Английская Википедия:Einstein notation
In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916.[1]
Introduction
Statement of convention
According to this convention, when an index variable appears twice in a single term and is not otherwise defined (see Free and bound variables), it implies summation of that term over all the values of the index. So where the indices can range over the set Шаблон:Math, <math display="block">y = \sum_{i = 1}^3 c_i x^i = c_1 x^1 + c_2 x^2 + c_3 x^3</math> is simplified by the convention to: <math display="block">y = c_i x^i</math>
The upper indices are not exponents but are indices of coordinates, coefficients or basis vectors. That is, in this context Шаблон:Math should be understood as the second component of Шаблон:Math rather than the square of Шаблон:Math (this can occasionally lead to ambiguity). The upper index position in Шаблон:Math is because, typically, an index occurs once in an upper (superscript) and once in a lower (subscript) position in a term (see Шаблон:Section link below). Typically, Шаблон:Math would be equivalent to the traditional Шаблон:Math.
In general relativity, a common convention is that
- the Greek alphabet is used for space and time components, where indices take on values 0, 1, 2, or 3 (frequently used letters are Шаблон:Math),
- the Latin alphabet is used for spatial components only, where indices take on values 1, 2, or 3 (frequently used letters are Шаблон:Math),
In general, indices can range over any indexing set, including an infinite set. This should not be confused with a typographically similar convention used to distinguish between tensor index notation and the closely related but distinct basis-independent abstract index notation.
An index that is summed over is a summation index, in this case "Шаблон:Math". It is also called a dummy index since any symbol can replace "Шаблон:Math" without changing the meaning of the expression (provided that it does not collide with other index symbols in the same term).
An index that is not summed over is a free index and should appear only once per term. If such an index does appear, it usually also appears in every other term in an equation. An example of a free index is the "Шаблон:Math" in the equation <math>v_i = a_i b_j x^j</math>, which is equivalent to the equation <math display="inline">v_i = \sum_j(a_{i} b_{j} x^{j})</math>.
Application
Einstein notation can be applied in slightly different ways. Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within a term.[2] When dealing with covariant and contravariant vectors, where the position of an index also indicates the type of vector, the first case usually applies; a covariant vector can only be contracted with a contravariant vector, corresponding to summation of the products of coefficients. On the other hand, when there is a fixed coordinate basis (or when not considering coordinate vectors), one may choose to use only subscripts; see Шаблон:Section link below.
Vector representations
Superscripts and subscripts versus only subscripts
In terms of covariance and contravariance of vectors,
- upper indices represent components of contravariant vectors (vectors),
- lower indices represent components of covariant vectors (covectors).
They transform contravariantly or covariantly, respectively, with respect to change of basis.
In recognition of this fact, the following notation uses the same symbol both for a vector or covector and its components, as in: <math display="block">\begin{align} v = v^i e_i = \begin{bmatrix} e_1 & e_2 & \cdots & e_n \end{bmatrix} \begin{bmatrix} v^1 \\ v^2 \\ \vdots \\ v^n \end{bmatrix} \\ w = w_i e^i = \begin{bmatrix} w_1 & w_2 & \cdots & w_n \end{bmatrix} \begin{bmatrix} e^1 \\ e^2 \\ \vdots \\ e^n \end{bmatrix} \end{align}</math>
where Шаблон:Math is the vector and Шаблон:Math are its components (not the Шаблон:Mathth covector Шаблон:Math), Шаблон:Math is the covector and Шаблон:Math are its components. The basis vector elements <math>e_i</math> are each column vectors, and the covector basis elements <math>e^i</math> are each row covectors. (See also Шаблон:Slink; duality, below and the examples)
In the presence of a non-degenerate form (an isomorphism Шаблон:Math, for instance a Riemannian metric or Minkowski metric), one can raise and lower indices.
A basis gives such a form (via the dual basis), hence when working on Шаблон:Math with a Euclidean metric and a fixed orthonormal basis, one has the option to work with only subscripts.
However, if one changes coordinates, the way that coefficients change depends on the variance of the object, and one cannot ignore the distinction; see Covariance and contravariance of vectors.
Mnemonics
In the above example, vectors are represented as Шаблон:Math matrices (column vectors), while covectors are represented as Шаблон:Math matrices (row covectors).
When using the column vector convention:
- "Upper indices go up to down; lower indices go left to right."
- "Covariant tensors are row vectors that have indices that are below (co-row-below)."
- Covectors are row vectors: <math display="block">\begin{bmatrix} w_1 & \cdots & w_k \end{bmatrix}.</math> Hence the lower index indicates which column you are in.
- Contravariant vectors are column vectors: <math display="block">\begin{bmatrix} v^1 \\ \vdots \\ v^k \end{bmatrix}</math> Hence the upper index indicates which row you are in.
