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

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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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@@ Line 1: / Line 1: @@
-<br>
+{{AccountNotLive}}
-{{Image|Fermi function.PNG|right|300px|Fermi occupancy function ''vs''. energy departure from Fermi level in volts for three temperatures; degeneracy factor ''g''&nbsp;&equiv;&nbsp;1.}}
 {{TOC|right}}
+==Coordinate system==
-==Fermi function==
+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>
-The '''Fermi function''' or, more completely, the '''Fermi-Dirac distribution function''' describes the occupancy of a electronic energy level in a system of electrons at equilibrium. The occupancy ''f(E)'' of an energy level of energy ''E'' at an [[absolute temperature]] ''T'' in [[Kelvin (unit)|kelvin]]s is given by:
+{{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}}
-:<math>
+</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]
-\begin{align}
+</ref><ref name=Fritzche>
-f(E) &= \frac{1}{1+g(E)\exp\left(\frac {E-E_F}{k_B T}\right )} \\
-&=\frac{1}{1+g(E)\exp\left(\frac {(E-E_F)/q}{k_B T/q}\right )}\\
+[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>
-\end{align}
+:<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>
-</math>
+==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>
-where ''g(E)'' is the ''degeneracy'' of the energy level, that is, the number of quantum states with that particular energy. Here ''E<sub>F</sub>'' is called the ''Fermi energy'' and ''k<sub>B</sub>'' is the [[Boltzmann constant]]. This occupancy function is plotted in the figure versus the energy ''E−E<sub>F</sub>'' in eV (electron volts). From the second form of the function above, it can be seen that the "natural" unit of energy is the ''thermal voltage k<sub>B</sub>T/q''; as temperature increases, so does this unit, accounting for the stretching out of the function along the energy axis with increasing temperature.
+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.''
-Notice that for an energy level with ''E&nbsp;=&nbsp;E<sub>F</sub>'' and a degeneracy factor ''g''&nbsp;=&nbsp;1, the occupancy is 1/2 regardless of temperature. The Fermi level ''E<sub>F</sub>'' thus can be referred to as the ''half-occupancy'' level.
+</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  }}
-===Dopant levels===
+</ref> The coordinates of a point '''r''' in an ''n''-dimensional space are simply an ordered set of ''n'' numbers:<ref name=Korn>
-Dopant impurities are used in [[semiconductor]]s to adjust the conductivity of the material. They introduce energy levels for electrons, and if they are ''acceptors'' become negatively charged when occupied. Typically an acceptor provides an energy level related to the valence band structure of the host material. A common case is two levels, one from the "heavy" hole valence band and one from the "light" hole valence band. Thus, the number of negatively charge acceptors, compared to the total number of acceptors, is:
-:<math>\frac {N_a^-}{N_a} = \frac {1}{1+4\exp\left(\frac {E-E_F}{k_B T}\right )} \ , </math>
+{{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}}
-where the degeneracy factor of 4 stems from the possibility of either a spin-up or a spin-down electron occupying the level ''E'', and the existence of ''two'' sources for holes of energy ''E'', one from the "heavy" hole band and one from the "light" hole band.
+</ref>
+:<math>\mathbf{r} =[x^1,\ x^2,\ \dots\ ,  x^n] \ .</math>
-In contrast to acceptors, ''donors'' become positively charged and tend to give up an electron. The number of positive donors compared to the total number of donors is then:
+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>
-:<math>\frac {N_d^+}{N_d} = \frac {1}{1+2\exp\left(\frac {E-E_F}{k_B T}\right )} \ , </math>
+{{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   }}
