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Definitions

Polarization Magnetization

Definition P

This definition of polarization as a "dipole moment per unit volume" is widely adopted, though in some cases it can bring to ambiguities and paradoxes .

Relation among E, P, D

In this equation, P is the (negative of the) field induced in the material when the "fixed" charges, the dipoles, shift in response to the total underlying field E, whereas D is the field due to the remaining charges, known as "free" charges .

Polarization ambiguity

Another problem in the definition of $\scriptstyle \mathbf {P}$ is related to the arbitrary choice of the "unit volume", or more precisely to the system's scale . For example, at microscopic scale a plasma can be regarded as a gas of free charges, thus $\scriptstyle \mathbf {P}$ should be zero. On the contrary, at a macroscopic scale the same plasma can be described as a continuous media, exhibiting a permittivity $\scriptstyle \varepsilon (\omega )\neq 1$ and thus a net polarization $\scriptstyle \mathbf {P} \neq \mathbf {0}$.

.

Magnetization

Physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume. It is represented by a pseudovector M.

Definition

Where dp is the elementary electric dipole moment.

Those definitions of P and M as a "moments per unit volume" are widely adopted, though in some cases they can bring to ambiguities and paradoxes.

Physics applications

To calculate the dipole moment m (A m²) using the formula: $\scriptstyle \mathbf {m} =\mathbf {M} V$, we have that $\scriptstyle \mathbf {M} =\mathbf {B_{r}} /\mu _{0}$, thus $\scriptstyle \mathbf {m} =\mathbf {B_{r}} V/\mu _{0}$, where:

• $\scriptstyle \mathbf {B_{r}}$ is the Residual Flux Density, expressed in Teslas (T).
• $\scriptstyle V$ is the volume (m³) of the magnet.
• $\scriptstyle \mu _{0}=4\pi \cdot 10^{-7}$ N/A² is the permeability of vacuum.

Magnetization current

The magnetization M makes a contribution to the current density J, known as the magnetization current or bound (volumetric) current

$\mathbf {J_{m}} =\nabla \times \mathbf {M}$

and for the bound surface current:

$\mathbf {K_{m}} =\mathbf {M} \times \mathbf {\hat {n}}$

so that the total current density that enters Maxwell's equations is given by

$\mathbf {J} =\mathbf {J_{f}} +\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}$

where J_f is the electric current density of free charges (also called the free current), the second term is the contribution from the magnetization, and the last term is related to the electric polarization P.

Applications

The cross product has applications in different contexts, e.g. it is used in computational geometry, physics and engineering.

A non-exhaustive list of examples is reported.

Angular momentum and torque

The angular momentum $\scriptstyle \mathbf {L}$ of a particle about a given origin is defined as:

$\mathbf {L} =\mathbf {r} \times \mathbf {p} \,$

where $\scriptstyle \mathbf {r}$ is the position vector of the particle relative to the origin, $\scriptstyle \mathbf {p}$ is the linear momentum of the particle.

In the same way, the moment $\scriptstyle \mathbf {M}$ of a force $\scriptstyle \mathbf {F} _{\mathrm {B} }$ applied at point B around point A is given as:

$\mathbf {M} _{\mathrm {A} }=\mathbf {r} _{\mathrm {AB} }\times \mathbf {F} _{\mathrm {B} }\,$

In Mechanics the moment of a force is also called torque and written as $\scriptstyle \mathbf {\tau }$

Since position $\scriptstyle \mathbf {r}$, linear momentum $\scriptstyle \mathbf {p}$ and force $\scriptstyle \mathbf {F}$ are all true vectors, both the angular momentum $\scriptstyle \mathbf {L}$ and the moment of a force $\scriptstyle \mathbf {M}$ are pseudovectors or axial vectors.

Rigid body

The cross product frequently appears in the description of rigid motions.

For two points P e Q on rigid body holds the law:

$\mathbf {v} _{P}-\mathbf {v} _{Q}=\mathbf {\omega } \times \left(\mathbf {r} _{P}-\mathbf {r} _{Q}\right)\,$

where $\scriptstyle \mathbf {r}$ is the point's position, $\scriptstyle \mathbf {v}$ is its velocity and $\scriptstyle \mathbf {\omega }$ is body's angular velocity.

Since position $\scriptstyle \mathbf {r}$ and velocity $\scriptstyle \mathbf {v}$ are true vectors, the angular velocity $\scriptstyle \mathbf {\omega }$ is a pseudovector or axial vector.

Lorentz force

See also: Lorentz force

The electromagnetic force exerted on a particle is:

$\mathbf {F} =q_{e}\,\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)$

where:

• $\mathbf {F}$ is the global electromagnetic force, also called Lorentz force
• $q_{e}$ is the particle's electric charge
• $\mathbf {E}$ is the electric field
• $\mathbf {v}$ is the particle's velocity
• $\mathbf {B}$ is the magnetic field

Since velocity $\scriptstyle \mathbf {v}$, force $\scriptstyle \mathbf {F}$ and electric field $\scriptstyle \mathbf {E}$ are all true vectors, the magnetic field $\scriptstyle \mathbf {B}$ is a pseudovector.

