Linear Vector Space In Mathematical Physics Pdf

Linear Vector Space

Let us denote the linear vector space formed by these automorphism, or to be more concrete, by regular matrices representing these automorphisms, as Autg.

From: Exterior Analysis , 2013

Basic Review

George A. Articolo , in Partial Differential Equations & Boundary Value Problems with Maple (Second Edition), 2009

0.2 Preparation for Linear Algebra

A linear vector space consists of a set of vectors or functions and the standard operations of addition, subtraction, and scalar multiplication. In solving ordinary and partial differential equations, we assume the solution space to behave like an ordinary linear vector space. A primary concern is whether or not we have enough of the correct vectors needed to span the solution space completely. We now investigate these notions as they apply directly to two-dimensional vector spaces and differential equations.

We use the simple example of the very familiar two-dimensional Euclidean vector space R2; this is the familiar (x, y) plane. The two standard vectors in the (x, y) plane are traditionally denoted as i and j. The vector i is a unit vector along the x-axis, and the vector j is a unit vector along the y-axis. Any point in the (x, y) plane can be reached by some linear combination, or superposition, of the two standard vectors i and j. We say the vectors "span" the space. The fact that only two vectors are needed to span the two-dimensional space R2 is not coincidental; three vectors would be redundant. One reason for this has to do with the fact that the two vectors i and j are "linearly independent"—that is, one cannot be written as a multiple of the other. The other reason has to do with the fact that in an n-dimensional Euclidean space, the minimum number of vectors needed to span the space is n.

A more formal mathematical definition of linear independence between two vectors or functions v1 and v2 reads as "The two vectors v1 and v2 are linearly independent if and only if the only solution to the linear equation

$c\text{1}\upsilon \text{1}+c2\upsilon 2=0$

is that both c1 and c2 are zero." Otherwise, the vectors are said to be linearly dependent.

In the simple case of the two-dimensional (x, y) space R2, linear independence can be geometrically understood to mean that the two vectors do not lie along the same direction (noncolinear). In fact, any set of two noncolinear vectors could also span the vector space of the (x, y) plane. There are an infinite number of sets of vectors that will do the job. One common connection between all sets, however, is that all the sets can be shown to be linearly dependent; that is, all the sets can be shown to be reducible to linear combinations of the standard i and j vectors.

For example, the two vector sets

$S1=\left\{i,j\right\}$

are both linearly independent sets of vectors that span the two-dimensional (x, y) space. Note that the vectors within each set are linearly independent, but the vectors between sets are linearly dependent.

A set of vectors S = {v1, v2, v3, …, vn} that are linearly independent and that span the space is called a set of "basis" vectors for that particular vector space. Thus, for the two-dimensional Euclidean space R2, the vectors i and j form a basis, and for the three-dimensional Euclidean space R3, vectors i, j, and k form a basis. The number of vectors in a basis is called the "dimension" of the vector space.

A set of basis vectors is fundamental to a particular vector space because any vector in that space can then be written as a unique superposition of those basis vectors. These concepts are important to us when we consider the solution space of both ordinary and partial differential equations. Another important concept in linear algebra is that of the inner product of two vectors in that particular vector space.

For the Euclidean space R3, if we let u and v be two different vectors in this space with components

$u=\left[u1,u2,u3\right]$

then the inner product of these two vectors is given as

$ip\left(u,\upsilon \right)=u1\upsilon 1+u2\upsilon 2+u3\upsilon 3$

Thus, the inner product is the sum of the product of the components of the two vectors. The inner product is sometimes also referred to as the "dot product."

If we take the square root of an inner product of a vector with itself, then we are evaluating the length of the vector, commonly called the "norm."

$norm\left(u\right)=\sqrt{ip\left(u,u\right)}$

Different vector spaces have different inner products. For example, we consider the vector space C[a, b] of all functions that are continuous over the finite closed interval [a, b]. Let f(x) and g(x) be two different vectors in this space. The inner product of these two vectors over the interval, with respect to the weight function w(x), is defined as the definite integral:

$ip\left(f,g\right)=\underset{a}{\overset{b}{\int }}f\left(x\right)g\left(x\right)w\left(x\right)dx$

From the basic definition of a definite integral, we see the inner product to be an (infinite) sum of the product of the components of the two vectors.

Similarly, in the space of continuous functions, if we take the square root of the inner product of a vector with itself, then we evaluate the length or norm of the vector to be

$norm\left(f\right)=\sqrt{{\int }_a^{b}f{\left(x\right)}^{2}w\left(x\right)dx}$

As an example, consider the two functions f(x) = sin(x) and g(x) = cos(x) over the finite closed interval [0, π] with a weight function w(x) = 1. The length or norm of f(x) is the definite integral

$norm\left(f\right)=\sqrt{{\int }_0^{\pi }sin{\left(x\right)}^{2}dx}$

which evaluates to

$norm\left(f\right)=\sqrt{\frac{\pi }{2}}$

Similarly, for g(x) the norm is the definite integral

$norm\left(g\right)=\sqrt{{\int }_0^{\pi }cos{\left(x\right)}^{2}dx}$

which evaluates to

$norm\left(g\right)=\sqrt{\frac{\pi }{2}}$

If we evaluate the inner product of the two functions f(x) and g(x), we get the definite integral

$ip\left(f,g\right)=\underset{0}{\overset{\pi }{\int }}cos\left(x\right)sin\left(x\right)dx$

which evaluates to

$ip\left(f,g\right)=0$

If the inner product between two vectors is zero, we say the two vectors are "orthogonal" to each other. Orthogonal vectors can also be shown to be linearly independent.

If we divide a vector by its length or norm, then we "normalize" the vector. For the preceding f(x) and g(x), the corresponding normalized vectors are

$f\left(x\right)=\sqrt{\frac{2}{\pi }}sin\left(x\right)$

A set that consists of vectors that are both normal and orthogonal is said to be an "orthonormal" set. For orthonormal sets, the inner product of two vectors in the set gives the value 1 if the vectors are alike or the value 0 if the vectors are not alike.

Two vectors φ_n(x) and φ_m(x), which are indexed by the positive integers n and m, are orthonormal with respect to the weight function w(x) over the interval [a, b] if the following relation holds:

$\underset{a}{\overset{b}{\int }}{\phi }_n\left(x\right){\phi }_m\left(x\right)w\left(x\right)dx=\delta \left(n,m\right)$

Here, δ(n, m) is the familiar Kronecker delta function whose value is 0 if n ≠ m and is 1 if n = m.

Orthonormal sets play a big role in the development of solutions to partial differential equations.

