Density functional theory/Introduction to functional analysis

Functional (mathematics)

In mathematics, and particularly in functional analysis and the Calculus of variations, a functional is a function from a vector space into its underlying scalar field, or a set of functions of the real numbers. In other words, it is a function that takes a vector as its input argument, and returns a scalar. Commonly the vector space is a space of functions, thus the functional takes a function for its input argument, then it is sometimes considered a function of a function. Its use originates in the calculus of variations where one searches for a function that minimizes a certain functional. A particularly important application in physics is searching for a state of a system that minimizes the energy functional.

Functional details

Duality

The mapping

$x_0\mapsto f(x_0)$

is a function, where $x_0$ is an argument of a function $f$. At the same time, the mapping of a function to the value of the function at a point

$f\mapsto f(x_0)$

is a functional, here $x_0$ is a parameter.

Provided that f is a linear function from a linear vector space to the underlying scalar field, the above linear maps are dual to each other, and in functional analysis both are called linear functionals.

Definite integral

Integrals such as

$f\mapsto I[f]=\underset{\mathrm{\Omega }}{\int }H(f(x),f^{\prime }(x),\dots )\mu (\text{d}x)$

form a special class of functionals. They map a function f into a real number, provided that H is real-valued. Examples include

• the area underneath the graph of a positive function f

$f\mapsto \underset{x_0}{\overset{x_1}{\int }}f(x)\mathrm{d}x$

• L^p norm of functions

$f\mapsto {(\int |f{|}^p\mathrm{d}x)}^{1/p}$

• the arclength of a curve in 2-dimensional Euclidean space

$f\mapsto \underset{x_0}{\overset{x_1}{\int }}\sqrt{1+|f^{\prime }(x){|}^2}\mathrm{d}x$

Vector scalar product

Given any vector $\overrightarrow{x}$ in a vector space $X$, the scalar product with another vector $\overrightarrow{y}$, denoted $\overrightarrow{x}\cdot \overrightarrow{y}$ or $⟨\overrightarrow{x},\overrightarrow{y}⟩$, is a scalar. The set of vectors such that this product is zero is a vector subspace of $X$, called the null space or kernel of $X$.

Local vs non-local

If a functional's value can be computed for small segments of the input curve and then summed to find the total value, a function is called local. Otherwise it is called non-local. For example:

$F(y)=\underset{x_0}{\overset{x_1}{\int }}y(x)\mathrm{d}x$

is local while

$F(y)=\frac{\underset{x_0}{\overset{x_1}{\int }}y(x)\mathrm{d}x}{\underset{x_0}{\overset{x_1}{\int }}(1+[y(x){]}^2)\mathrm{d}x}$

is non-local. This occurs commonly when integrals occur separately in the numerator and denominator of an equation such as in calculations of center of mass.

 Linear functionals

Linear functionals first appeared in functional analysis, the study of vector spaces of functions.  A typical example of a linear functional is integration: the linear transformation defined by the Riemann integral

$I(f)=\underset{a}{\overset{b}{\int }}f(x)dx$

is a linear functional from the vector space C[a,b] of continuous functions on the interval [a, b] to the real numbers.  The linearity of I(f) follows from the standard facts about the integral:

$I(f+g)=\underset{a}{\overset{b}{\int }}(f(x)+g(x))dx=\underset{a}{\overset{b}{\int }}f(x)dx+\underset{a}{\overset{b}{\int }}g(x)dx=I(f)+I(g)$

$I(\alpha f)=\underset{a}{\overset{b}{\int }}\alpha f(x)dx=\alpha \underset{a}{\overset{b}{\int }}f(x)dx=\alpha I(f).$

Functional derivative

The functional derivative is defined first; Then the functional differential is defined in terms of the functional derivative.

Functional derivative

Given a manifold M representing (continuous/smooth/with certain boundary conditions/etc.) functions ρ and a functional F defined as

$F:M\to ℝ\text{or}F:M\to ℂ,$

the functional derivative of Template:Mathρ], denoted Template:Mathρ, is defined by

$\begin{array}{cc}\int \frac{\delta F}{\delta \rho (x)}\varphi (x)dx& =\underset{\epsilon \to 0}{\lim }\frac{F[\rho +\epsilon \varphi ]-F[\rho ]}{\epsilon }\\ & ={[\frac{d}{dϵ}F[\rho +ϵ\varphi ]]}_{ϵ=0},\end{array}$

where $\varphi$ is an arbitrary function. $ϵ\varphi$ is called the variation of ρ.

