每日彙整: 2018-04-05

樹莓派、樹莓派之學習、樹莓派之教育

STEM 隨筆︰古典力學︰運動學【二．六．一】

如果說︰

需求為創造之母。

那麼『並矢張量』

Dyadics

In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.

There are numerous ways to multiply two Euclidean vectors. The dot product takes in two vectors and returns a scalar, while the cross product returns a pseudovector. Both of these have various significant geometric interpretations and are widely used in mathematics, physics, and engineering. The dyadic product takes in two vectors and returns a second order tensor called a dyadic in this context. A dyadic can be used to contain physical or geometric information, although in general there is no direct way of geometrically interpreting it.

The dyadic product is distributive over vector addition, and associative with scalar multiplication. Therefore, the dyadic product is linear in both of its operands. In general, two dyadics can be added to get another dyadic, and multiplied by numbers to scale the dyadic. However, the product is not commutative; changing the order of the vectors results in a different dyadic.

The formalism of dyadic algebra is an extension of vector algebra to include the dyadic product of vectors. The dyadic product is also associative with the dot and cross products with other vectors, which allows the dot, cross, and dyadic products to be combined together to obtain other scalars, vectors, or dyadics.

It also has some aspects of matrix algebra, as the numerical components of vectors can be arranged into row and column vectors, and those of second order tensors in square matrices. Also, the dot, cross, and dyadic products can all be expressed in matrix form. Dyadic expressions may closely resemble the matrix equivalents.

The dot product of a dyadic with a vector gives another vector, and taking the dot product of this result gives a scalar derived from the dyadic. The effect that a given dyadic has on other vectors can provide indirect physical or geometric interpretations.

Dyadic notation was first established by Josiah Willard Gibbs in 1884. The notation and terminology are relatively obsolete today. Its uses in physics include continuum mechanics and electromagnetism.

In this article, upper-case bold variables denote dyadics (including dyads) whereas lower-case bold variables denote vectors. An alternative notation uses respectively double and single over- or underbars.

 

『記號法』之興，豈無由乎？

人們講︰

好記號法使概念表達簡練清晰！

比方從『分數表示』 \frac{y}{x} ，通『割線斜率』 \frac{\Delta y}{\Delta x} ，達『切線微分』\frac{dy}{dx} 。

為什麼不覺得它的『定義』及『術語』︰

Definitions and terminology

Dyadic, outer, and tensor products

A dyad is a tensor of order two and rank one, and is the result of the dyadic product of two vectors (complex vectors in general), whereas a dyadic is a general tensor of order two (which may be full rank or not).

There are several equivalent terms and notations for this product:

  • the dyadic product of two vectors \displaystyle \mathbf {a} and  \displaystyle \mathbf {b} is denoted by \displaystyle \mathbf {a} \mathbf {b} (no symbol; no multiplication signs, crosses, dots etc.)
  • the outer product of two column vectors \displaystyle \mathbf {a} and \displaystyle \mathbf {b} is denoted and defined as \displaystyle \mathbf {a} \otimes \mathbf {b} or \displaystyle \mathbf {a} \mathbf {b} ^{T} , where \displaystyle T means transpose,
  • the tensor product of two vectors \displaystyle \mathbf {a} and \displaystyle \mathbf {b} is denoted \displaystyle \mathbf {a} \otimes \mathbf {b},

In the dyadic context they all have the same definition and meaning, and are used synonymously, although the tensor product is an instance of the more general and abstract use of the term.

Dirac’s bra–ket notation makes the use of dyads and dyadics intuitively clear, see Cahill (2013).

Three-dimensional Euclidean space

To illustrate the equivalent usage, consider three-dimensional Euclidean space, letting:

\displaystyle \mathbf {a} =a_{1}\mathbf {i} +a_{2}\mathbf {j} +a_{3}\mathbf {k}

\displaystyle \mathbf {b} =b_{1}\mathbf {i} +b_{2}\mathbf {j} +b_{3}\mathbf {k}

be two vectors where i, j, k (also denoted e1, e2, e3) are the standard basis vectors in this vector space (see also Cartesian coordinates). Then the dyadic product of a and b can be represented as a sum:
\displaystyle {\begin{array}{llll}\mathbf {ab} =&a_{1}b_{1}\mathbf {ii} &+a_{1}b_{2}\mathbf {ij} &+a_{1}b_{3}\mathbf {ik} \\&+a_{2}b_{1}\mathbf {ji} &+a_{2}b_{2}\mathbf {jj} &+a_{2}b_{3}\mathbf {jk} \\&+a_{3}b_{1}\mathbf {ki} &+a_{3}b_{2}\mathbf {kj} &+a_{3}b_{3}\mathbf {kk} \end{array}}
or by extension from row and column vectors, a 3×3 matrix (also the result of the outer product or tensor product of a and b):
\displaystyle \mathbf {ab} \equiv \mathbf {a} \otimes \mathbf {b} \equiv \mathbf {ab} ^{\mathrm {T} }={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}{\begin{pmatrix}b_{1}&b_{2}&b_{3}\end{pmatrix}}={\begin{pmatrix}a_{1}b_{1}&a_{1}b_{2}&a_{1}b_{3}\\a_{2}b_{1}&a_{2}b_{2}&a_{2}b_{3}\\a_{3}b_{1}&a_{3}b_{2}&a_{3}b_{3}\end{pmatrix}}.
A dyad is a component of the dyadic (a monomial of the sum or equivalently an entry of the matrix) — the dyadic product of a pair of basis vectors scalar multiplied by a number.