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Exterior Algebra

Erdoğan S. Şuhubi , in Exterior Analysis, 2013

1.3 Multilinear Functionals

Let (U ₁, U ₂, …,U_k ) be ordered k -tuple of linear vector spaces defined over the same field of scalars $\mathbb{F}$ . Let us consider a scalar-valued function $\mathcal{T}:U_1\times U_2\times \cdots \times U_k\to \mathbb{F}$ on the Cartesian product of these vector spaces. If the function, $\mathcal{T}\left(u_{\left(1\right)},u_{\left(2\right)},\dots ,u_{\left(k\right)}\right)\in \mathbb{F}$ , where u _(α) ∈ U _α , α = 1, 2,…, k, is linear in each one of its arguments, that is, if the following relations

(1.3.1) $\begin{array}{l}\mathcal{T}\left(\dots ,u_{\left(i\right)}+{\upsilon }_{\left(i\right)},\dots \right)=\mathcal{T}\left(\dots ,u_{\left(i\right)},\dots \right)+\mathcal{T}\left(\dots ,{\upsilon }_{\left(i\right)},\dots \right)\\ \mathcal{T}\left(\dots ,\alpha u_{\left(i\right)},\dots \right)=\alpha \mathcal{T}\left(\dots ,u_{\left(i\right)},\dots \right),\alpha \in \mathbb{F}\end{array}$

are satisfied for all 1   ≤ i ≤ k, then the function $\mathcal{T}$ is called a multilinear functional (or a k-linear functional ). In finite-dimensional vector spaces whose dimensions and bases are n ₁,.,n_k and {e _i ^(α)}   ∈ U _α , i  =   1,   …, n _α ,α  =   1,   …, k, we can then write $u_{\left(\alpha \right)}={\displaystyle \sum _{i=1}^{n_{\alpha }}{u}_{\left(\alpha \right)}^i}{e}_i^{\left(\alpha \right)}$ , without having recourse to the summation convention. Multilinearity then leads to the following value of the functional at vectors u ₍₁₎  ∈ U ₁, u ₍₂₎  ∈ U ₂,   …, u _(k)  ∈ U _k

(1.3.2) $\mathcal{T}\left(u_{\left(1\right)},u_{\left(2\right)},\dots ,u_{\left(k\right)}\right)={\displaystyle \sum _{i_1=1}^{n_1}{\displaystyle \sum _{i_2=1}^{n_2}\dots {\displaystyle \sum _{i_k=1}^{n_k}{t}_{i_1{i}_2\dots i_k}}{u}_{\left(1\right)}^{i_1}{u}_{\left(2\right)}^{i_2}\cdots u_{\left(k\right)}^{i_k}}}$

Where n ₁ × n ₂ × ... × n _k number of scalar $t_{i_1{i}_2\cdots i_k}$ are defined by

(1.3.3) $t_{i_1{i}_2\dots i_k}=\mathcal{T}\left(e_{i_1}^{\left(1\right)},e_{i_2}^{\left(2\right)},\dots ,e_{i_k}^{\left(k\right)}\right)\in \mathbb{F}$

We thus conclude that the set of scalars $\left\{t_{i_1{i}_2\dots i_k}\right\}$ completely determines the action of a k -linear functional on any set of k number of vectors u ₍₁₎  ∈ U ₁, u ₍₂₎  ∈ U ₂,   …, u _(k)  ∈ U _k . We can thus say that they unambiguously characterise a multilinear functional.

Let us now suppose that U ₁ = U ₂ =⋯ = U_k = U ⁽ⁿ⁾. The value of a multilinear functional $\mathcal{T}:U^k\to \mathbb{F}$ on vectors u ₍₁₎,u ₍₂₎, ..., u_(k) ∈ U can now be found from (1.3.2) and (1.3.3) as follows

(1.3.4) $\begin{array}{l}\mathcal{T}\left(u_{\left(1\right)},u_{\left(2\right)},\dots ,u_{\left(k\right)}\right)=t_{i_1{i}_2\dots i_k}{u}_{\left(1\right)}^{i_1}{u}_{\left(2\right)}^{i_2}\cdots u_{\left(k\right)}^{i_k},\\ t_{i_1{i}_2\dots i_k}=\mathcal{T}\left(e_{i_1},e_{i_2},\dots ,e_{\mathit{ik}}\right),1\le i_1,i_2,\cdots ,i_k\le n\end{array}$

where we experience no difficulty in resorting to the summation convention because the range of all indices is the same now, from 1 to n. In this case, we can introduce a more advantageous representation of a multilinear functional as an operator. To this end, we shall first introduce the tensor product of two vector spaces.

Let U and V be two linear vector spaces defined on the same field of scalars $\mathbb{F}$ . As is well known, the Cartesian product U × V of these spaces is formed by ordered pairs (u,υ), where u ∈ U and υ ∈ V. There is initially no algebraic structure on this product set. However, by making use of known operations on vector spaces U and V, we may define appropriate operations on the set U × V so that it may be equipped with a structure of a linear vector space. The resulting vector space will be called the tensor product of spaces U and V and will be denoted by W  = U  ⊗ V. Let us choose operations of vector addition and scalar multiplication on W in such a way that tensor product of vectors u  ⊗ υ  ∈ U  ⊗ V has to satisfy the following bilinearity conditions:

(i).

u  ⊗   (υ ₁  + υ ₂)   = u  ⊗ υ ₁  ⊗ υ ₂,

(ii).

(u ₁  + u ₂)   ⊗ υ  = u ₁  ⊗ υ  + u ₂  ⊗ υ,

(iii).

$\left(\mathit{\alpha u}\right)\otimes \upsilon =u\otimes \left(\mathit{\alpha \upsilon }\right)=\alpha \left(u\otimes \upsilon \right),\alpha \in \mathbb{F}.$

Let us note that the same symbol + in the foregoing expressions represents,in fact, different addition operations in three different vector spaces U, V and W. We can thus write

$\left(u_1+u_2\right)\otimes \left({\upsilon }_1+{\upsilon }_2\right)=u_1\otimes {\upsilon }_1+u_1\otimes {\upsilon }_2+u_2\otimes {\upsilon }_1+u_2\otimes {\upsilon }_2.$

The space W is then defined as the collection of all finite sums ${\displaystyle \sum _i{u}_i\otimes {\upsilon }_i}$ where u_i ∈ U and υ_i ∈ V . If we consider finite-dimensional vector spacesU ^(m)and V ⁽ⁿ⁾with respective bases {e_i } and {f_j }, a vector w  ∈ W is evidently expressible as ω  = ω ^ij e _i   ⊗ f _j . Hence, W is an mn-dimensionalvector space with a basis {e _i   ⊗ f _j }. The tensor product can evidently be extendedon Cartesian products of arbitrary number of vector spaces.Let us now consider the n-dimensional dual space U ^⁎ of an n-dimensionalvector space U. It is quite clear that an element, or a vector, of thetensor product ⊗ ^k U* can now be represented by

(1.3.5) $\mathcal{T}=t_{i_1{i}_2...i_k}{f}^{i_1}\otimes f^{i_2}\otimes ...\otimes f^{i_k}$

where{fⁱ } is the reciprocal basis in U ^* corresponding to the basis {e _i } in U. We define the value of the element $\mathcal{T}$ on an ordered k-tuple of vectors (u ₍₁₎,u ₍₂₎,   …,u _(k))   ∈ U ^k as