Functional differential

The differential (or variation or first variation) of the functional Template:Math[ρ] is,

$\delta F=\int \frac{\delta F}{\delta \rho (x)}\delta \rho (x)dx,$

where Template:MathρTemplate:Math is the variation of ρTemplate:Math.Template:Clarify This is similar in form to the total differential of a function Template:Math(ρ₁, ρ₂, ..., ρ_n),

$dF=\sum _{i=1}^n\frac{\partial F}{\partial {\rho }_i}d{\rho }_i,$

where ρ₁, ρ₂, ... , ρ_n are independent variables. Comparing the last two equations, the functional derivative Template:MathρTemplate:Math has a role similar to that of the partial derivative Template:Mathρ_i , where the variable of integration Template:Math is like a continuous version of the summation index Template:Math.

Formal description

The definition of a functional derivative may be made more mathematically precise and formal by defining the space of functions more carefully. For example, when the space of functions is a Banach space, the functional derivative becomes known as the Fréchet derivative, while one uses the Gâteaux derivative on more general locally convex spaces. Note that the well-known Hilbert spaces are special cases of Banach spaces. The more formal treatment allows many theorems from ordinary calculus and analysis to be generalized to corresponding theorems in functional analysis, as well as numerous new theorems to be stated.

Properties

Like the derivative of a function, the functional derivative satisfies the following properties, where Template:Math[ρ] and Template:Math[ρ] are functionals:

• Linear:

$\frac{\delta (\lambda F+\mu G)}{\delta \rho (x)}=\lambda \frac{\delta F}{\delta \rho (x)}+\mu \frac{\delta G}{\delta \rho (x)},\lambda ,\mu$   constant,

• Product rule:

$\frac{\delta (FG)}{\delta \rho (x)}=\frac{\delta F}{\delta \rho (x)}G+F\frac{\delta G}{\delta \rho (x)},$

• Chain rules:

If Template:Math is a differentiable function, then

$\frac{\delta F[f(\rho )]}{\delta \rho (x)}=\frac{\delta F[f(\rho )]}{\delta f(\rho (x))}\frac{df(\rho (x))}{d\rho (x)},$

$\frac{\delta f(F[\rho ])}{\delta \rho (x)}=\frac{df(F[\rho ])}{dF[\rho ]}\frac{\delta F[\rho ]}{\delta \rho (x)}.$

Determining functional derivatives

We give a formula to determine functional derivatives for a common class of functionals that can be written as the integral of a function and its derivatives. This is a generalization of the Euler–Lagrange equation: indeed, the functional derivative was introduced in physics within the derivation of the Lagrange equation of the second kind from the principle of least action in Lagrangian mechanics (18th century). The first three examples below are taken from density functional theory (20th century), the fourth from statistical mechanics (19th century).

Formula

Given a functional

$F[\rho ]=\int f(𝒓,\rho (𝒓),\mathrm{\nabla }\rho (𝒓))d𝒓,$

and a function Template:Math(Template:Math) that vanishes on the boundary of the region of integration, from a previous section Definition,

$\begin{array}{cc}\int \frac{\delta F}{\delta \rho (𝒓)}\varphi (𝒓)d𝒓& ={[\frac{d}{d\epsilon }\int f(𝒓,\rho +\epsilon \varphi ,\mathrm{\nabla }\rho +\epsilon \mathrm{\nabla }\varphi )d𝒓]}_{\epsilon =0}\\ & =\int (\frac{\partial f}{\partial \rho }\varphi +\frac{\partial f}{\partial \mathrm{\nabla }\rho }\cdot \mathrm{\nabla }\varphi )d𝒓\\ & =\int [\frac{\partial f}{\partial \rho }\varphi +\mathrm{\nabla }\cdot \left(\frac{\partial f}{\partial \mathrm{\nabla }\rho }\varphi \right)-(\mathrm{\nabla }\cdot \frac{\partial f}{\partial \mathrm{\nabla }\rho })\varphi ]d𝒓\\ & =\int [\frac{\partial f}{\partial \rho }\varphi -(\mathrm{\nabla }\cdot \frac{\partial f}{\partial \mathrm{\nabla }\rho })\varphi ]d𝒓\\ & =\int (\frac{\partial f}{\partial \rho }-\mathrm{\nabla }\cdot \frac{\partial f}{\partial \mathrm{\nabla }\rho })\varphi (𝒓)d𝒓.\end{array}$