Just as the standard basis (and unit) vectors i, j, k, have the representations:

\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\\0\end{pmatrix}},\mathbf {j} ={\begin{pmatrix}0\\1\\0\end{pmatrix}},\mathbf {k} ={\begin{pmatrix}0\\0\\1\end{pmatrix}}

(which can be transposed), the standard basis (and unit) dyads have the representation:
\displaystyle \mathbf {ii} ={\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}},\mathbf {ij} ={\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}},\mathbf {ik} ={\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\end{pmatrix}}
\displaystyle \mathbf {ji} ={\begin{pmatrix}0&0&0\\1&0&0\\0&0&0\end{pmatrix}},\mathbf {jj} ={\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix}},\mathbf {jk} ={\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix}}
\displaystyle \mathbf {ki} ={\begin{pmatrix}0&0&0\\0&0&0\\1&0&0\end{pmatrix}},\mathbf {kj} ={\begin{pmatrix}0&0&0\\0&0&0\\0&1&0\end{pmatrix}},\mathbf {kk} ={\begin{pmatrix}0&0&0\\0&0&0\\0&0&1\end{pmatrix}}
For a simple numerical example in the standard basis:
\displaystyle {\begin{aligned}\mathbf {A} &=2\mathbf {ij} +{\frac {\sqrt {3}}{2}}\mathbf {ji} -8\pi \mathbf {jk} +{\frac {2{\sqrt {2}}}{3}}\mathbf {kk} \\&=2{\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}}+{\frac {\sqrt {3}}{2}}{\begin{pmatrix}0&0&0\\1&0&0\\0&0&0\end{pmatrix}}-8\pi {\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix}}+{\frac {2{\sqrt {2}}}{3}}{\begin{pmatrix}0&0&0\\0&0&0\\0&0&1\end{pmatrix}}\\&={\begin{pmatrix}0&2&0\\{\sqrt {3}}/2&0&-8\pi \\0&0&{\frac {2{\sqrt {2}}}{3}}\end{pmatrix}}\end{aligned}}

N-dimensional Euclidean space

If the Euclidean space is N–dimensional, and
\displaystyle \mathbf {a} =\sum _{i=1}^{N}a_{i}\mathbf {e} _{i}=a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+\cdots a_{N}\mathbf {e} _{N}
\displaystyle \mathbf {b} =\sum _{j=1}^{N}b_{j}\mathbf {e} _{j}=b_{1}\mathbf {e} _{1}+b_{2}\mathbf {e} _{2}+\cdots b_{N}\mathbf {e} _{N}
where ei and ej are the standard basis vectors in N-dimensions (the index i on ei selects a specific vector, not a component of the vector as in ai), then in algebraic form their dyadic product is:
\displaystyle \mathbf {ab} =\sum _{j=1}^{N}\sum _{i=1}^{N}a_{i}b_{j}{\mathbf {e} }_{i}\mathbf {e} _{j}.
This is known as the nonion form of the dyadic. Their outer/tensor product in matrix form is:
\displaystyle \mathbf {ab} =\mathbf {ab} ^{\mathrm {T} }={\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{N}\end{pmatrix}}{\begin{pmatrix}b_{1}&b_{2}&\cdots &b_{N}\end{pmatrix}}={\begin{pmatrix}a_{1}b_{1}&a_{1}b_{2}&\cdots &a_{1}b_{N}\\a_{2}b_{1}&a_{2}b_{2}&\cdots &a_{2}b_{N}\\\vdots &\vdots &\ddots &\vdots \\a_{N}b_{1}&a_{N}b_{2}&\cdots &a_{N}b_{N}\end{pmatrix}}.
A dyadic polynomial A, otherwise known as a dyadic, is formed from multiple vectors ai and bj:
\displaystyle \mathbf {A} =\sum _{i}\mathbf {a} _{i}\mathbf {b} _{i}=\mathbf {a} _{1}\mathbf {b} _{1}+\mathbf {a} _{2}\mathbf {b} _{2}+\mathbf {a} _{3}\mathbf {b} _{3}+\cdots
A dyadic which cannot be reduced to a sum of less than N dyads is said to be complete. In this case, the forming vectors are non-coplanar,[dubious – discuss] see Chen (1983).