$\mathcal{T}\left(u_{\left(1\right)},\dots ,u_{\left(k\right)}\right)=t_{i_1\dots i_k}{u}_{\left(1\right)}^{j_1}\dots u_{\left(k\right)}^{\mathit{jk}}{f}^{i_1}\left(e_{j_1}\right)\cdots f^{i_k}\left(e_{\mathit{jk}}\right)$

In view of (1.2.7), we then find that

$\mathcal{T}\left(u_{\left(1\right)},u_{\left(2\right)},\dots ,u_{\left(k\right)}\right)=t_{i_1}{}_{i_2\cdots i_k}{u}_{\left(1\right)}^{i_1}{u}_{\left(2\right)}^{i_2}\cdots u_{\left(k\right)}^{i_k}.$

We immediately see that the above relation leads to (1.3.4) ₂ for vectors $e_{i_1},e_{i_2},\dots ,e_{i_k}$ . Hence (1.3.5) does in fact play the part of k -linear functionalon U^k and the tensor product ⊗ ^k U* is the vector space in which such k-linear functionals inhabit. We say that the elements of this vector space are k-covariant tensors and the number k is known as the order of the tensor.The scalar coefficients $t_{i_1{i}_2\dots i_k}$ are then called the components of such a tensor with respect to bases. $f^{i_1}\otimes \cdots \otimes f^{i_k}$ It is easily observed that thetensorproduct . $f^{i_1}\otimes \cdots \otimes f^{i_k}$ of basis vectors constitutes a basis for the space ⊗ ^k U*. Indeed the value of the zero element in ⊗ ^k U*

$t_{i_1{i}_2\cdots i_k}{f}^{i_1}\otimes f^{i_2}\otimes \cdots \otimes f^{i_k}=0$

on vectors $e_{j_1},e_{j_2},\cdots ,e_{j_k}\in U$ vanishes naturally so that one obtains

$t_{i_1{i}_2\cdots i_k}{f}^{i_1}\left(e_{j_1}\right)f^{i_2}\left(e_{j_2}\right)\cdots f^{i_k}\left(e_{j_k}\right)=t_{j_1{j}_2\cdots j_k}=0$

for all coefficients. Hence, the dimension of this vector space is n^k . Obviously,the sum of two tensors of the same kind and multiplication of a tensorby a scalar are again the following tensors of the same kind:

$\begin{array}{l}{\mathcal{T}}_1+{\mathcal{T}}_2=\left(t_{i_1{i}_2\cdots i_k}^{\left(1\right)}+t_{i_1{i}_2\cdots i_k}^{\left(2\right)}\right)f^{i_1}\otimes f^{i_2}\otimes \cdots \otimes f^{i_k}\\ \alpha \mathcal{T}=\left(\alpha t_{i_1{i}_2\cdots i_k}\right)f^{i_1}\otimes f^{i_2}\otimes \cdots \otimes f^{i_k}.\end{array}$

This is of course a direct consequence of ⊗ ^k U* being a linear vector space.

We can now naturally define the tensorial product of a k-covariant tensorand an l-covariant tensor by

${\mathcal{T}}_1\otimes {\mathcal{T}}_2=t_{i_1\cdots i_k}^{\left(1\right)}{t}_{j_1\cdots j_i}^{\left(2\right)}{f}^{i_1}\otimes \cdots \otimes f^{i_k}\otimes f^{j_1}\otimes \cdots \otimes f^{\mathit{jl}}.$

The result is obviously a (k + l)-covariant tensor.

Let us now change the basis {e_i }in the vector space U to another basis {e _i ^'} as in (1.2.11). We know that the reciprocal basis {fⁱ }in the dual space U^* changes to a reciprocal basis {f^'i }through the relations (1.2.14). Consequently,the same tensor $\mathcal{T}$ is represented with respect to two different basesas follows

$\begin{array}{l}\mathcal{T}=t_{j_1{j}_2\dots j_k}{f}^{j_1}\otimes f^{j_2}\otimes \cdots \otimes f^{j_k}=t_{i_1{i}_2\dots i_k}^{\mathit{\text{'}}}{f}^{\text{'}{i}_1}\otimes f^{\text{'}{i}_2}\otimes \cdots \otimes f^{\mathit{\text{'}}}{}^{\mathit{ik}}\\ =t_{j_1{j}_2\cdots j_k}{b}_{i_1}^{j_1}{b}_{i_2}^{j_2}\cdots b_{\mathit{ik}}^{\mathit{jk}}{f}^{\mathit{\text{'}}}{}^{i_1}\otimes f^{\mathit{\text{'}}}{}^{i_2}\otimes \cdots \otimes f^{\mathit{\text{'}}}{}^{i_k}\end{array}$

from which we immediately deduce that the following rule of transformation between components of a k-covariant tensor must be valid:

(1.3.6) $t_{i_1{i}_2\cdots i_k}^{\mathit{\text{'}}}=b_{i_1}^{j_1}{b}_{i_2}^{j_2}\cdots b_{i_k}^{j_k}{t}_{j_1{j}_2\cdots j_k}.$

In a similar fashion we may define a multilinear (k-linear) functionalon the dual space U* of a vector space. Such a functional $\mathcal{T}:{\left(U^{*}\right)}^k\to \mathbb{F}$ assigns a scalar number $\mathcal{T}\left(f^{\left(1\right)},f^{\left(2\right)},\dots ,f^{\left(k\right)}\right)\in \mathbb{F}$ to an ordered k-tuple of linear functionals (f ⁽¹⁾,f ⁽²⁾,   …,f ^(k))   ∈   (U*) ^k and obeys the rules

$\begin{array}{l}\mathcal{T}\left(\cdots ,f^{\left(i\right)}+g^{\left(i\right)},\dots \right)=\mathcal{T}(\dots ,f^{\left(i\right)},\dots )+\mathcal{T}(\dots ,g^{\left(i\right)},\dots )\\ \mathcal{T}\left(\dots ,\alpha f^{\left(i\right)},\dots \right)=\alpha \mathcal{T}(\dots ,f^{\left(i\right)},\dots ),\alpha \in \mathbb{F}.\end{array}$

By resorting to the reciprocal basis {f ⁱ }   ∈ U* corresponding to the basis {e_i } ∈ U we can of course write $f^{\left(m\right)}={\alpha }_i^{\left(m\right)}{f}^i,{\alpha }_i^{\left(m\right)}\in \mathbb{F},1\le m\le k$ and we obtain

(1.3.7) $\begin{array}{l}\mathcal{T}\left(f^{\left(1\right)},f^{\left(2\right)},\dots ,f^{\left(k\right)}\right)=t^{i_1{i}_2\dots i_k}{\alpha }_{i_1}^{\left(1\right)}{\alpha }_{i_2}^{\left(2\right)}\cdots {\alpha }_{i_k}^{\left(k\right)},\\ t^{i_1{i}_2\dots i_k}=\mathcal{T}\left(f^{i_1},f^{i_2},\dots ,f^{i_k}\right).\end{array}$

