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Coordinate system

The coordinates of a point r in an n-dimensional real numerical space ℝⁿ or a complex n-space ℂⁿ are simply an ordered set of n real or complex numbers:^[1]^[2]^[3]

\mathbf {r} =[x^{1},\ x^{2},\ \dots \ ,x^{n}]\ .

Coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system.^[4]

Manifolds

A coordinate system in mathematics is a facet of geometry or of algebra, in particular, a property of manifolds (for example, in physics, configuration spaces or phase spaces).^[5]^[6] The coordinates of a point r in an n-dimensional space are simply an ordered set of n numbers:^[1]

\mathbf {r} =[x^{1},\ x^{2},\ \dots \ ,x^{n}]\ .

In a general Banach space, these numbers could be (for example) coefficients in a functional expansion like a Fourier series. In a physical problem, they could be spacetime coordinates or normal mode amplitudes. In a robot design, they could be angles of relative rotations, linear displacements, or deformations of joints.^[7] Here we will suppose these coordinates can be related to a Cartesian coordinate system by a set of functions:

x^{j}=x^{j}(x,\ y,\ z,\ \dots )\ ,

j=1,\ \dots \ ,\ n\

where x, y, z, etc. are the n Cartesian coordinates of the point. Given these functions, coordinate surfaces are defined by the relations:

x^{j}(x,y,z,\dots )=\mathrm {constant} \ ,

j=1,\ \dots \ ,\ n\ .

The intersection of these surfaces define coordinate lines. At any selected point, tangents to the intersecting coordinate lines at that point define a set of basis vectors {e₁, e₂, …, e_n} at that point. That is:

\mathbf {e} _{i}(\mathbf {r} )=\lim _{\epsilon \rightarrow 0}{\frac {\mathbf {r} \left(x^{1},\ \dots ,\ x^{i}+\epsilon ,\ \dots ,\ x^{n}\right)-\mathbf {r} \left(x^{1},\ \dots ,\ x^{i},\ \dots ,\ x^{n}\right)}{\epsilon }}\ ,

which can be normalized to be of unit length. For more detail see curvilinear coordinates.

Coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system.^[4] If the basis vectors are orthogonal at every point, the coordinate system is an orthogonal coordinate system.

An important aspect of a coordinate system is its metric g_ik, which determines the arc length ds in the coordinate system in terms of its coordinates:^[8]

(ds)^{2}=g_{ik}\ dx^{i}\ dx^{k}\ ,

where repeated indices are summed over.

As is apparent from these remarks, a coordinate system is a mathematical construct, part of an axiomatic system. There is no necessary connection between coordinate systems and physical motion (or any other aspect of reality). However, coordinate systems can be used to describe motion by interpreting one coordinate as time. Thus, Lorentz transformations and Galilean transformations may be viewed as coordinate transformations.

Notes

↑ ^1.0 ^1.1 Granino Arthur Korn, Theresa M. Korn (2000). Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review. Courier Dover Publications, p. 169. ISBN 0486411478.
↑ Morita
↑ Fritzche
↑ ^4.0 ^4.1 Wilford Zdunkowski & Andreas Bott (2003). Dynamics of the Atmosphere. Cambridge University Press. ISBN 052100666X.
↑ According to Hawking and Ellis: "A manifold is a space locally similar to Euclidean space in that it can be covered by coordinate patches. This structure allows differentiation to be defined, but does not distinguish between different coordinate systems. Thus, the only concepts defined by the manifold structure are those that are independent of the choice of a coordinate system." Stephen W. Hawking & George Francis Rayner Ellis (1973). The Large Scale Structure of Space-Time. Cambridge University Press, p. 11. ISBN 0521099064. A mathematical definition is: A connected Hausdorff space M is called an n-dimensional manifold if each point of M is contained in an open set that is homeomorphic to an open set in Euclidean n-dimensional space.
↑ Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu (2001). Geometry of Differential Forms. American Mathematical Society Bookstore, p. 12. ISBN 0821810456.
↑ Katsu Yamane (2004). Simulating and Generating Motions of Human Figures. Springer, 12–13. ISBN 3540203176.
↑ A. I. Borisenko, I. E. Tarapov, Richard A. Silverman (1979). “§2.8.4 Arc length. Metric coefficients”, Vector and Tensor Analysis with Applications, Reprint of Prentice-Hall 1968 ed. Courier Dover Publications, pp. 86 ff. ISBN 0486638332.