The second line is obtained using the total derivative, where Template:Mathρ is a derivative of a scalar with respect to a vector. The third line was obtained by use of a product rule for divergence. The fourth line was obtained using the divergence theorem and the condition that Template:Math on the boundary of the region of integration. Since Template:Math is also an arbitrary function, applying the fundamental lemma of calculus of variations to the last line, the functional derivative is

$\frac{\delta F}{\delta \rho (𝒓)}=\frac{\partial f}{\partial \rho }-\mathrm{\nabla }\cdot \frac{\partial f}{\partial \mathrm{\nabla }\rho }$

where ρ = ρ(Template:Math) and Template:Math, ρ, ∇ρ). This formula is for the case of the functional form given by Template:Math[ρ] at the beginning of this section. For other functional forms, the definition of the functional derivative can be used as the starting point for its determination. (See the example Coulomb potential energy functional.)

The above equation for the functional derivative can be generalized to the case that includes higher dimensions and higher order derivatives. The functional would be,

$F[\rho (𝒓)]=\int f(𝒓,\rho (𝒓),\mathrm{\nabla }\rho (𝒓),{\mathrm{\nabla }}^{(2)}\rho (𝒓),\dots ,{\mathrm{\nabla }}^{(N)}\rho (𝒓))d𝒓,$

where the vector Template:Math, and Template:Math is a tensor whose Template:Math components are partial derivative operators of order Template:Math,

${\left[{\mathrm{\nabla }}^{(i)}\right]}_{{\alpha }_1{\alpha }_2\cdots {\alpha }_i}=\frac{{\partial }^i}{\partial r_{{\alpha }_1}\partial r_{{\alpha }_2}\cdots \partial r_{{\alpha }_i}}\text{where}{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_i=1,2,\cdots ,n.$

An analogous application of the definition of the functional derivative yields

$\begin{array}{cc}\frac{\delta F[\rho ]}{\delta \rho }& =\frac{\partial f}{\partial \rho }-\mathrm{\nabla }\cdot \frac{\partial f}{\partial (\mathrm{\nabla }\rho )}+{\mathrm{\nabla }}^{(2)}\cdot \frac{\partial f}{\partial \left({\mathrm{\nabla }}^{(2)}\rho \right)}+\dots +(-1{)}^N{\mathrm{\nabla }}^{(N)}\cdot \frac{\partial f}{\partial \left({\mathrm{\nabla }}^{(N)}\rho \right)}\\ & =\frac{\partial f}{\partial \rho }+\sum _{i=1}^N(-1{)}^i{\mathrm{\nabla }}^{(i)}\cdot \frac{\partial f}{\partial \left({\mathrm{\nabla }}^{(i)}\rho \right)}.\end{array}$

In the last two equations, the Template:Math components of the tensor $\frac{\partial f}{\partial \left({\mathrm{\nabla }}^{(i)}\rho \right)}$ are partial derivatives of Template:Math with respect to partial derivatives of ρ,

${\left[\frac{\partial f}{\partial \left({\mathrm{\nabla }}^{(i)}\rho \right)}\right]}_{{\alpha }_1{\alpha }_2\cdots {\alpha }_i}=\frac{\partial f}{\partial {\rho }_{{\alpha }_1{\alpha }_2\cdots {\alpha }_i}}\text{where}{\rho }_{{\alpha }_1{\alpha }_2\cdots {\alpha }_i}\equiv \frac{{\partial }^i\rho }{\partial r_{{\alpha }_1}\partial r_{{\alpha }_2}\cdots \partial r_{{\alpha }_i}},$

and the tensor scalar product is,

${\mathrm{\nabla }}^{(i)}\cdot \frac{\partial f}{\partial \left({\mathrm{\nabla }}^{(i)}\rho \right)}=\sum _{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_i=1}^n\frac{{\partial }^i}{\partial r_{{\alpha }_1}\partial r_{{\alpha }_2}\cdots \partial r_{{\alpha }_i}}\frac{\partial f}{\partial {\rho }_{{\alpha }_1{\alpha }_2\cdots {\alpha }_i}}.$