The ensemble of scalar numbers $t^{i_1{i}_2\dots i_k},1\le i_1,i_2,\dots ,i_k\le n$ entirely determines the action of a multilinear functional $\mathcal{T}$ on (U  *) ^k . Let us nowdefine an element in the tensor product ⊗ ^k U by

$\mathcal{T}=t_{i_1{i}_2\dots i_k}{e}_i{}_{{}_1}\otimes e_{i_2}\otimes \cdots \otimes e_{i_k}.$

$\mathcal{T}$ is called a k-contravariant tensor . It is evident that the linearly independentselements $e_{i_1}\otimes e_{i_2}\cdots \otimes e_{i_k}$ constitute a basis for the vector space ⊗ ^k U. n ^k number of scalars $t^{i_1{i}_2\dots i_k}$ are said to be components of this tensorwith respect to bases $e_{i_1}\otimes \cdots \otimes e_{i_k}$ . Let us define the value of the tensor $\mathcal{T}$ on k linear functionals f ⁽¹⁾,f ⁽²⁾,…,f ^(k) by the relation

$\mathcal{T}\left(f^{\left(1\right)},f^{\left(2\right)},\dots ,f^{\left(k\right)}\right)=t^{i_1{i}_2\dots i_k}{f}^{\left(1\right)}\left(e_i{}_{{}_1}\right)f^{\left(2\right)}\left(e_{i_2}\right)\cdots f^{\left(k\right)}\left(e_{i_k}\right).$

In view of (1.2.6) we find that

$\mathcal{T}\left(f^{\left(1\right)},f^{\left(2\right)},\cdots f^{\left(k\right)}\right)=t^{{}_{i_1{i}_2\cdots i_k}}{\alpha }_{i_1}^{\left(1\right)}{\alpha }_{i_2}^{\left(2\right)}{\cdots }_{i_k}^{\left(k\right)}.$

It is clear that the product of a k-contravariant tensor and an l-contravarianttensor is a (k + l)-contravariant tensor. We now consider a change of basisin the vector space U. We then obtain

$\begin{array}{l}\mathcal{T}=t^{j_1{j}_2\cdots j_k}{e}_{j_1}\otimes e_{j_2}\otimes \cdots \otimes e_{j_k}=t^{\mathit{\text{'}}{i}_1{i}_2\cdots i_k}{e}_{i_1}^{\mathit{\text{'}}}\otimes e_{i_2}^{\mathit{\text{'}}}\otimes \cdots \otimes e_{i_k}^{\mathit{\text{'}}}\\ =t^{j_1{j}_2\cdots j_k}{a}_{j_1}^{i_1}{a}_{j_2}^{i_2}\cdots a_{j_k}^{i_k}{e}_{i_1}^{\mathit{\text{'}}}\otimes e_{i_2}^{\mathit{\text{'}}}\otimes \cdots \otimes e_{i_k}^{\mathit{\text{'}}}\end{array}$

from which we deduce the following rule of transformation for componentsof a contravariant tensor

(1.3.8) $t^{\mathit{\text{'}}}{}^{i_1{i}_2\cdots i_k}=a_{j_1}^{i_1}{a}_{j_2}^{i_2}\cdots a_{j_k}^{i_k}{t}^{j_1{j}_2\cdots j_k}.$

We can also easily define tensors of mixed type. A k-contravariant and l-covariant mixed tensor is an element of the vector space ⊗ ^k U  ⊗ ^l U  * and can be written in the form

$\begin{array}{l}\mathcal{T}=t_{j_1{j}_2\cdots j_l}^{i_1{i}_2\cdots i_k}{e}_{i_1}\otimes e_{i_2}\otimes \cdots \otimes e_{i_k}\otimes f^{j_1}\otimes f^{j_2}\otimes \cdots \otimes f^{j_l},\\ t_{j_1{j}_2\cdots j_l}^{i_1{i}_2\cdots i_k}=\mathcal{T}\left(f^{i_1},f^{i_2},\cdots ,f^{i_k},e_{j_1},e_{j_2},\cdots ,e_{j_l}\right),\\ 1\le i_1,i_2,\cdots ,i_k\le n,1\le j_1,j_2,\cdots ,j_l\le n.\end{array}$

The value of this tensor on linear functionals f ⁽¹⁾, f ⁽²⁾,   ⋯, f ^(k)  ∈ U  * and Vectors u ₍₁₎, u ₍₂₎,   ⋯, u _(l)  ∈ U is given by

$\mathcal{T}\left(f^{\left(1\right)},\cdots ,f^{\left(k\right)},u_{\left(1\right)},\dots ,u_{\left(l\right)}\right)=t_{j_1{j}_2\cdots j_l}^{i_1{i}_2\cdots i_k}{\alpha }_{i_1}^{\left(1\right)}{\alpha }_{i_2}^{\left(2\right)}\cdots {\alpha }_{i_k}^{\left(k\right)}{u}_{\left(1\right)}^{j_1}{u}_{\left(2\right)}^{j_2}\cdots u_{\left(l\right)}^{j_l}.$

It is quite obvious that we do not have to select the ordering in the tensorproducts in the foregoing way. We may, of course, consider a differentordering such as U  ⊗ U  *   ⊗ U  *   ⊗ U  ⊗ U  *   ⊗   ⋯. The indices of componentsof this type of a tensor occupy accordingly proper upper and lower positions.It is evident that different ordering of spaces in the tensor product willgive rise to different types of tensors of the same order.

If, in a mixed tensor of order k + l, we remove the tensor product betweenthe functional $f^{j_m}$ and the vector $e_{i_n}$ , then the relation $f^{j_m}\left(e_{i_n}\right)={\delta }_{i_n}^{j_m}$ between reciprocal basis vectors reduces the order of the tensor. We thusobtain a (k − 1)-contravariant and (l − 1)-covariant tensor, in other words, atensor of order k + l − 2 defined by the relation

${\mathcal{T}}_c=t_{j_1\cdots i_n\cdots j_l}^{i_1\cdots i_n\cdots i_k}{e}_{i_1}\otimes \cdots \otimes e_i{}_{{}_{n-1}}\otimes e_{i_{n+1}}\otimes \cdots \otimes e_{i_k}\otimes f^{j_1}\otimes \cdots \otimes f^{j_{m-1}}\otimes f^{j_{m+1}}\otimes \cdots \otimes f^{j_l}.$

This operation is called a contraction . The components of the contractedtensor are given as follows:

${}_c{t}_{j_1\cdots j_{m-1}{j}_{m+1}\cdots j_l}^{i_1\cdots i_{n-1}{i}_{n+1}\cdots i_k}=t_{j_1\cdots j_{m-1}i{j}_{m+1}\cdots j_l}^{i_1\cdots i_{n-1}i{i}_{n+1}\cdots i_k}.$

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Fast Translations: Basic Theory and O(p3) Methods

NAIL A. GUMEROV , RAMANI DURAISWAMI , in Fast Multipole Methods for the Helmholtz Equation in Three Dimensions, 2004

7.1.1.2 Linear operators

Operators are defined as mappings from one linear vector space to another (both spaces can be the same). If we have a function ψ ∈ B(Ω) and can correspond it to some other function ψ′ ∈ B(Ω′), then such a mapping is an operator A acting from the space B(Ω) to the space B(Ω′). This can be written as

(7.1.1) $\psi \prime =A\left[\psi \right],A:B\left(\Omega \right)\to B\left(\Omega \prime \right),\psi \in B\left(\Omega \right),\psi \prime \in B\left(\Omega \prime \right)$

The operator is called linear if it satisfies the following property:

(7.1.2) $A\left[\alpha {\psi }_1+\beta {\psi }_2\right]=\alpha A\left[{\psi }_1\right]+\beta A\left[{\psi }_2\right],\alpha ,\beta \in ℂ.$

Important examples of linear operators are as follows.