Generalizations

Skew-symmetric matrix

If the cross product is defined as a binary operation, it takes in input just 2 vectors. If its output is not required to be a vector or a pseudovector but a matrix, then it can be generalized in an arbitrary number of dimensions .

In Mechanics, for example, the angular velocity can be interpreted either as a pseudovector $\scriptstyle \omega$ or as a anti-symmetric matrix or skew-symmetric tensor $\scriptstyle \Omega$. In the latter case, the velocity law for a rigid body looks:

$\mathbf {v} _{P}-\mathbf {v} _{Q}={\Omega }\cdot \left(\mathbf {r} _{P}-\mathbf {r} _{Q}\right)\,$

where $\scriptstyle \Omega$ is formally defined from the rotation matrix $\scriptstyle R^{N\times N}$ associated to body's frame: $\Omega \triangleq {\frac {dR}{dt}}R^{T}$. In three-dimensions holds:

$\Omega =[\omega ]_{\times }={\begin{bmatrix}\,\,0&\!-\omega _{3}&\,\,\,\omega _{2}\\\,\,\,\omega _{3}&0&\!-\omega _{1}\\\!-\omega _{2}&\,\,\omega _{1}&\,\,0\end{bmatrix}}$

In Quantum Mechanics the angular momentum $\scriptstyle L$ is often represented as an anti-symmetric matrix or tensor. More precisely, it is the result of cross product involving position $\scriptstyle \mathbf {x}$ and linear momentum $\scriptstyle \mathbf {p}$:

$L_{ij}=x_{i}p_{j}-p_{i}x_{j}$

Since both $\scriptstyle \mathbf {x}$ and $\scriptstyle \mathbf {p}$ can have an arbitrary number $\scriptstyle N$ of components, that kind of cross product can be extended to any dimension, holding the "physical" interpretation of the operation.

See the "Alternative ways to compute the cross product" section for numerical details.

Cloaking device

- Sistemare la citazione [1] del lavoro di Alù e Monticone (mancano i riferimenti alla pubblicazione)

- Nella sezione "Metamaterial cloaking", dopo la frase:

"Using transformation optics it is possible to design the optical parameters of a "cloak" so that it guides light around some region, rendering it invisible over a certain band of wavelengths."

aggiungere il riferimento all'articolo di Pendry e Smith, su "Controlling EM fields"

Poco dopo: "There are several theories of cloaking, giving rise to different types of invisibility." aggiungere riferimenti a tesi, Alù/Engheta, Tachi.

-References numerare la citazione di Ulf e Smith e sistemare dopo quella di Pendry su "Controlling EM fields"

Invisibility

Sezione "Pratical efforts" "Engineers and scientists have performed various kinds of research to investigate the possibility of finding ways to create real optical invisibility (cloaks) for objects. Methods are typically based on implementing the theoretical techniques of transformation optics, which have given rise to several theories of cloaking." Aggiungere un riferimento alla tesi, a Pendry,Smith, Galdi e Alù.

Sezione "External links": aggiungere link alla Tachi (vedi pagina del "Cloaking Device")

Cloak of invisibility

Sezione "References"

Sezione "Further readings" Aggiungere un riferimento alla tesi. aggiungere link alla Tachi

• Inami, M.; Kawakami, N.; Susumu, T. (2003). "Optical camouflage using retro-reflective projection technology" (PDF). Proceedings of the 2nd IEEE/ACM International Symposium on Mixed and Augmented Reality. IEEE Computer Society: 348–349. doi:10.1109/ISMAR.2003.1240754. ISBN 0-7695-2006-5. S2CID 44776407.

• Gonano, C.A. (2016). A perspective on metasurfaces, circuits, holograms and invisibility (PDF). Politecnico di Milano, Italy.

Sezione "External links": aggiungere link al video di Alù su "The quest for invisibility"

• On The Quest To Invisibility - Metamaterials and Cloaking (video), Prof. Andrea Alù at TEDxAustin, 2013.

aggiungere link alla Tachi (vedi pagina del "Cloaking Device")

Esempio per le note

Esempio 1:

Esempio 2: Paolo Maldini, Fabio Cannavaro, Dino Zoff.

