Template:Math depicted as a projected vector field on the bottom plane.

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

<math display="block">\big(\nabla f(x)\big)\cdot \mathbf{v} = D_{\mathbf v}f(x)</math>

where the right-hand side is the directional derivative and there are many ways to represent it. Formally, the derivative is dual to the gradient; see relationship with derivative.

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).

The magnitude and direction of the gradient vector are independent of the particular coordinate representation.<ref>Template:Harvtxt</ref><ref>Template:Harvtxt</ref>

Cartesian coordinatesEdit

In the three-dimensional Cartesian coordinate system with a Euclidean metric, the gradient, if it exists, is given by

<math display="block">\nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k},</math>

where Template:Math, Template:Math, Template:Math are the standard unit vectors in the directions of the Template:Math, Template:Math and Template:Math coordinates, respectively. For example, the gradient of the function <math display="block">f(x,y,z)= 2x+3y^2-\sin(z)</math> is <math display="block">\nabla f(x, y, z) = 2\mathbf{i}+ 6y\mathbf{j} -\cos(z)\mathbf{k}.</math> or <math display="block">\nabla f(x, y, z) = \begin{bmatrix}

2 \\
6y \\
-\cos z

\end{bmatrix}. </math>

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; this article follows the convention of the gradient being a column vector, while the derivative is a row vector.

Cylindrical and spherical coordinatesEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

In cylindrical coordinates, the gradient is given by:<ref name="Schey-1992" />

<math display="block">\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,</math>

where Template:Math is the axial distance, Template:Math is the azimuthal or azimuth angle, Template:Math is the axial coordinate, and Template:Math, Template:Math and Template:Math are unit vectors pointing along the coordinate directions.

In spherical coordinates with a Euclidean metric, the gradient is given by:<ref name="Schey-1992">Template:Harvnb.</ref>

<math display="block">\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 \sin\theta}\frac{\partial f}{\partial \varphi}\mathbf{e}_\varphi,</math>

where Template:Math is the radial distance, Template:Math is the azimuthal angle and Template:Math is the polar angle, and Template:Math, Template:Math and Template:Math are again local unit vectors pointing in the coordinate directions (that is, the normalized covariant basis).

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

General coordinatesEdit

We consider general coordinates, which we write as Template:Math, where Template:Mvar 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 Template:Math refers to the second component—not the quantity Template:Math squared. The index variable Template:Math refers to an arbitrary element Template:Math. Using Einstein notation, the gradient can then be written as:

<math display="block">\nabla f = \frac{\partial f}{\partial x^{i}}g^{ij} \mathbf{e}_j</math> (Note that its dual is <math display="inline">\mathrm{d}f = \frac{\partial f}{\partial x^{i}}\mathbf{e}^i</math>),

where <math>\mathbf{e}^i = \mathrm{d}x^i</math> and <math>\mathbf{e}_i = \partial \mathbf{x}/\partial x^i</math> refer to the unnormalized local covariant and contravariant bases respectively, <math>g^{ij}</math> 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 <math>\hat{\mathbf{e}}_i</math> and <math>\hat{\mathbf{e}}^i</math>, using the scale factors (also known as Lamé coefficients) <math>h_i= \lVert \mathbf{e}_i \rVert = \sqrt{g_{i i}} = 1\, / \lVert \mathbf{e}^i \rVert</math> :

<math display="block">\nabla f = \frac{\partial f}{\partial x^{i}}g^{ij} \hat{\mathbf{e}}_{j}\sqrt{g_{jj}} = \sum_{i=1}^n \, \frac{\partial f}{\partial x^{i}} \frac{1}{h_i} \mathbf{\hat{e}}_i</math> (and <math display="inline">\mathrm{d}f = \sum_{i=1}^n \, \frac{\partial f}{\partial x^{i}} \frac{1}{h_i} \mathbf{\hat{e}}^i</math>),

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, <math>\mathbf{\hat{e}}_i</math>, <math>\mathbf{\hat{e}}^i</math>, and <math>h_i</math> are neither contravariant nor covariant.

The latter expression evaluates to the expressions given above for cylindrical and spherical coordinates.