Differential operators, D. An example is ∂/∂z which performs partial differentiation with respect to z (we also used ∂ _z = k ⁻¹∂/∂z and sometimes to stress that this operator acts in functional space, we use the notation D_z = ∂ _z ). Differential operators include ∇, ∇², or differentiation in a direction s, s·∇. These operators map B(Ω) → B(Ω), so that the domain of definition does not change and we have

(7.1.3) $\psi \prime =D\left[\psi \right],D:B\left(\Omega \right)\to B\left(\Omega \right).$

Rotation operators, Rot. These operators are generated by rotation of the basis vectors with matrix Q, so we can write this operator as Rot(Q). If Ω is the interior or exterior of a sphere, then rotation does not change the domain of definition, and we have

(7.1.4) $\begin{array}{lll}\psi \prime =Rot(Q)[\psi ],& \psi \prime (r)=\psi (\widehat{r}),& \widehat{r}=Qr,\\ Rot(Q):B(\Omega )\to B(\Omega ),& \Omega :r\gtrless a.& \end{array}$

Translation operators, T. These operators arise due to a shift in the function argument by a translation vector t, so we can write them as T(t). The translation operator shifts the domain of definition of the transformed function ψ′(r) = ψ(r +t) (indeed if the function ψ(r) has only a singularity at r = -t, then ψ′(r) has the same singularity at r = 0). In general, we have

(7.1.5) $\begin{array}{lll}\psi \prime =T(t)[\psi ],\hfill & \psi \prime (r)=\psi (r\prime ),\hfill & r\prime =r+t,\hfill \\ T(t):B(\Omega )\to B(\Omega \prime ),\hfill & \hfill & \hfill \end{array}$

where Ω′ is obtained by the shift of the domain Ω.

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Lie Groups: General Theory

R. Gilmore , in Encyclopedia of Mathematical Physics, 2006

Solvable Algebra

If $\mathfrak{g}$ is a Lie algebra, the linear vector space obtained by taking all possible commutators of the operators in $\mathfrak{g}$ is called the "derived" algebra: $[\mathfrak{g},\mathfrak{g}]={\mathfrak{g}}^{(1)}\subseteq \mathfrak{g}$ . If ${\mathfrak{g}}^{(1)}=\mathfrak{g}$ , there is no point in continuing this process. If ${\mathfrak{g}}^{(1)}\subset \mathfrak{g}$ , it is useful to define $\mathfrak{g}={\mathfrak{g}}^{(0)}$ and to continue this process by defining ${\mathfrak{g}}^{(2)}$ as the derived algebra of ${\mathfrak{g}}^{(1)}:{\mathfrak{g}}^{(2)}=[{\mathfrak{g}}^{(1)},{\mathfrak{g}}^{(1)}]$ . We can continue in this way, defining ${\mathfrak{g}}^{(n+1)}$ as the algebra derived from ${\mathfrak{g}}^{(n)}$ . Ultimately (for finite-dimensional Lie algebras), either ${\mathfrak{g}}^{(n+1)}=0$ or ${\mathfrak{g}}^{(n+1)}={\mathfrak{g}}^{(n)}$ for some n. If the former case occurs,

$\mathfrak{g}={\mathfrak{g}}^{(0)}\supset {\mathfrak{g}}^{(1)}\supset {\mathfrak{g}}^{(2)}\supset \cdots \supset {\mathfrak{g}}^{(n)}\supset {\mathfrak{g}}^{(n+1)}=0$

the Lie algebra ${\mathfrak{g}}^{(0)}$ is called solvable. Each algebra ${\mathfrak{g}}^{(i)}$ is an invariant subalgebra of ${\mathfrak{g}}^{(j)},i>j$ .

Example

The Lie algebra spanned by the boson number, creation, annihilation, and identity operators is solvable. The series of derived algebras has dimensions 4, 3, 1, 0.

${\mathfrak{g}}^{(0)}$	${\mathfrak{g}}^{(1)}$	${\mathfrak{g}}^{(2)}$	${\mathfrak{g}}^{(3)}$
a † a	−	−	−
a †	a †	−	−
a	a	−	−
I	I	I	−

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Electromagnetic Theory and Optics

Avijit Lahiri , in Basic Optics, 2016

Digression: tensors and tensor fields

For a given r and given t, a vector such as E(r, t ) is an element of a real three-dimensional linear vector space which we denote as ${\mathcal{R}}^{(3)}$ . A tensor of rank 2 is then an element of a nine-dimensional vector space $\mathcal{T}$ that includes the direct product ${\mathcal{R}}^{(3)}\times {\mathcal{R}}^{(3)}$ and, in addition, contains all possible linear combinations of direct products of pairs of vectors. If ${\widehat{n}}_1,{\widehat{n}}_2,{\widehat{n}}_3$ constitute an orthonormal basis in ${\mathcal{R}}^{(3)}$ , then an orthonormal basis in $\mathcal{T}$ will be made up of the objects ${\widehat{n}}_i{\widehat{n}}_j(i,j=1,2,3),$ and a tensor of rank 2 can be expressed as a linear combination of the form ${\sum }_{i,j}{C}_{ij}{\widehat{n}}_i{\widehat{n}}_j.$ Thus, with reference to this basis, the tensor under consideration is completely described by the 3 × 3 matrix with elements C _ij . The matrix (and also the tensor) is termed 'symmetric' if C _ij = C _ji (i, j = 1, 2, 3). The matrix is said to be positive definite if all its eigenvalues are positive. Now consider any of the above field vectors (say, E(r, t)) at a given time instant but at all possible points r. This means a vector associated with every point in some specified region in space. The set of all these vectors is termed a vector field in the region under consideration. The vector field is, moreover, time dependent since the field vectors depend, in general, on t. Similarly, one can have a tensor field such as the permittivity tensor [ϵ] or the permeability tensor [μ] in an inhomogeneous anisotropic medium in which the electric and magnetic material properties vary from point to point in addition to being direction dependent. While these can, in general, even be time-dependent tensor fields, we will, in this book, consider media with time-independent properties alone.