Abstract description
The virtue of Einstein notation is that it represents the invariant quantities with a simple notation.
In physics, a scalar is invariant under transformations of basis. In particular, a Lorentz scalar is invariant under a Lorentz transformation. The individual terms in the sum are not. When the basis is changed, the components of a vector change by a linear transformation described by a matrix. This led Einstein to propose the convention that repeated indices imply the summation is to be done.
As for covectors, they change by the inverse matrix. This is designed to guarantee that the linear function associated with the covector, the sum above, is the same no matter what the basis is.
The value of the Einstein convention is that it applies to other vector spaces built from Шаблон:Math using the tensor product and duality. For example, Шаблон:Math, the tensor product of Шаблон:Math with itself, has a basis consisting of tensors of the form Шаблон:Math. Any tensor Шаблон:Math in Шаблон:Math can be written as: <math display="block">\mathbf{T} = T^{ij}\mathbf{e}_{ij}.</math>
Шаблон:Math, the dual of Шаблон:Math, has a basis Шаблон:Math, Шаблон:Math, ..., Шаблон:Math which obeys the rule <math display="block">\mathbf{e}^i (\mathbf{e}_j) = \delta^i_j.</math> where Шаблон:Math is the Kronecker delta. As <math display="block">\operatorname{Hom}(V, W) = V^* \otimes W</math> the row/column coordinates on a matrix correspond to the upper/lower indices on the tensor product.
Common operations in this notation
In Einstein notation, the usual element reference <math>A_{mn}</math> for the <math>m</math>-th row and <math>n</math>-th column of matrix <math>A</math> becomes <math>{A^m}_{n}</math>. We can then write the following operations in Einstein notation as follows.
Inner product
Using an orthogonal basis, the inner product (vector dot product) is the sum of corresponding components multiplied together: <math display="block">\mathbf{u} \cdot \mathbf{v} = u_j v^j</math>
This can also be calculated by multiplying the covector on the vector.
Vector cross product
Again using an orthogonal basis (in 3 dimensions), the cross product intrinsically involves summations over permutations of components: <math display="block">\mathbf{u} \times \mathbf{v} = {\varepsilon^i}_{jk} u^j v^k \mathbf{e}_i</math> where <math display="block">{\varepsilon^i}_{jk} = \delta^{il} \varepsilon_{ljk}</math>
Шаблон:Math is the Levi-Civita symbol, and Шаблон:Math is the generalized Kronecker delta. Based on this definition of Шаблон:Math, there is no difference between Шаблон:Math and Шаблон:Math but the position of indices.
Matrix-vector multiplication
The product of a matrix Шаблон:Math with a column vector Шаблон:Math is: <math display="block">\mathbf{u}_{i} = (\mathbf{A} \mathbf{v})_{i} = \sum_{j=1}^N A_{ij} v_{j}</math> equivalent to <math display="block">u^i = {A^i}_j v^j </math>
This is a special case of matrix multiplication.
Matrix multiplication
The matrix product of two matrices Шаблон:Math and Шаблон:Math is: <math display="block">\mathbf{C}_{ik} = (\mathbf{A} \mathbf{B})_{ik} =\sum_{j=1}^N A_{ij} B_{jk}</math>
equivalent to <math display="block">{C^i}_k = {A^i}_j {B^j}_k</math>
Trace
For a square matrix Шаблон:Math, the trace is the sum of the diagonal elements, hence the sum over a common index Шаблон:Math.
Outer product
The outer product of the column vector Шаблон:Math by the row vector Шаблон:Math yields an Шаблон:Math matrix Шаблон:Math: <math display="block">{A^i}_j = u^i v_j = {(u v)^i}_j</math>
Since Шаблон:Math and Шаблон:Math represent two different indices, there is no summation and the indices are not eliminated by the multiplication.
Raising and lowering indices
Given a tensor, one can raise an index or lower an index by contracting the tensor with the metric tensor, Шаблон:Math. For example, taking the tensor Шаблон:Math, one can lower an index: <math display="block">g_{\mu\sigma} {T^\sigma}_\beta = T_{\mu\beta}</math>
Or one can raise an index: <math display="block">g^{\mu\sigma} {T_\sigma}^\alpha = T^{\mu\alpha}</math>
See also
- Tensor
- Abstract index notation
- Bra–ket notation
- Penrose graphical notation
- Levi-Civita symbol
- DeWitt notation
Notes
- Шаблон:Note labelThis applies only for numerical indices. The situation is the opposite for abstract indices. Then, vectors themselves carry upper abstract indices and covectors carry lower abstract indices, as per the example in the introduction of this article. Elements of a basis of vectors may carry a lower numerical index and an upper abstract index.
References
Bibliography
External links