Examples

Thomas–Fermi kinetic energy functional

The Thomas–Fermi model of 1927 used a kinetic energy functional for a noninteracting uniform electron gas in a first attempt of density-functional theory of electronic structure:

$T_{\mathrm{T}\mathrm{F}}[\rho ]=C_{\mathrm{F}}\int {\rho }^{5/3}(𝐫)d𝐫.$

Since the integrand of Template:Math[ρ] does not involve derivatives of ρTemplate:Math, the functional derivative of Template:Math[ρ] is,

$\begin{array}{cc}\frac{\delta T_{\mathrm{T}\mathrm{F}}}{\delta \rho (𝒓)}& =C_{\mathrm{F}}\frac{\partial {\rho }^{5/3}(𝐫)}{\partial \rho (𝐫)}\\ & =\frac{5}{3}{C}_{\mathrm{F}}{\rho }^{2/3}(𝐫).\end{array}$

Coulomb potential energy functional

For the electron-nucleus potential, Thomas and Fermi employed the Coulomb potential energy functional

$V[\rho ]=\int \frac{\rho (𝒓)}{|𝒓|}d𝒓.$

Applying the definition of functional derivative,

$\begin{array}{cc}\int \frac{\delta V}{\delta \rho (𝒓)}\varphi (𝒓)d𝒓& ={[\frac{d}{d\epsilon }\int \frac{\rho (𝒓)+\epsilon \varphi (𝒓)}{|𝒓|}d𝒓]}_{\epsilon =0}\\ & =\int \frac{1}{|𝒓|}\varphi (𝒓)d𝒓.\end{array}$

So,

$\frac{\delta V}{\delta \rho (𝒓)}=\frac{1}{|𝒓|}.$

For the classical part of the electron-electron interaction, Thomas and Fermi employed the Coulomb potential energy functional

$J[\rho ]=\frac{1}{2}\iint \frac{\rho (𝐫)\rho ({𝐫}^{\prime })}{|𝐫-{𝐫}^{\prime }|}d𝐫d{𝐫}^{\prime }.$

From the definition of the functional derivative,

$\begin{array}{cc}\int \frac{\delta J}{\delta \rho (𝒓)}\varphi (𝒓)d𝒓& ={[\frac{d}{dϵ}J[\rho +ϵ\varphi ]]}_{ϵ=0}\\ & ={\left[\frac{d}{dϵ}(\frac{1}{2}\iint \frac{[\rho (𝒓)+ϵ\varphi (𝒓)][\rho ({𝒓}^{\prime })+ϵ\varphi ({𝒓}^{\prime })]}{|𝒓-{𝒓}^{\prime }|}d𝒓d{𝒓}^{\prime })\right]}_{ϵ=0}\\ & =\frac{1}{2}\iint \frac{\rho ({𝒓}^{\prime })\varphi (𝒓)}{|𝒓-{𝒓}^{\prime }|}d𝒓d{𝒓}^{\prime }+\frac{1}{2}\iint \frac{\rho (𝒓)\varphi ({𝒓}^{\prime })}{|𝒓-{𝒓}^{\prime }|}d𝒓d{𝒓}^{\prime }\\ \end{array}$

The first and second terms on the right hand side of the last equation are equal, since Template:Math and Template:Math in the second term can be interchanged without changing the value of the integral. Therefore,

$\int \frac{\delta J}{\delta \rho (𝒓)}\varphi (𝒓)d𝒓=\int (\int \frac{\rho ({𝒓}^{\prime })}{|𝒓-{𝒓}^{\prime }|}d{𝒓}^{\prime })\varphi (𝒓)d𝒓$

and the functional derivative of the electron-electron coulomb potential energy functional Template:Math[ρ] is,

$\frac{\delta J}{\delta \rho (𝒓)}=\int \frac{\rho ({𝒓}^{\prime })}{|𝒓-{𝒓}^{\prime }|}d{𝒓}^{\prime }.$