Relationship with derivativeTemplate:AnchorEdit

{{#invoke:sidebar|collapsible | class = plainlist | titlestyle = padding-bottom:0.25em; | pretitle = Part of a series of articles about | title = Calculus | image = <math>\int_{a}^{b} f'(t) \, dt = f(b) - f(a)</math> | listtitlestyle = text-align:center; | liststyle = border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa; | expanded = vector | abovestyle = padding:0.15em 0.25em 0.3em;font-weight:normal; | above =

Fundamental theorem

Template:Startflatlist

Template:Endflatlist Template:Startflatlist

Template:Endflatlist

| list2name = differential | list2titlestyle = display:block;margin-top:0.65em; | list2title = Template:Bigger | list2 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | heading1 = Definitions
 | content1 =

 | heading2 = Concepts
 | content2 =

 | heading3 = Rules and identities
 | content3 =

}}

| list3name = integral | list3title = Template:Bigger | list3 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

| heading2 = Definitions

 | content2 =

 | heading3 = Integration by
 | content3 =

}}

| list4name = series | list4title = Template:Bigger | list4 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

 | heading2 = Convergence tests
 | content2 =

}}

| list5name = vector | list5title = Template:Bigger | list5 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

 | heading2 = Theorems
 | content2 =

}}

| list6name = multivariable | list6title = Template:Bigger | list6 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | heading1 = Formalisms
 | content1 =

 | heading2 = Definitions
 | content2 =

}}

| list7name = advanced | list7title = Template:Bigger | list7 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

}}

| list8name = specialized | list8title = Template:Bigger | list8 =

| list9name = miscellanea | list9title = Template:Bigger | list9 =

}}

Relationship with total derivativeTemplate:AnchorEdit

The gradient is closely related to the total derivative (total differential) <math>df</math>: they are transpose (dual) to each other. Using the convention that vectors in <math>\R^n</math> are represented by column vectors, and that covectors (linear maps <math>\R^n \to \R</math>) are represented by row vectors,Template:Efn the gradient <math>\nabla f</math> and the derivative <math>df</math> are expressed as a column and row vector, respectively, with the same components, but transpose of each other:

<math display="block">\nabla f(p) = \begin{bmatrix}\frac{\partial f}{\partial x_1}(p) \\ \vdots \\ \frac{\partial f}{\partial x_n}(p) \end{bmatrix} ;</math> <math display="block">df_p = \begin{bmatrix}\frac{\partial f}{\partial x_1}(p) & \cdots & \frac{\partial f}{\partial x_n}(p) \end{bmatrix} .</math>

While these both have the same components, they differ in what kind of mathematical object they represent: at each point, the derivative is a cotangent vector, a linear form (or covector) which expresses how much the (scalar) output changes for a given infinitesimal change in (vector) input, while at each point, the gradient is a tangent vector, which represents an infinitesimal change in (vector) input. In symbols, the gradient is an element of the tangent space at a point, <math>\nabla f(p) \in T_p \R^n</math>, while the derivative is a map from the tangent space to the real numbers, <math>df_p \colon T_p \R^n \to \R</math>. The tangent spaces at each point of <math>\R^n</math> can be "naturally" identifiedTemplate:Efn with the vector space <math>\R^n</math> itself, and similarly the cotangent space at each point can be naturally identified with the dual vector space <math>(\R^n)^*</math> of covectors; thus the value of the gradient at a point can be thought of a vector in the original <math>\R^n</math>, not just as a tangent vector.

Computationally, given a tangent vector, the vector can be multiplied by the derivative (as matrices), which is equal to taking the dot product with the gradient: <math display="block"> (df_p)(v) = \begin{bmatrix}\frac{\partial f}{\partial x_1}(p) & \cdots & \frac{\partial f}{\partial x_n}(p) \end{bmatrix} \begin{bmatrix}v_1 \\ \vdots \\ v_n\end{bmatrix} = \sum_{i=1}^n \frac{\partial f}{\partial x_i}(p) v_i = \begin{bmatrix}\frac{\partial f}{\partial x_1}(p) \\ \vdots \\ \frac{\partial f}{\partial x_n}(p) \end{bmatrix} \cdot \begin{bmatrix}v_1 \\ \vdots \\ v_n\end{bmatrix} = \nabla f(p) \cdot v</math>