Choquet-Bruhat Bishop O'Neill Warner

[Korn-1] 1.0 ^1.1 Granino Arthur Korn, Theresa M. Korn (2000). Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review. Courier Dover Publications, p. 169. ISBN 0486411478.

[Morita-2] Morita

[Fritzche-3] Fritzche

[Zdunkowski-4] 4.0 ^4.1 Wilford Zdunkowski & Andreas Bott (2003). Dynamics of the Atmosphere. Cambridge University Press. ISBN 052100666X.

[Hawking-5] According to Hawking and Ellis: "A manifold is a space locally similar to Euclidean space in that it can be covered by coordinate patches. This structure allows differentiation to be defined, but does not distinguish between different coordinate systems. Thus, the only concepts defined by the manifold structure are those that are independent of the choice of a coordinate system." Stephen W. Hawking & George Francis Rayner Ellis (1973). The Large Scale Structure of Space-Time. Cambridge University Press, p. 11. ISBN 0521099064. A mathematical definition is: A connected Hausdorff space M is called an n-dimensional manifold if each point of M is contained in an open set that is homeomorphic to an open set in Euclidean n-dimensional space.

[Morita2-6] Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu (2001). Geometry of Differential Forms. American Mathematical Society Bookstore, p. 12. ISBN 0821810456.

[Yamane-7] Katsu Yamane (2004). Simulating and Generating Motions of Human Figures. Springer, 12–13. ISBN 3540203176.

[Borisenko-8] A. I. Borisenko, I. E. Tarapov, Richard A. Silverman (1979). “§2.8.4 Arc length. Metric coefficients”, Vector and Tensor Analysis with Applications, Reprint of Prentice-Hall 1968 ed. Courier Dover Publications, pp. 86 ff. ISBN 0486638332.