The second functional derivative is

$\frac{{\delta }^{2}J[\rho ]}{\delta \rho ({𝐫}^{\prime })\delta \rho (𝐫)}=\frac{\partial }{\partial \rho ({𝐫}^{\prime })}\left(\frac{\rho ({𝐫}^{\prime })}{|𝐫-{𝐫}^{\prime }|}\right)=\frac{1}{|𝐫-{𝐫}^{\prime }|}.$

Weizsäcker kinetic energy functional

In 1935 von Weizsäcker proposed to add a gradient correction to the Thomas-Fermi kinetic energy functional to make it suit better a molecular electron cloud:

$T_{\mathrm{W}}[\rho ]=\frac{1}{8}\int \frac{\mathrm{\nabla }\rho (𝐫)\cdot \mathrm{\nabla }\rho (𝐫)}{\rho (𝐫)}d𝐫=\int t_{\mathrm{W}}d𝐫,$

where

$t_{\mathrm{W}}\equiv \frac{1}{8}\frac{\mathrm{\nabla }\rho \cdot \mathrm{\nabla }\rho }{\rho }\text{and}\rho =\rho (𝒓).$

Using a previously derived formula for the functional derivative,

$\begin{array}{cc}\frac{\delta T_{\mathrm{W}}}{\delta \rho (𝒓)}& =\frac{\partial t_{\mathrm{W}}}{\partial \rho }-\mathrm{\nabla }\cdot \frac{\partial t_{\mathrm{W}}}{\partial \mathrm{\nabla }\rho }\\ & =-\frac{1}{8}\frac{\mathrm{\nabla }\rho \cdot \mathrm{\nabla }\rho }{{\rho }^2}-(\frac{1}{4}\frac{{\mathrm{\nabla }}^2\rho }{\rho }-\frac{1}{4}\frac{\mathrm{\nabla }\rho \cdot \mathrm{\nabla }\rho }{{\rho }^2})\text{where}{\mathrm{\nabla }}^2=\mathrm{\nabla }\cdot \mathrm{\nabla },\end{array}$

and the result is,

$\frac{\delta T_{\mathrm{W}}}{\delta \rho (𝒓)}=\frac{1}{8}\frac{\mathrm{\nabla }\rho \cdot \mathrm{\nabla }\rho }{{\rho }^2}-\frac{1}{4}\frac{{\mathrm{\nabla }}^2\rho }{\rho }.$

Entropy

The entropy of a discrete random variable is a functional of the probability mass function.

$\begin{array}{c}H[p(x)]=-\sum _{x}p(x)\log p(x)\end{array}$

Thus,

$\begin{array}{cc}\sum _x\frac{\delta H}{\delta p(x)}\varphi (x)& ={[\frac{d}{dϵ}H[p(x)+ϵ\varphi (x)]]}_{ϵ=0}\\ & ={[-\frac{d}{d\epsilon }\sum _x[p(x)+\epsilon \varphi (x)]\log [p(x)+\epsilon \varphi (x)]]}_{\epsilon =0}\\ & =-\sum _x[1+\log p(x)]\varphi (x).\end{array}$

Thus,

$\frac{\delta H}{\delta p(x)}=-1-\log p(x).$

Exponential

Let

$F[\phi (x)]=e^{\int \phi (x)g(x)dx}.$

Using the delta function as a test function,

$\begin{array}{cc}\frac{\delta F[\phi (x)]}{\delta \phi (y)}& =\underset{\epsilon \to 0}{\lim }\frac{F[\phi (x)+\epsilon \delta (x-y)]-F[\phi (x)]}{\epsilon }\\ & =\underset{\epsilon \to 0}{\lim }\frac{e^{\int (\phi (x)+\epsilon \delta (x-y))g(x)dx}-e^{\int \phi (x)g(x)dx}}{\epsilon }\\ & =e^{\int \phi (x)g(x)dx}\underset{\epsilon \to 0}{\lim }\frac{e^{\epsilon \int \delta (x-y)g(x)dx}-1}{\epsilon }\\ & =e^{\int \phi (x)g(x)dx}\underset{\epsilon \to 0}{\lim }\frac{e^{\epsilon g(y)}-1}{\epsilon }\\ & =e^{\int \phi (x)g(x)dx}g(y).\end{array}$

Thus,

$\frac{\delta F[\phi (x)]}{\delta \phi (y)}=g(y)F[\phi (x)].$

This is particularly useful in calculating the correlation functions from the partition function in quantum field theory.