Differential or (exterior) derivativeEdit

The best linear approximation to a differentiable function <math display="block">f : \R^n \to \R</math> at a point <math>x</math> in <math>\R^n</math> is a linear map from <math>\R^n</math> to <math>\R</math> which is often denoted by <math>df_x</math> or <math>Df(x)</math> and called the differential or total derivative of <math>f</math> at <math>x</math>. The function <math>df</math>, which maps <math>x</math> to <math>df_x</math>, is called the total differential or exterior derivative of <math>f</math> and is an example of a differential 1-form.

Much as the derivative of a function of a single variable represents the slope of the tangent to the graph of the function,<ref>Template:Harvtxt</ref> the directional derivative of a function in several variables represents the slope of the tangent hyperplane in the direction of the vector.

The gradient is related to the differential by the formula <math display="block">(\nabla f)_x\cdot v = df_x(v)</math> for any <math>v\in\R^n</math>, where <math>\cdot</math> is the dot product: taking the dot product of a vector with the gradient is the same as taking the directional derivative along the vector.

If <math>\R^n</math> is viewed as the space of (dimension <math>n</math>) column vectors (of real numbers), then one can regard <math>df</math> as the row vector with components <math display="block">\left( \frac{\partial f}{\partial x_1}, \dots, \frac{\partial f}{\partial x_n}\right),</math> so that <math>df_x(v)</math> is given by matrix multiplication. Assuming the standard Euclidean metric on <math>\R^n</math>, the gradient is then the corresponding column vector, that is, <math display="block">(\nabla f)_i = df^\mathsf{T}_i.</math>

Linear approximation to a functionEdit

The best linear approximation to a function can be expressed in terms of the gradient, rather than the derivative. The gradient of a function <math>f</math> from the Euclidean space <math>\R^n</math> to <math>\R</math> at any particular point <math>x_0</math> in <math>\R^n</math> characterizes the best linear approximation to <math>f</math> at <math>x_0</math>. The approximation is as follows:

<math display="block">f(x) \approx f(x_0) + (\nabla f)_{x_0}\cdot(x-x_0)</math>

for <math>x</math> close to <math>x_0</math>, where <math>(\nabla f)_{x_0}</math> is the gradient of <math>f</math> computed at <math>x_0</math>, and the dot denotes the dot product on <math>\R^n</math>. This equation is equivalent to the first two terms in the multivariable Taylor series expansion of <math>f</math> at <math>x_0</math>.

Relationship with Template:VanchorEdit

Let Template:Math be an open set in Template:Math. If the function Template:Math is differentiable, then the differential of Template:Math is the Fréchet derivative of Template:Math. Thus Template:Math is a function from Template:Math to the space Template:Math such that <math display="block">\lim_{h\to 0} \frac{|f(x+h)-f(x) -\nabla f(x)\cdot h|}{\|h\|} = 0,</math> where · is the dot product.

As a consequence, the usual properties of the derivative hold for the gradient, though the gradient is not a derivative itself, but rather dual to the derivative:

Linearity: The gradient is linear in the sense that if Template:Math and Template:Math are two real-valued functions differentiable at the point Template:Math, and Template:Mvar and Template:Mvar are two constants, then Template:Math is differentiable at Template:Math, and moreover <math display="block">\nabla\left(\alpha f+\beta g\right)(a) = \alpha \nabla f(a) + \beta\nabla g (a).</math>
Product rule: If Template:Math and Template:Math are real-valued functions differentiable at a point Template:Math, then the product rule asserts that the product Template:Math is differentiable at Template:Math, and <math display="block">\nabla (fg)(a) = f(a)\nabla g(a) + g(a)\nabla f(a).</math>
Chain rule: Suppose that Template:Math is a real-valued function defined on a subset Template:Math of Template:Math, and that Template:Math is differentiable at a point Template:Math. There are two forms of the chain rule applying to the gradient. First, suppose that the function Template:Math is a parametric curve; that is, a function Template:Math maps a subset Template:Math into Template:Math. If Template:Math is differentiable at a point Template:Math such that Template:Math, then <math display="block">(f\circ g)'(c) = \nabla f(a)\cdot g'(c),</math> where ∘ is the composition operator: Template:Math.