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-{{Image|MOS Capacitor.PNG|right|250px|Cross section of MOS capacitor showing charge layers}}
+{{AccountNotLive}}
 {{TOC|right}}
-==MOS Capacitor==
+==Coordinate system==
-The '''MOS capacitor''' or '''metal-oxide semiconductor''' capacitor is a two terminal device consisting of three layers: a metal ''gate'' electrode, a separating insulator (often an oxide layer), and a semiconducting layer called the ''body''. The device operates using the [[field effect]], that is, the modulation of the surface conductivity of the semiconductor body by means of an applied voltage between the gate and the body.
-==Operation==
+The coordinates of a point '''r''' in an ''n''-dimensional real numerical space ℝ<sup>n</sup> or a complex ''n''-space ℂ<sup>n</sup>  are simply an ordered set of ''n'' real or complex numbers:<ref name=Korn>
-{{Image|Seimiconductor band bending.PNG|right|350px|''Field effect'': Top panels: An applied voltage bends bands, depleting holes from surface (left). The charge inducing the bending is balanced by a layer of negative acceptor-ion charge (right). Bottom panel: A larger applied voltage further depletes holes but conduction band lowers enough in energy to populate an inversion layer.}}
-A device based upon a uniformly doped ''p''-type body is described. The ''n''-type case is similar. The device has four regions of operation depending upon the applied voltage between gate and body:
-#''Accumulation'': For a negative gate bias, holes are drawn to the semiconductor-insulator interface. A conducting surface extends from the bulk all the way to the interface, and the surface conductivity is enhanced by the accumulation of holes at the interface.
-#''Flat bands'': For a specific value of voltage, the bulk hole density is exactly the same from the bulk to the interface. The body is everywhere charge neutral because the hole density exactly balances the acceptor density. The ''flatband voltage'' corresponds to a zero potential drop in the body, but that does not necessarily correspond to zero applied voltage because of the [[contact potential]] between the metal gate and the body that leads to band bending even when zero voltage is applied.
-#''Depletion'': For a positive gate voltage, the holes are pushed away from the positive charge on the gate electrode, and a surface layer depleted of holes is formed extending from the interface to the depth necessary to make the exposed, immobile, negative acceptor ion charge exactly balance the positive charge on the gate. Increase in positive charge on the gate with increasing voltage is balanced by expansion of the depletion layer, increasing the acceptor charge.
-#''Inversion'': For positive gate voltages above a ''threshold voltage'', a surface inversion layer of electrons forms in a narrow layer near the interface. This conducting inversion layer is separated form the ''p''-type neutral bulk by the intervening insulating depletion layer of immobile acceptor charge. Once the gate voltage increases beyond the threshold voltage, additional positive charge on the gate is compensated by increased inversion layer electron charge, and the depletion layer depth no longer expands.
-These various operation regions all are subsumed under the notion of the [[field effect]], the modulation of conductivity by an applied electric field. The figure illustrates the charge balance for the cases of depletion (top panels) and inversion (bottom panels). On the left side of the figure, the band edges are plotted as a function of depth into the capacitor. The electron population of the bands depends upon how close they are to the [[Fermi function#Fermi level|Fermi level]] (the horizontal dashed line): because this separation varies with depth, so does the occupancy of the bands. For example, the valence bands are bent far below the Fermi level near the interface, so the energy levels in the valence band near the interface are filled with electrons, and there are no holes. At greater depth from the interface, however, the valence band edge becomes the bulk value, and is close enough to the Fermi level to allow some electron vacancies (holes, in other words) in the valence band. Consequently there are holes present in the bulk region, which are of a density equal to that of the negatively charged acceptor impurities in this region, resulting in a neutral bulk. The right panels in the figure illustrate the charge densities as a function of depth that are a consequence of the band bending.
+{{cite book |title=Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review |author=Granino Arthur Korn, Theresa M. Korn |pages=p. 169 |url=http://books.google.com/books?id=xHNd5zCXt-EC&pg=PA169&dq=curvilinear+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3psSqwpBtA3U40e46VPPaMNMEw4g#PPA169,M1
+|isbn=0486411478 |year=2000 |publisher=Courier Dover Publications}}
-===Contact potential===
+</ref><ref name=Morita>[http://books.google.com/books?id=5N33Of2RzjsC&pg=PA12&dq=geometry++axiom+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3Vi7xsLiYiWCK0erF6X2gczHOkJA#v=onepage&q&f=false Morita]