Thus, in terms of the Cartesian components, relations (1.1f) and (1.1g) can be written as

(1.2a) $\begin{array}{ll}\hfill D_i& =\sum _j{ϵ}_{ij}{E}_j,\hfill \end{array}$

(1.2b) $\begin{array}{ll}\hfill B_i& =\sum _j{\mu }_{ij}{H}_j.\hfill \end{array}$

As mentioned above, the electric permittivity and magnetic permeability tensors ([ϵ], [μ]) reduce, in the case of an isotropic medium, to scalars (corresponding to constant multiples of the identity matrix) and the above relations simplify to

(1.3a) $\begin{array}{ll}\hfill \mathbf{D}& =ϵ\mathbf{E},D_i=ϵE_i(i=1,2,3),\hfill \end{array}$

(1.3b) $\begin{array}{ll}\hfill \mathbf{B}& =\mu \mathbf{H},B_i=\mu H_i(i=1,2,3).\hfill \end{array}$

It is not unusual for an optically anisotropic medium, with a permittivity tensor [ϵ], to be characterized by a scalar permeability μ (approximately μ ₀, the permeability of free space). In this book I use the SI system of units, in which the permittivity and permeability of free space are, respectively, ϵ ₀ = 8.85 × 10⁻¹² C² N⁻¹ m⁻² and μ ₀ = 4π × 10⁻⁷ NA⁻².

In general, for linear media with time-independent properties, the following situations may be encountered: (1) isotropic homogeneous media, for which ϵ and μ are scalar constants independent of r; (2) isotropic inhomogeneous media for which ϵ and μ are scalars but vary from point to point; (3) anisotropic homogeneous media where [ϵ] and [μ] are tensors independent of the position vector r; and (4) anisotropic inhomogeneous media in which [ϵ] and [μ] are tensor fields. As mentioned above, in most situations relating to optics one can, for simplicity, assume [μ] to be a scalar constant, μ ≈ μ ₀.

However, in reality, the relation between E and D is of a more complex nature (that between B and H may, in principle, be similarly complex), even for a linear, homogeneous, isotropic medium with time-independent properties, than is apparent from Eq. (1.3a) since ϵ is, in general, a frequency-dependent object. A time-dependent field vector can be analyzed into its Fourier components, each component corresponding to some specific angular frequency ω. A relation such as Eq. (1.3a) can be used only in situations where this frequency dependence of the electric (and also magnetic) properties of the medium under consideration can be ignored (ie, when dispersion effects are not important). In this book we will generally assume the media are nondispersive, taking into account dispersion effects only in certain specific contexts (see Section 1.17).

One more constitutive equation holds for a conducting medium:

(1.4) $\begin{array}{l}\hfill \mathbf{j}=[\sigma ]\mathbf{E},\end{array}$

where, in general, the conductivity [σ] is once again a second-rank symmetric tensor which, for numerous situations of practical relevance, reduces to a scalar. The conductivity may also be frequency dependent, as will be discussed briefly in Section 1.17.2.7.

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Tensor Fields on Manifolds

Erdoğan S. Şuhubi , in Exterior Analysis, 2013

4.2 Cotangent Bundle

We consider an m-dimensional smooth manifold M and the tangent space T_p (M) at a point p ∈ M . As is well known, the dual of the tangent space is a linear vector space formed by all linear functionals on the tangent space [see p. 11]. We denote this m-dimensional dual space by T _p ^∗(M) and we also call it the cotangent space at the point p. When we choose the natural basis of the tangent space at the point p as the vectors {∂/∂xⁱ : i = 1, … , m} generated by the local coordinates in the chart containing the point p, we have seen on p. 125 that reciprocal basis vectors in the dual space are given by linear functionals as differentials {dxⁱ : i = 1, … , m} so that the following relations

(4.2.1) $d{x}^i\left(\frac{\partial }{\partial x^j}\right)=〈d{x}^i,\frac{\partial }{\partial x^j}〉={\delta }_j^i$

are satisfied. Hence, at a point p ∈ M, a vector V ∈ T_p (M) and a linear functional ω  ∈ T _p ^*(M) can be expressed as

(4.2.2) $V={\upsilon }^i\frac{\partial }{\partial x^i},\omega ={\omega }_{i}d{x}^i,{\upsilon }^i,{\omega }_i\in \mathbb{R}\cdot$

The value of the functional ω on the vector V at p then happens to be

(4.2.3) $\begin{array}{l}\omega \left(V\right)=〈\omega ,V〉=〈{\omega }_{i}d{x}^i,{\upsilon }^j\frac{\partial }{\partial x^j}〉\\ ={\omega }_i{\upsilon }^j{\delta }_j^i={\omega }_i{\upsilon }^i\in \mathbb{R}\cdot \end{array}$

We shall call elements of the dual space T _p ^*(M) as 1-forms at the point p. Next, we define the set

(4.2.4) $T^{*}\left(M\right)=\underset{p\in M}{\cup }{T}_p^{*}\left(M\right)=\left\{\left(p,\omega \right):p\in M,\omega \in T_p^{*}\left(M\right)\right\}.$

By repeating exactly our approach in Sec. 2.8, we see that T*(M) can be endowed with a differentiable structure making it a 2m-dimensional smooth manifold which will be called henceforth as the cotangent bundle . The local coordinates of T*(M) are evidently given by {x ¹,…, x^m , ω ₁, … , ω_m }. A section of the bundle T*(M) as we have already done in p. 130 characterises this time a 1-form field on the smooth manifold M. In terms of local coordinates in the relevant chart, this field is of course expressible as follows

(4.2.5) $\omega \left(p\right)={\omega }_i\left(\mathbf{x}\right)d{x}^i\in T*\left(M\right),\mathbf{x}\mathbf{=}\phi \left(p\right)\cdot$

Different charts containing the point p gives rise to a coordinate transformation given by invertible functions yⁱ = yⁱ (x^j ). When we write the 1-form ω in different local coordinates, the relation

$\omega \left(p\right)={\omega }_{j}d{x}^j={{\omega }^{\prime }}_{i}d{y}^i={{\omega }^{\prime }}_i\frac{\partial y^i}{\partial x^j}d{x}^j$

leads to the following relations between components of ω in two different coordinate systems

${\omega }_j={{\omega }^{\prime }}_i\frac{\partial y^i}{\partial x^j}\mathrm{or}{{\omega }^{\prime }}_i=\frac{\partial x^j}{\partial y^i}{\omega }_j\cdot$

Because of this transformation rule, the elements of the cotangent bundle are usually called covariant vector or covector fields . We have already seen that the transformation rule between components of vectors in two different charts in the tangent bundle are given by [see (2.6.9)]

${{\upsilon }^{\prime }}^i=\frac{\partial y^i}{\partial x^j}{\upsilon }^j\cdot$

That is the reason why we call vectors in the tangent bundle as contravariant vector fields.