-where now the degeneracy factor is 2 (because of the spin-up or spin-down possibilities for occupancy) and there is typically only one energy level associated with the conduction band.<ref name= Parker>
+</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>
-[http://books.google.com/books?id=yJtXOdECL6AC&pg=PA734&dq=%22Fermi+function%22&hl=en&ei=_fotTeb4E42CsQOYqPG1Bg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CC0Q6AEwAQ#v=onepage&q=%22Fermi%20function%22&f=false p. 735]
+where ''x'', ''y'', ''z'', ''etc.'' are the ''n'' Cartesian coordinates of the point. Given these functions,  '''coordinate surfaces''' are defined by the relations:
-</ref><ref name=Morkoç>[http://books.google.com/books?id=AqhH9xaiZtgC&pg=PA132&dq=Fermi+%22degeneracy+factor%22&hl=en&ei=Ah4uTee_DJK-sAPsw8ixBg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCYQ6AEwAA#v=onepage&q=Fermi%20%22degeneracy%20factor%22&f=false §2.3.1] </ref><ref name=Willardson>
+:<math> x^j (x, y, z, \dots) = \mathrm{constant}\ , </math>&ensp; &ensp; <math> j = 1, \ \dots \ , \ n\  .</math>
-[http://books.google.com/books?id=-aCAjT8J6mYC&pg=PA151&dq=%22bound+state+degeneracy+factor%22&hl=en&ei=PCAuTZ_HNoG4sAP2xaSQBg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCYQ6AEwAA#v=onepage&q=%22bound%20state%20degeneracy%20factor%22&f=false Appendix B, p. 151]
-</ref><ref name=Reisch>[http://books.google.com/books?id=EMZ3EI52pIQC&pg=PA126&dq=Fermi+%22degeneracy+factor%22&hl=en&ei=Ah4uTee_DJK-sAPsw8ixBg&sa=X&oi=book_result&ct=result&resnum=10&ved=0CFoQ6AEwCQ#v=onepage&q=Fermi%20%22degeneracy%20factor%22&f=false p. 126] </ref><ref name=Seebauer>[http://books.google.com/books?id=1Ea3b6Ndkk4C&pg=PA10&dq=Fermi+%22degeneracy+factor%22&hl=en&ei=uiEuTaHdK470swP4kZiXBw&sa=X&oi=book_result&ct=result&resnum=5&ved=0CDkQ6AEwBDgK#v=onepage&q=Fermi%20%22degeneracy%20factor%22&f=false p. 11 ]</ref>
-==Fermi level==
+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:
-The Fermi level in the Fermi function represents for a system of independent electrons a very special case of the more general notion of an [[electrochemical potential]]. The [[chemical potential]] of a chemical species is the work required to add a particle of that species to an ensemble of particles at constant temperature and pressure. The ''electrochemical potential'' is the same quantity, but for a charged particle that has both chemical and electrical interactions.<ref name= Schmickler>
-[http://books.google.com/books?id=c2fYKNzO01QC&pg=PA13&dq=%22electrochemical+potential%22&hl=en&ei=qSkvTdLqII64sAONlenRCQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDMQ6AEwAw#v=onepage&q=%22electrochemical%20potential%22&f=false p. 13]
+:<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>
-</ref>
+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>
-For a system of independent electrons, this energy is the Fermi energy.
+where repeated indices are summed over.
-==Fermi surface==
+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.
-At zero temperature, the energy levels up to the Fermi energy are full, and those below are empty. The energy of an electron in a solid is a function of the ''wavevector'' '''k'''. For a "gas" of electrons in space, the energy is given by:
-:<math>E = \frac{(\hbar \mathbf{k})^2}{2 m} \ , </math>
-where h is [[Planck's constant]] divided by 2π, and ''m'' is the electron mass. For such an assembly, the equation ''E&nbsp;=&nbsp;E<sub>F</sub>'' defines a spherical surface in the wavevector space. This surface, separating the filled from the empty energy levels, is called the ''Fermi surface''.
-In a crystal the dependence of energy upon wavevector is more complicated and the Fermi surface can be a complex surface indeed.
 ==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

Manifolds

Notes

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

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Coordinate system

Manifolds

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