More generally, if instead Template:Math, then the following holds: <math display="block">\nabla (f\circ g)(c) = \big(Dg(c)\big)^\mathsf{T} \big(\nabla f(a)\big),</math> where Template:Math^T denotes the transpose Jacobian matrix.

For the second form of the chain rule, suppose that Template:Math is a real valued function on a subset Template:Math of Template:Math, and that Template:Math is differentiable at the point Template:Math. Then <math display="block">\nabla (h\circ f)(a) = h'\big(f(a)\big)\nabla f(a).</math>

Further properties and applicationsEdit

Level setsEdit

Template:See also A level surface, or isosurface, is the set of all points where some function has a given value.

If Template:Math is differentiable, then the dot product Template:Math of the gradient at a point Template:Math with a vector Template:Math gives the directional derivative of Template:Math at Template:Math in the direction Template:Math. It follows that in this case the gradient of Template:Math is orthogonal to the level sets of Template:Math. For example, a level surface in three-dimensional space is defined by an equation of the form Template:Math. The gradient of Template:Math 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 Template:Math such that Template:Math is nowhere zero. The gradient of Template:Math is then normal to the hypersurface.

Similarly, an affine algebraic hypersurface may be defined by an equation Template:Math, where Template:Math is a polynomial. The gradient of Template:Math 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 theoremEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

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.

Gradient is direction of steepest ascentEdit

The gradient of a function <math>f \colon \R^n \to \R</math> at point Template:Math is also the direction of its steepest ascent, i.e. it maximizes its directional derivative:

Let <math> v \in \R^n</math> be an arbitrary unit vector. With the directional derivative defined as

<math display="block">\nabla_v f (x) = \lim_{h \rightarrow 0} \frac{f(x + vh) - f(x)}{h},</math>

we get, by substituting the function <math>f(x + vh)</math> with its Taylor series,

<math display="block">\nabla_v f (x) = \lim_{h \rightarrow 0} \frac{(f(x) + \nabla f \cdot vh + R) - f(x)}{h},</math>

where <math>R</math> denotes higher order terms in <math>vh</math>.

Dividing by <math>h</math>, and taking the limit yields a term which is bounded from above by the Cauchy-Schwarz inequality<ref>Template:Cite book</ref>

Choosing <math>v^* = \nabla f/|\nabla f|</math> maximizes the directional derivative, and equals the upper bound

<math display="block">|\nabla_{v^*} f (x)| = |(\nabla f)^2/|\nabla f|| = |\nabla f|.</math>

GeneralizationsEdit

JacobianEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The Jacobian matrix is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds.<ref>Template:Harvtxt</ref><ref>Template:Harvtxt</ref> A further generalization for a function between Banach spaces is the Fréchet derivative.

Suppose Template:Math is a function such that each of its first-order partial derivatives exist on Template:Math. Then the Jacobian matrix of Template:Math is defined to be an Template:Math matrix, denoted by <math>\mathbf{J}_\mathbb{f}(\mathbb{x})</math> or simply <math>\mathbf{J}</math>. The Template:Mathth entry is <math display="inline">\mathbf J_{ij} = {\partial f_i} / {\partial x_j}</math>. Explicitly <math display="block">\mathbf J = \begin{bmatrix}

   \dfrac{\partial \mathbf{f}}{\partial x_1} & \cdots & \dfrac{\partial \mathbf{f}}{\partial x_n} \end{bmatrix}

= \begin{bmatrix}

   \nabla^\mathsf{T} f_1 \\  
   \vdots \\
   \nabla^\mathsf{T} f_m   
   \end{bmatrix}

= \begin{bmatrix}

   \dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n}\\
   \vdots & \ddots & \vdots\\
   \dfrac{\partial f_m}{\partial x_1} & \cdots & \dfrac{\partial f_m}{\partial x_n} \end{bmatrix}.</math>