-The flat band condition is examined further here, to explain why a zero applied voltage does not necessarily imply flat bands. The metal gate contains electrons, as does the semiconductor body. For this discussion, the insulator is imagined as a charge-free inert layer. Suppose an electron is taken from the metal and transferred to the semiconductor. There are three possibilities: no work is done, some work is done to transfer the electron, or the electron provides energy during the transfer. In the first case, zero applied voltage corresponds to flat bands, but in the other two cases a short circuit between metal and semiconductor leads to a net transfer of electrons making the gate negatively charged in the first case and the semiconductor in the second case. That is, without any applied voltage, a charge transfer occurs, and this transfer results in a potential difference at zero applied voltage, the ''contact potential''. Thus, to achieve flat bands (zero potential drop across the semiconductor), a ''flatband voltage'' is necessary.
+</ref><ref name=Fritzche>
-The underlying cause of this charge transfer at a microscopic level is complicated in detail by the effects of interfaces including interface traps, and by charges in the insulating layer related to various defects. Within a few lattice spacings of an interface (≈1nm, say), the atomic arrangement differs from the bulk materials on either side of an interface, and the chemistry associated with such distortion can lead to charges (and related potentials) near this interface, even if the junction is free from defects. Two consequences are that the band bending may depend upon the crystal orientation of the surfaces forming the interfaces, and the band bending is not dictated entirely by the behavior of the metal and semiconductor idealized as bulk media. However, the driving force at an atomic level is the ability of some atoms to strip electrons from other atoms of a different species, a phenomenon called [[electronegativity]] of the atoms.<ref name=electronegativity>
+[http://books.google.com/books?id=jSeRz36zXIMC&pg=PA155&dq=complex+%22coordinate+system%22&hl=en&ei=LA2JTYD1MYfWtQP2j92NDA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCkQ6AEwAA#v=onepage&q=complex%20%22coordinate%20system%22&f=false Fritzche]</ref>
+:<math>\mathbf{r} =[x^1,\ x^2,\ \dots\ ,  x^n] \ .</math>
+Coordinate surfaces, coordinate lines, and [[Basis (linear algebra)|basis vectors]] are components of a '''coordinate system'''.<ref name=Zdunkowski>{{cite book |title=Dynamics of the Atmosphere |page=84  |isbn=052100666X |year=2003 |author=Wilford Zdunkowski & Andreas Bott |publisher=Cambridge University Press |url=http://books.google.com/books?id=GuYvC21v3g8C&pg=RA1-PA84&dq=%22curvilinear+coordinate+system%22&lr=&as_brr=0&sig=ACfU3U2g2k7kY5u-CVcJ1pH5ZxsbEb9Rig  }}</ref>
-See, for example, {{cite book |title=Silicon Materials Science and Technology X, Issue 2 |editor=Howard R. Huff, H. Iwai, H. Richter, eds |author=Hisham Z Massoud |url=http://books.google.com/books?id=KqIa5SCP93sC&pg=PA195 |pages=pp. 195 ''ff'' |chapter=Growth kinetics and electrical properties of ultrathin silicon-dioxide layers |isbn=156677439X |publisher=The Electrochemical Society |edition=Tenth symposium on silicon material science and technology |year=2006}}
+==Manifolds==
+A coordinate system in mathematics is a facet of [[geometry]] or of [[algebra]], in particular, a property of [[Manifold (geometry)|manifold]]s (for example, in physics, [[configuration space]]s or [[phase space]]s).<ref name=Hawking>
+According to Hawking and Ellis: "A manifold is a space locally similar to Euclidean space in that it can be covered by coordinate patches. This structure allows differentiation to be defined, but does not distinguish between different coordinate systems. Thus, the only concepts defined by the manifold structure are those that are independent of the choice of a coordinate system." {{cite book |title=The Large Scale Structure of Space-Time |author=Stephen W. Hawking & George Francis Rayner Ellis |isbn=0521099064 |year=1973 |publisher=Cambridge University Press |pages=p. 11 |url=http://books.google.com/books?id=QagG_KI7Ll8C&pg=PA59&dq=manifold+%22The+Large+Scale+Structure+of+Space-Time%22&lr=&as_brr=0&sig=ACfU3U1q-iaRTBDo6J8HMEsyPeFi8cJNWg#PPA11,M1
+}} A mathematical definition is: ''A connected [[Hausdorff space]] ''M'' is called an ''n''-dimensional manifold if each point of ''M'' is contained in an open set that is homeomorphic to an open set in Euclidean ''n''-dimensional space.''
+</ref><ref name=Morita2>
+{{cite book |title=Geometry of Differential Forms |author=Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu |pages=p. 12 |url=http://books.google.com/books?id=5N33Of2RzjsC&pg=PA12&dq=geometry++axiom+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3Vi7xsLiYiWCK0erF6X2gczHOkJA#PPA12,M1
+|isbn=0821810456 |year=2001 |publisher=American Mathematical Society Bookstore  }}
+</ref> The coordinates of a point '''r''' in an ''n''-dimensional space are simply an ordered set of ''n'' numbers:<ref name=Korn>