See also

• Bivector
• Cartesian product – A product of two sets
• Dot Product
• Exterior algebra
• Multiple cross products – Products involving more than three vectors
• Pseudovector
• × (the symbol)

Bibliografia

{{cite book| "nome" | "cognome" | "titolo" | "anno" | "casa_ed." | "città"}}

oppure, con uno schema simile a quello di Bibtex:

{{cite book |first="nome" |last="cognome" |title="titolo" |year="anno" |publisher="casa_ed." |location="città" }}

• . 1982. {{cite book}}: Missing or empty |title= (help); Text "Ballanti" ignored (help); Text "Federico" ignored (help); Text "Lato Side Editori" ignored (help); Text "Led Zeppelin" ignored (help); Text "Roma" ignored (help)
• . 1988. {{cite book}}: Missing or empty |title= (help); Text "Arcana Editrice" ignored (help); Text "Davis" ignored (help); Text "Led Zeppelin: Il Martello degli Dei" ignored (help); Text "Milano" ignored (help); Text "Stephen" ignored (help)
• Levi-Civita, T.; Amaldi, U. (1949). Lezioni di meccanica razionale (in italiano). Vol. 1. Bologna: Zanichelli editore.{{cite book}}: CS1 maint: unrecognized language (link)

• Morando, A.P.; Leva, S. (1998). Note di teoria dei Campi Vettoriali (in italiano). Bologna: Esculapio.{{cite book}}: CS1 maint: unrecognized language (link)

• Gonano, Carlo Andrea (12 2011). Estensione in N-D di prodotto vettore e rotore e loro applicazioni (PDF) (in italiano). Politecnico di Milano. {{cite book}}: Check date values in: |date= (help)CS1 maint: unrecognized language (link)

• McDavid, A.W.; McMullen, C.D. (10 2006). Generalizing Cross Products and Maxwell's Equations to Universal Extra Dimensions (PDF). {{cite book}}: Check date values in: |date= (help)

References

• Cajori, Florian (1929). A History Of Mathematical Notations Volume II. Open Court Publishing. p. 134. ISBN 978-0-486-67766-8.
• E. A. Milne (1948) Vectorial Mechanics, Chapter 2: Vector Product, pp 11 –31, London: Methuen Publishing.
• Wilson, Edwin Bidwell (1901). Vector Analysis: A text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs. Yale University Press.
• S. Severini; A. Settimi (2013). "On the Divergenceless Property of the Magnetic Induction Field". Physics Research International. 2013. Hindawi: 1–5. doi:10.1155/2013/292834.

• A.W. McDavid; C.D. McMullen (2006). "Generalizing Cross Products and Maxwell's Equations to Universal Extra Dimensions" (PDF). {{cite journal}}: Cite journal requires |journal= (help)

• A.T. de Hoop (2012). "Lorentz-covariant electromagnetic fields in (N + 1)-spacetime — An axiomatic approach to special relativity". Wave Motion. 49 (8). Elsevier: 737–744. doi:10.1016/j.wavemoti.2012.05.002.

• C.A. Gonano; R.E. Zich (2013). "Magnetic monopoles and Maxwell's equations in N dimensions". Electromagnetics in Advanced Applications (ICEAA), 2013 International Conference on. IEEE: 1544–1547. doi:10.1109/ICEAA.2013.6632510. ISBN 978-1-4673-5707-4. S2CID 14068218.

Voci correlate

• Ghiaccio
• Acqua di mare
• Pioggia
• Acquedotto

Collegamenti esterni

Esempio 1: * http://www.google.com/

• http://www.google.com/

Esempio 2: * {{en}} [http://www.google.co.uk/ Il mio testo]

• [undefined] Error: {{Lang-xx}}: no text (help) Il mio testo
• http://www.minerva.unito.it/Home/IndiceSito.htm
• (in Italian) Umberto Scotti, Il prodotto vettoriale, dispensa rapida (4 pagine), Università di Napoli Federico II.
• (in Italian) Fernando Scarlassara, Calcolo vettoriale, dispensa base (15 slide illustrate), Università degli Studi di Padova.
• (in Italian) Carlo Andrea Gonano, Estensione in N-D di prodotto vettore e rotore e loro applicazioni, Politecnico di Milano, dicembre 2011.
• [undefined] Error: {{Lang-xx}}: no text (help) A.W. McDavid e C.D. McMullen, Generalizing Cross Products and Maxwell's Equations to Universal Extra Dimensions, ottobre 2006.
• (in English) A.W. McDavid e C.D. McMullen, Generalizing Cross Products and Maxwell's Equations to Universal Extra Dimensions, October 2006.

• {en} A.W. McDavid e C.D. McMullen, Generalizing Cross Products and Maxwell's Equations to Universal Extra Dimensions, October 2006.

External links

• "Cross product", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Cross Product". MathWorld.
• A quick geometrical derivation and interpretation of cross products
• C.A. Gonano and R.E. Zich (2014). Cross product in N Dimensions - the doublewedge product, Polytechnic of Milan, Italy.
• C.A. Gonano and R.E. Zich (2014). Cross product in N Dimensions - the doublewedge product, Polytechnic University of Milan, Italy.
• Z.K. Silagadze (2002). Multi-dimensional vector product. Journal of Physics. A35, 4949 (it is only possible in 7-D space)
• Real and Complex Products of Complex Numbers
• An interactive tutorial created at Syracuse University - (requires java)
• W. Kahan (2007). Cross-Products and Rotations in Euclidean 2- and 3-Space. University of California, Berkeley (PDF).

Category:Bilinear maps Category:Binary operations Category:Vector calculus Category:Analytic geometry