Gradient of a vector fieldEdit

Template:See also 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 Template:Math is defined by:

<math display="block">\nabla \mathbf{f}=g^{jk}\frac{\partial f^i}{\partial x^j} \mathbf{e}_i \otimes \mathbf{e}_k,</math>

(where the Einstein summation notation is used and the tensor product of the vectors Template:Math and Template:Math is a dyadic tensor of type (2,0)). Overall, this expression equals the transpose of the Jacobian matrix:

<math display="block">\frac{\partial f^i}{\partial x^j} = \frac{\partial (f^1,f^2,f^3)}{\partial (x^1,x^2,x^3)}.</math>

In curvilinear coordinates, or more generally on a curved manifold, the gradient involves Christoffel symbols:

<math display="block">\nabla \mathbf{f}=g^{jk}\left(\frac{\partial f^i}{\partial x^j}+{\Gamma^i}_{jl}f^l\right) \mathbf{e}_i \otimes \mathbf{e}_k,</math>

where Template:Math are the components of the inverse metric tensor and the Template:Math are the coordinate basis vectors.

Expressed more invariantly, the gradient of a vector field Template:Math can be defined by the Levi-Civita connection and metric tensor:<ref>Template:Harvnb.</ref>

<math display="block">\nabla^a f^b = g^{ac} \nabla_c f^b ,</math>

where Template:Math is the connection.

Riemannian manifoldsEdit

For any smooth function Template:Mvar on a Riemannian manifold Template:Math, the gradient of Template:Math is the vector field Template:Math such that for any vector field Template:Math, <math display="block">g(\nabla f, X) = \partial_X f,</math> that is, <math display="block">g_x\big((\nabla f)_x, X_x \big) = (\partial_X f) (x),</math> where Template:Math denotes the inner product of tangent vectors at Template:Math defined by the metric Template:Math and Template:Math is the function that takes any point Template:Math to the directional derivative of Template:Math in the direction Template:Math, evaluated at Template:Math. In other words, in a coordinate chart Template:Math from an open subset of Template:Math to an open subset of Template:Math, Template:Math is given by: <math display="block">\sum_{j=1}^n X^{j} \big(\varphi(x)\big) \frac{\partial}{\partial x_{j}}(f \circ \varphi^{-1}) \Bigg|_{\varphi(x)},</math> where Template:Math denotes the Template:Mathth component of Template:Math in this coordinate chart.

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

<math display="block">\nabla f = g^{ik} \frac{\partial f}{\partial x^k} {\textbf e}_i .</math>

Generalizing the case Template:Math, the gradient of a function is related to its exterior derivative, since <math display="block">(\partial_X f) (x) = (df)_x(X_x) .</math> More precisely, the gradient Template:Math is the vector field associated to the differential 1-form Template:Math using the musical isomorphism <math display="block">\sharp=\sharp^g\colon T^*M\to TM</math> (called "sharp") defined by the metric Template:Math. The relation between the exterior derivative and the gradient of a function on Template:Math is a special case of this in which the metric is the flat metric given by the dot product.

NotesEdit

Template:Notelist

ReferencesEdit

Template:Reflist

External linksEdit

Template:Sister project

{{#invoke:citation/CS1|citation

|CitationClass=web }}

Template:Springer.
{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Gradient%7CGradient.html}} |title = Gradient |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

Template:Calculus topics

Gradient

Contents

MotivationEdit

NotationEdit

DefinitionEdit

Cartesian coordinatesEdit

Cylindrical and spherical coordinatesEdit

General coordinatesEdit

Relationship with derivativeTemplate:AnchorEdit

Relationship with total derivativeTemplate:AnchorEdit

Differential or (exterior) derivativeEdit

Linear approximation to a functionEdit

Relationship with Template:VanchorEdit

Further properties and applicationsEdit

Level setsEdit

Conservative vector fields and the gradient theoremEdit

Gradient is direction of steepest ascentEdit

GeneralizationsEdit

JacobianEdit

Gradient of a vector fieldEdit

Riemannian manifoldsEdit

See alsoEdit

NotesEdit

ReferencesEdit

Further readingEdit

External linksEdit