+{{cite book |title=Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review |author=Granino Arthur Korn, Theresa M. Korn |pages=p. 169 |url=http://books.google.com/books?id=xHNd5zCXt-EC&pg=PA169&dq=curvilinear+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3psSqwpBtA3U40e46VPPaMNMEw4g#PPA169,M1
+|isbn=0486411478 |year=2000 |publisher=Courier Dover Publications}}
 </ref>
+:<math>\mathbf{r} =[x^1,\ x^2,\ \dots\ ,  x^n] \ .</math>
+In a general [[Banach space]], these numbers could be (for example) coefficients in a functional expansion like a [[Fourier series]]. In a physical problem, they could be [[spacetime]] coordinates or [[normal mode]] amplitudes. In a [[Robotics|robot design]], they could be angles of relative rotations, linear displacements, or deformations of [[linkage (mechanical)|joints]].<ref name=Yamane>
+{{cite book |author=Katsu Yamane |title=Simulating and Generating Motions of Human Figures |isbn=3540203176 |year=2004 |publisher=Springer  |pages=12–13 |url=http://books.google.com/books?id=tNrMiIx3fToC&pg=PA12&dq=generalized+coordinates+%22kinematic+chain%22&lr=&as_brr=0&sig=ACfU3U3LRGJJTAHs21CHdOvuu08vw0cAuw#PPA13,M1   }}
+</ref> Here we will suppose these coordinates can be related to a [[Cartesian coordinate]] system by a set of functions:
+:<math>x^j = x^j (x,\  y,\  z,\  \dots)\ , </math>&ensp; &ensp; <math> j = 1, \ \dots \ , \ n\  </math>
+where ''x'', ''y'', ''z'', ''etc.'' are the ''n'' Cartesian coordinates of the point. Given these functions,  '''coordinate surfaces''' are defined by the relations:
+:<math> x^j (x, y, z, \dots) = \mathrm{constant}\ , </math>&ensp; &ensp; <math> j = 1, \ \dots \ , \ n\  .</math>
+The intersection of these surfaces define '''coordinate lines'''. At any selected point, tangents to the intersecting coordinate lines at that point define a set of '''basis vectors''' {'''e'''<sub>1</sub>, '''e'''<sub>2</sub>, …, '''e'''<sub>n</sub>} at that point. That is:
+:<math>\mathbf{e}_i(\mathbf{r}) =\lim_{\epsilon \rightarrow 0} \frac{\mathbf{r}\left(x^1,\  \dots,\  x^i+\epsilon,\  \dots ,\  x^n \right) - \mathbf{r}\left(x^1,\  \dots,\  x^i,\  \dots ,\  x^n \right)}{\epsilon }\ ,</math>
+which can be normalized to be of unit length. For more detail see [[curvilinear coordinates]].
+Coordinate surfaces, coordinate lines, and [[Basis (linear algebra)|basis vectors]] are components of a '''coordinate system'''.<ref name=Zdunkowski>{{cite book |title=Dynamics of the Atmosphere |page=84  |isbn=052100666X |year=2003 |author=Wilford Zdunkowski & Andreas Bott |publisher=Cambridge University Press |url=http://books.google.com/books?id=GuYvC21v3g8C&pg=RA1-PA84&dq=%22curvilinear+coordinate+system%22&lr=&as_brr=0&sig=ACfU3U2g2k7kY5u-CVcJ1pH5ZxsbEb9Rig  }}</ref> If the basis vectors are orthogonal at every point, the coordinate system is an [[Orthogonal coordinates|orthogonal coordinate system]].
+An important aspect of a coordinate system is its [[Metric (mathematics)|metric]] ''g''<sub>ik</sub>, which determines the [[arc length]] ''ds'' in the coordinate system in terms of its coordinates:<ref name=Borisenko>{{cite book |title=Vector and Tensor Analysis with Applications |author= A. I. Borisenko, I. E. Tarapov, Richard A. Silverman |page=86 |url=http://books.google.com/books?id=CRIjIx2ac6AC&pg=PA86&dq=coordinate+metric&lr=&as_brr=0&sig=ACfU3U1osXaT2hg7Md57cJ9katl3ttL43Q
+|isbn=0486638332 |publisher=Courier Dover Publications |year=1979 |pages=pp. 86 ''ff'' |chapter=§2.8.4 Arc length. Metric coefficients |edition=Reprint of Prentice-Hall 1968 ed   }}</ref>
+:<math>(ds)^2 = g_{ik}\ dx^i\ dx^k \ , </math>
+where repeated indices are summed over.
+As is apparent from these remarks, a coordinate system is a mathematical construct, part of an [[axiomatic system]]. There is no necessary connection between coordinate systems and physical motion (or any other aspect of reality). However, coordinate systems can be used to describe motion by interpreting one coordinate as time. Thus, [[Lorentz transformation]]s and [[Galilean transformation]]s may be viewed as [[coordinate transformation]]s.
 ==Notes==
 <references/>
+[http://books.google.com/books?id=hUWEXphqLo8C&pg=PA111&dq=manifold+%22coordinate+system%22&hl=en&ei=I5GGTbWsPIz2tgOmoIzoAQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CEQQ6AEwBA#v=onepage&q=manifold%20%22coordinate%20system%22&f=false Choquet-Bruhat]
+[http://books.google.com/books?id=sRaSuentwngC&pg=PA2&dq=manifold+%22coordinate+system%22&hl=en&ei=I5GGTbWsPIz2tgOmoIzoAQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDIQ6AEwAQ#v=onepage&q=manifold%20%22coordinate%20system%22&f=false Bishop]
+[http://books.google.com/books?id=CGk1eRSjFIIC&pg=PA3&dq=manifold+%22coordinate+system%22&hl=en&ei=I5GGTbWsPIz2tgOmoIzoAQ&sa=X&oi=book_result&ct=result&resnum=7&ved=0CE8Q6AEwBg#v=onepage&q=manifold%20%22coordinate%20system%22&f=false O'Neill]
+[http://books.google.com/books?id=iaeUqc2yQVQC&pg=PA31&dq=manifold+%22coordinate+system%22&hl=en&ei=I5GGTbWsPIz2tgOmoIzoAQ&sa=X&oi=book_result&ct=result&resnum=9&ved=0CFgQ6AEwCA#v=onepage&q=manifold%20%22coordinate%20system%22&f=false Warner]

User:John R. Brews/Draft: Difference between revisions

Latest revision as of 11:20, 14 September 2024

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Coordinate system

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