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Representations for Morphological Image Operators and Analogies with Linear Operators

Petros Maragos , in Advances in Imaging and Electron Physics, 2013

2.2.1 Algebraic Definitions for Linear Operators

A mapping $L:X\to Y$ between two linear spaces over the same scalar field is called a linear operator if it preserves addition and scalar multiplication. This is equivalent to

$L\left(a_1{x}_1+\cdots +a_n{x}_n\right)=a_{1}L\left(x_1\right)+\cdots +a_{n}L\left(x_n\right)$

for all $x_1,\mathrm{\dots },x_n\in X$ , all scalars $a_1,\mathrm{\dots },a_n$ and all finite n. An operator that does not satisfy the above is called nonlinear. The null space and range of L are defined as

$\text{Null}\left(L\right)\triangleq \left\{x\in X:L\left(x\right)=\mathbf{0}\right\},\text{Ran}\left(L\right)\triangleq \left\{L\left(x\right):x\in X\right\}.$

The null space is a linear subspace of X, whereas the range is a linear subspace of Y. For any linear operator $L:X\to Y$ between two linear spaces, the dimensions of its null and range spaces are related as follows to the dimension of the domain space:

$\text{dim(Nul}(L))+\text{dim(Ran}(L))=\text{dim}\left(\text{Dom}\left(L\right)\right).$

Two linear spaces X and Y over the same scalar field are called isomorphic if there exists an invertible linear operator $L:X\to Y$ . Such a mapping L is called an isomorphism between the two linear spaces. The inverse mapping $L^{-1}:Y\to X$ is also a linear operator. It can be shown that a linear operator L is an isomorphism iff $\text{Nul}\left(L\right)=\left\{\mathbf{0}\right\}$ .

All linear spaces over the same field are isomorphic iff they have the same dimension. Hence, all real (resp. complex) finite-dimensional linear spaces are isomorphic to ${\mathbb{R}}^n$ (resp. ${\mathbb{C}}^n$ ) for some n. Thus, finite-dimensional linear spaces are essentially linear vector spaces, if by "vector" ^§ we agree to mean a finite tuple of scalars.

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Preliminaries

Dan C. Marinescu , Gabriela M. Marinescu , in Classical and Quantum Information, 2012

1.4 POSTULATES OF QUANTUM MECHANICS

A model of a physical system is an abstraction based on correspondence rules that relate the entities manipulated by the model to the physical objects or systems in the real world. Once such rules are established, we can operate only with the abstractions according to a set of transformation rules. To ensure the usefulness of the model and its ability to describe physical reality, we have to validate the model and compare its prediction with the physical reality. To ensure expressiveness, the ability of the model to describe the physical system, the correspondence and the transformation rules must be kept as simple as possible, but, at the same time, complete—in other words, capable of capturing the relevant properties of the physical system and of its dynamics, its evolution in time.

Distinguishability and system dynamics require the model to abstract the concepts of observable and of state of the physical object. An observable is a property of the system state that can be revealed as a result of some physical transformation. The state at time t is a synthetic characterization of the object that could be revealed by the measurement of relevant observables at time t.

The model must also abstract the concept of measurement; it should describe the relation between the state of the object before and after the measurement and specify how to interpret the results of a measurement, how to map the range of possible results to abstractions. In the physical world, we often have to deal with a collection of physical objects. If A, B, C, … are the abstractions of the objects a, b, c, …, respectively, we need another transformation rule to specify how to construct {A, B, C, …}, the abstraction corresponding to the collection {a, b, c, …}. Last, but not least, we need transformation rules to describe the system dynamics, the evolution of the system in time.

Quantum mechanics is a model of the physical world at all scales; it describes more accurately than classical physics systems at the atomic and subatomic scale. This model allows us to abstract our knowledge of a quantum system, to describe the state of single and composite quantum systems, the effect of a measurement on the system's state, and the dynamics of quantum systems. A quantum state summarizes our knowledge about a quantum system at a given moment in time, it allows us to describe what we know, as well as, what we do not know, about the system. An impressive number of experiments have produced results consistent with the prediction of quantum mechanics and so far there is no experimental evidence to disprove it; thus, we shall use this model to study the properties of quantum information.

The correspondence and transformation rules are captured by the postulates of quantum mechanics (Figure 1.2). We find it useful to expand the traditional three postulates of quantum mechanics, the state postulate, the dynamics postulate, and the measurement postulate, to emphasize some aspects important for quantum information processing:

FIGURE 1.2. The postulates of quantum mechanics.

1.

A quantum system, Q, is described in an n-dimensional Hilbert space, H_n , where n is finite. The Hilbert space H_n is a linear vector space over the field of complex numbers with an inner product. The dimension, n, of the Hilbert space is equal to the maximum number of reliably distinguishable states the system Q can be in.

2.

A state |ψ〉 of the quantum system Q corresponds to a direction (or ray) in H_n . In Section 1.11, we shall see that the most general representation of a quantum state is any density operator over an n-dimensional Hilbert space with n finite. The density operator is Hermitian, has non-negative eigenvalues, and has a trace equal to unity.

3.

When the internal conditions and the environment of a quantum system are completely specified and no measurements are performed on the system, the system's evolution is described by a unitary transformation in H_n defined by the Hamiltonian operator. A unitary transformation U is linear and preserves the inner product. The spontaneous evolution of an unobserved quantum system with the density matrix, ρ, is

$\rho \mapsto U_{\rho }{U}^{†},$

with U^†, the adjoint of U.

4.

Given two independently prepared quantum systems, Q described in H_n and S described in H_n , the bipartite system consisting of both Q and S is described, in a Hilbert space, H_n ⊗ H_m, the tensor product of the two Hilbert spaces.

5.

A measurement of the quantum system Q in the state |ψ described in H_n corresponds to a resolution of H_n to orthogonal subspaces, {H ^j }, and a projection of the system's state to these subspaces, {Pj}, such that the sum of the projections is ∑ P _j = 1. The measurement produces the result, j, with the probability

$Pr\text{ob}(j)=\left|{\text{P}}_j\right|\psi 〉{|}^2.$

The state after the measurement is

$|\phi 〉=\frac{{\text{P}}_j|\psi 〉}{\left|{\text{P}}_j\right|\psi 〉|}=\frac{{\text{P}}_j|\psi 〉}{\sqrt{Pr\text{ob}(j)}}.$

Manipulation of coherent quantum states is at the heart of quantum computing and quantum communication. A quantum computation involves a single entity and consists of unitary transformations of the quantum state. Quantum communication involves multiple entities and involves the transmission of quantum states over noisy communication channels.

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Convex Functions, Partial Orderings, and Statistical Applications

In Mathematics in Science and Engineering, 1992

1.28 Remarks

(a) We shall say that f′ is increasing on U if for x, y ∈ U we have

$(f^{\prime }(\text{x})-f^{\prime }(y))(\text{x}-y)\ge 0,$

and that f′ is strictly increasing on U if this inequality is strict for all x ≠ y.