Classification

The following table classifies dyadics:

  Determinant Adjugate Matrix and its rank
Zero = 0 = 0 = 0; rank 0: all zeroes
Linear = 0 = 0 ≠ 0; rank 1: at least one non-zero element and all 2 × 2 subdeterminants zero (single dyadic)
Planar = 0 ≠ 0 (single dyadic) ≠ 0; rank 2: at least one non-zero 2 × 2 subdeterminant
Complete ≠ 0 ≠ 0 ≠ 0; rank 3: non-zero determinant

Identities

The following identities are a direct consequence of the definition of the tensor product:[1]

  1. Compatible with scalar multiplication:
    \displaystyle (\alpha \mathbf {a} )\mathbf {b} =\mathbf {a} (\alpha \mathbf {b} )=\alpha (\mathbf {a} \mathbf {b} ) for any scalar \displaystyle \alpha.
  2. Distributive over vector addition:
    \displaystyle \mathbf {a} (\mathbf {b} +\mathbf {c} )=\mathbf {a} \mathbf {b} +\mathbf {a} \mathbf {c}
    \displaystyle (\mathbf {a} +\mathbf {b} )\mathbf {c} =\mathbf {a} \mathbf {c} +\mathbf {b} \mathbf {c}

 

有何動人之處？？

蓋『符號』構成『系統』，並非孤立存在。既經不同『學習過程』洗禮，往往習以為常，或難客觀論辨矣！！

且借

Dyadic algebra

Product of dyadic and vector

There are four operations defined on a vector and dyadic, constructed from the products defined on vectors.

  Left Right
Dot product \displaystyle \mathbf {c} \cdot \left(\mathbf {a} \mathbf {b} \right)=\left(\mathbf {c} \cdot \mathbf {a} \right)\mathbf {b} \displaystyle \left(\mathbf {a} \mathbf {b} \right)\cdot \mathbf {c} =\mathbf {a} \left(\mathbf {b} \cdot \mathbf {c} \right)
Cross product \displaystyle \mathbf {c} \times \left(\mathbf {ab} \right)=\left(\mathbf {c} \times \mathbf {a} \right)\mathbf {b} \displaystyle \left(\mathbf {ab} \right)\times \mathbf {c} =\mathbf {a} \left(\mathbf {b} \times \mathbf {c} \right)

………

 

舉例而言

\displaystyle \mathbf {a} =a_{1}\mathbf {i} +a_{2}\mathbf {j} +a_{3}\mathbf {k}

\displaystyle = (\mathbf {a} \cdot \mathbf {i} )\mathbf {i} +(\mathbf {a} \cdot \mathbf {j} )\mathbf {j} +(\mathbf {a} \cdot \mathbf {k} )\mathbf {k}

將之改寫為︰

\displaystyle = \mathbf {a} \cdot (\mathbf {i} \mathbf {i} + \mathbf {j} \mathbf {j} + \mathbf {k} \mathbf {k}).

到底有什麼益處耶？？！！

反思

\displaystyle \mathbf {a} \ ?\neq \mathbf {a} \cdot (\mathbf {i^{'}} \mathbf {i^{'}} + \mathbf {j^{'}} \mathbf {j^{'}} + \mathbf {k^{'}} \mathbf {k^{'}})

，果不能還其『向量本色』嘛！！？？

或可推知

狄拉克符號

狄拉克符號或狄拉克標記（英語：Dirac notation）是量子力學中廣泛應用於描述量子態的一套標準符號系統。在這套系統中，每一個量子態都被描述為希爾伯特空間中的態向量，定義為括量（ket）：\displaystyle |\psi \rangle；每一個括量的共軛轉置定義為其包量（bra）：\displaystyle \langle \psi |。

此標記法為狄拉克於1939年將「bracket」（括號）這個詞拆開後所造的。[1]在中國方面，一些舊有的教科書和文獻中也將其譯為「刁矢」和「刃矢」、或「彳矢」和「亍矢」，現已棄用。

……

Bra–ket notation

In quantum mechanics, bra–ket notation is a standard notation for describing quantum states. It can also be used to denote abstract vectors and linear functionals in mathematics. The notation begins with using angle brackets, ⟨ and ⟩, and a vertical bar, |, to denote the scalar product of vectors or the action of a linear functional on a vector in a complex vector space. The scalar product or action is written as

\displaystyle \langle \phi \mid \psi \rangle .

The right part is called the ket /kɛt/; it is a vector, typically represented as a column vector and written
\displaystyle |\psi \rangle .
The left part is called the bra, /brɑː/; it is the Hermitian conjugate of the ket with the same label, typically represented as a row vector and is written
\displaystyle \langle \phi |.
A combination of bras, kets, and operators is interpreted using matrix multiplication. A bra and a ket with the same label are Hermitian conjugates of each other.

Bra-ket notation was introduced in 1939 by Paul Dirac[1][2] and is also known as the Dirac notation.

The bra-ket notation has a precursor in Hermann Grassmann‘s use of the notation \displaystyle [\phi \mid \psi ]  for his inner products nearly 100 years earlier.[3]

………

 

『直覺性』之由來也◎