Functional derivative of a function

A function can be written in the form of an integral like a functional. For example,

$\rho (𝒓)=F[\rho ]=\int \rho ({𝒓}^{\prime })\delta (𝒓-{𝒓}^{\prime })d{𝒓}^{\prime }.$

Since the integrand does not depend on derivatives of ρ, the functional derivative of ρTemplate:Math is,

$\begin{array}{cc}\frac{\delta \rho (𝒓)}{\delta \rho ({𝒓}^{\prime })}\equiv \frac{\delta F}{\delta \rho ({𝒓}^{\prime })}& =\frac{\partial }{\partial \rho ({𝒓}^{\prime })}[\rho ({𝒓}^{\prime })\delta (𝒓-{𝒓}^{\prime })]\\ & =\delta (𝒓-{𝒓}^{\prime }).\end{array}$

Application in calculus of variations

In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. In an integrand Template:Math of a functional, if a function Template:Math is varied by adding to it another function Template:Math that is arbitrarily small, and the resulting Template:Math is expanded in powers of Template:Math, the coefficient of Template:Math in the first order term is called the functional derivative.

For example, consider the functional

$J[f]=\underset{a}{\overset{b}{\int }}L[x,f(x),f{}^{\prime }(x)]dx,$

where Template:Math. If Template:Math is varied by adding to it a function Template:Math, and the resulting integrand Template:Math is expanded in powers of Template:Math, then the change in the value of Template:Math to first order in Template:Math can be expressed as follows:

$\delta J=\underset{a}{\overset{b}{\int }}\frac{\delta J}{\delta f(x)}\delta f(x)dx.$

The coefficient of Template:Math, denoted as Template:Math, is called the functional derivative of Template:Math with respect to Template:Math at the point Template:Math. For this example functional, the functional derivative is the left hand side of the Euler-Lagrange equation,

$\frac{\delta J}{\delta f(x)}=\frac{\partial L}{\partial f}-\frac{d}{dx}\frac{\partial L}{\partial f^{\prime }}.$

Using the delta function as a test function

In physics, it's common to use the Dirac delta function $\delta (x-y)$ in place of a generic test function $\varphi (x)$, for yielding the functional derivative at the point $y$ (this is a point of the whole functional derivative as a partial derivative is a component of the gradient):

$\frac{\delta F[\rho (x)]}{\delta \rho (y)}=\underset{\epsilon \to 0}{\lim }\frac{F[\rho (x)+\epsilon \delta (x-y)]-F[\rho (x)]}{\epsilon }.$

This works in cases when $F[\rho (x)+\epsilon f(x)]$ formally can be expanded as a series (or at least up to first order) in $\epsilon$. The formula is however not mathematically rigorous, since $F[\rho (x)+\epsilon \delta (x-y)]$ is usually not even defined.

The definition given in a previous section is based on a relationship that holds for all test functions Template:Math, so one might think that it should hold also when Template:Math is chosen to be a specific function such as the delta function. However, the latter is not a valid test function.

In the definition, the functional derivative describes how the functional $F[\phi (x)]$ changes as a result of a small change in the entire function $\phi (x)$. The particular form of the change in $\phi (x)$ is not specified, but it should stretch over the whole interval on which $x$ is defined. Employing the particular form of the perturbation given by the delta function has the meaning that $\phi (x)$ is varied only in the point $y$. Except for this point, there is no variation in $\phi (x)$.

Functional equation

The traditional usage also applies when one talks about a functional equation, meaning an equation between functionals: an equation $F=G$ between functionals can be read as an 'equation to solve', with solutions being themselves functions. In such equations there may be several sets of variable unknowns, like when it is said that an additive function $f$ is one satisfying the functional equation

$f(x+y)=f\left(x\right)+f\left(y\right)$.

Functional derivative and functional integration

Functional derivatives are used in Lagrangian mechanics. They are derivatives of functionals: i.e. they carry information on how a functional changes, when the function changes by a small amount. See also calculus of variations.

Richard Feynman used functional integrals as the central idea in his sum over the histories formulation of quantum mechanics. This usage implies an integral taken over some function space.

References

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See also

• Linear functional
• Optimization (mathematics)
• Tensor

References

General

• Template:Springer
• Template:MathWorld
• Template:Lang Algebra

Citations