(b) A function f: U → M (U ⊆ L; L, M are normed linear vector spaces) is Fréchet differentiable at x ₀ (x ₀ ∈ U) if there exists a linear transformation T: L → M such that

$\underset{\text{x}\to {\text{x}}_0}{\lim }\frac{f(\text{x})-f({\text{x}}_0)-T(\text{x}-{\text{x}}_0)}{\Vert \text{x}-{\text{x}}_0\Vert }=0,$

which is equivalent to

$f(\text{x})=f({\text{x}}_0)+T(\text{x}-{\text{x}}_0)+o\left(\Vert \text{x}-{\text{x}}_0\Vert \right)$

as x → x ₀. The linear transformation T is called the Fréchet derivative and is denoted by f′(x ₀).

(c) A similar derivative of the Fréchet derivative is called the second Fréchet derivative. This derivative is a symmetric bilinear transformation defined on L x L, i.e., f″ _{k, h} (x) = f″ _{h, k} (x) (h, k ∈ L). Note that if f: U → $ℝ$ is continuously differentiable on the open convex set U ⊆ L and f″(x) exists throughout U, then for any x, x ₀ ∈ U there is an s ∈ (0, 1) such that

(1.27) $f(\text{x})=f({\text{x}}_0)+f_h^{\text{'}}({\text{x}}_0)+\frac{1}{2}{f}_{h,h}^{"}({\text{x}}_0+sh),$

where h = x − x ₀.

(d) A symmetric bilinear transformation B(h, k) defined on L x L is positive (nonnegative) definite if for every h ∈ L (h ≠ 0), we have

$B(h,h)>0(B(h,h)\ge 0).$

(e) The following definition is also valid: A (continuous real-valued) function f is operator convex on (λ, v) if f(αa + βb) ≤ αf(a) + βf(b) for positive reals α, β such that α + β = 1 and operators a, b with their spectra in (λ, v). (See Davis, 1957, for a brief survey of operator functions and Ando, 1978, for further comments on classes of operator functions.)

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Mathematical preliminaries

J.E. Akin , in Finite Element Analysis with Error Estimators, 2005

2.1 Introduction

The earliest forms of finite element analysis were based on physical intuition with little recourse to higher mathematics. As the range of applications expanded, for example to the theory of plates and shells, some physical approaches failed and some succeeded. The use of higher mathematics such as variational calculus explained why the successful methods worked. At the same time the mathematicians were attracted by this new field of study. In the last few years the mathematical theory of finite element analysis has grown quite large. Since the state of the art now depends heavily on error estimators and error indicators it is necessary for an engineer to be aware of some basic mathematical topics of finite element analysis. We will consider load vectors and solution vectors, and residuals of various weak forms. All of these require us to define some method to 'measure' these entities. For the above linear vectors with discrete coefficients, V ^T = [V ₁ V ₂ … V_n ], we might want to use a measure like the root mean square, RMS:

$RM{S}^2=\frac{1}{n}\sum _{i=1}^n{V}_i^2=\frac{1}{2}{V}^{T}V$

which we will come to call a norm of the linear vector space. Other quantities vary with spatial position and appear in integrals over the solution domain and/or its boundaries. We will introduce various other norms to measure these integral quantities.

The finite element method always involves integrals so it is useful to review some integral identities such as Gauss' Theorem (Divergence Theorem):

$\underset{\Omega }{\int }\nabla \cdot u\text{d}\Omega =\underset{\Gamma }{\int }u\cdot n\text{d}\Gamma =\underset{\Gamma }{\int }\frac{\partial \text{u}}{\partial \text{n}}\text{d}\Gamma$

which is expressed in Cartesian tensor form as

$\underset{\Omega }{\int }{u}_{i,i}d\Omega =\underset{\Gamma }{\int }{u}_i{n}_{i}d\Gamma$

where there is an implied summation over subscripts that occurs an even number of times and a comma denotes partial differentiation with respect to the directions that follow it.

That is, ( ), _i = ∂( )/∂x _i . The above theorem can be generalized to a tensor with any number of subscripts:

$\underset{\Omega }{\int }{A}_{ijk\mathrm{\dots }q,r}d\Omega =\underset{\Gamma }{\int }{A}_{ijk\mathrm{\dots }q}{n}_{r}d\Gamma .$

We will often have need for one of the Green's Theorems:

$\underset{\Omega }{\int }\left(\nabla \text{A}\cdot \nabla \text{B}+\text{A}{\nabla }^2\text{B}\right)\text{d}\Omega =\underset{\Gamma }{\int }\text{A}\frac{\partial \text{B}}{\partial \text{n}}\text{d}\Gamma$

And

$\underset{\Gamma }{\int }\left(A{\nabla }^{2}B-B{\nabla }^{2}A\right)d\Omega =\underset{\Gamma }{\int }\left(A\nabla \text{B}-\text{B}\nabla \text{A}\right)\cdot n\text{d}\Gamma$

which in Cartesian tensor form are

$\underset{\Omega }{\int }\left(A_{,i}{B}_{,i}+A{B}_{,ii}\right)d\Omega =\underset{\Gamma }{\int }A{B}_{,i}{n}_{i}d\Gamma$

and

$\underset{\Omega }{\int }\left(A{B}_{,ii}-B{A}_{,ii}\right)d\Omega =\underset{\Gamma }{\int }\left(A{B}_{,i}-B{A}_{,i}\right)n_{i}d\Gamma .$

We need these relations to derive the Galerkin weak form statements and to manipulate the associated error estimators. Usually, we are interested in removing the highest derivative term in an integral and use the second from last equation in the form

(2.1) $\underset{\Omega }{\int }A{B}_{,ii}dg\Omega =\underset{\Gamma }{\int }A{B}_{,i}{n}_{,i}d\Gamma -\underset{\Omega }{\int }{A}_{,i}{B}_{,i}d\Omega .$

In one-dimensional applications this process is called integration by parts:

$\underset{a}{\overset{b}{\int }}pdq={pq|}_a^b-\underset{a}{\overset{b}{\int }}qdp.$

Error estimator proofs utilize inequalities like the Schwarz inequality

(2.2) $\left|a\cdot b\right|\le \left|a\right|\left|b\right|$

and the triangle inequality

(2.3) $\left|a+b\right|\le \left|a\right|+\left|b\right|.$

Finite element error estimates often use the Minkowski inequality

(2.4) ${\left[\sum _{i=1}^n{\left|x_i\pm y_i\right|}^p\right]}^{1/p}\le {\left[\sum _{i=1}^n{\left|x_i\right|}^p\right]}^{1/p}+{\left[\sum _{i=1}^n{\left|y_i\right|}^p\right]}^{1/p},1<p<\infty ,$

and the corresponding integral inequality

(2.5) ${\left[\underset{\Omega }{\int }{\left|x\pm y\right|}^{p}d\Omega \right]}^{1/p}\le {\left[\underset{\Omega }{\int }{\left|x\right|}^{p}d\Omega \right]}^{1/2}+{\left[\underset{\Omega }{\int }{\left|y\right|}^{p}d\Omega \right]}^{1/p},1<p<\infty ,$

We begin the preliminary concepts by introducing linear spaces. These are a collection of objects for which the operations of addition and scalar multiplication are defined in a simple and logical fashion.

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Linear Vector Space In Mathematical Physics Pdf

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