Residue theorem: Difference between revisions
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{{distinguish|Inertial frame of reference}} | |||
{{Classical mechanics |Fundamentals}} | |||
{{Other uses|Framing (disambiguation){{!}}Framing}} | |||
An i - | In [[physics]], a '''frame of reference''' (or '''reference frame''') may refer to a [[coordinate system]] used to represent and measure properties of objects, such as their [[position (vector)|position]] and [[orientation (geometry)|orientation]], at different moments of [[time]]. It may also refer to a [[Cartesian coordinate system|set of axes]] used for such representation. In a weaker sense, a reference frame does not specify coordinates, but only defines the same 3-[[dimension (mathematics and physics)|dimensional]] [[space]] for all moments of time such that the frame can distinguish objects [[rest (physics)|at rest]] from those that are [[motion (physics)|moving]]. | ||
In [[Theory of relativity|Einsteinian relativity]], reference frames are used to specify the relationship between a moving [[Observer (special relativity)|observer]] and the phenomenon or phenomena under observation. In this context, the phrase often becomes "'''observational frame of reference'''" (or "'''observational reference frame'''"), which implies that the observer is at rest in the frame, although not necessarily located at its [[origin (mathematics)|origin]]. A relativistic reference frame includes (or implies) the [[coordinate time]], which does not correspond across different frames [[relative motion|moving relatively]] each other. The situation thus differs from [[Galilean relativity]], where all possible coordinate times are essentially equivalent. | |||
== Different aspects of "frame of reference" == | |||
The need to distinguish between the various meanings of "frame of reference" has led to a variety of terms. For example, sometimes the type of coordinate system is attached as a modifier, as in ''Cartesian frame of reference''. Sometimes the state of motion is emphasized, as in ''[[Rotating reference frame|rotating frame of reference]]''. Sometimes the way it transforms to frames considered as related is emphasized as in ''Galilean frame of reference''. Sometimes frames are distinguished by the scale of their observations, as in ''macroscopic'' and ''microscopic frames of reference''.<ref name=macroscopic>The distinction between macroscopic and microscopic frames shows up, for example, in electromagnetism where [[Constitutive equation|constitutive relations]] of various time and length scales are used to determine the current and charge densities entering [[Maxwell's equations]]. See, for example, {{cite book |title=Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media |author=Kurt Edmund Oughstun |page=165 |url=http://books.google.com/books?id=behRnNRiueAC&pg=PA165&dq=macroscopic+frame++electromagnetism|isbn=0-387-34599-X |year=2006 |publisher=Springer}}. These distinctions also appear in thermodynamics. See {{cite book |title=Classical Theory |author=Paul McEvoy |page=205 |url=http://books.google.com/books?id=dj0wFIxn-PoC&pg=PA206&dq=macroscopic+frame#PPA205,M1 |isbn=1-930832-02-8 |year=2002 |publisher=MicroAnalytix}}.</ref> | |||
In this article, the term ''observational frame of reference'' is used when emphasis is upon the ''state of motion'' rather than upon the coordinate choice or the character of the observations or observational apparatus. In this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. On the other hand, a ''coordinate system'' may be employed for many purposes where the state of motion is not the primary concern. For example, a coordinate system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, of course, the formulation of many problems in physics employs ''[[generalized coordinates]]'', ''[[normal modes]]'' or ''[[eigenvectors]]'', which are only indirectly related to space and time. It seems useful to divorce the various aspects of a reference frame for the discussion below. We therefore take observational frames of reference, coordinate systems, and observational equipment as independent concepts, separated as below: | |||
*An observational frame (such as an [[inertial frame]] or [[non-inertial frame of reference]]) is a physical concept related to state of motion. | |||
*A coordinate system is a mathematical concept, amounting to a choice of language used to describe observations.<ref name =Pontriagin> | |||
In very general terms, a coordinate system is a set of arcs ''x''<sup>i</sup> = ''x''<sup>i</sup> (''t'') in a complex [[Lie group]]; see {{cite book |author=Lev Semenovich Pontri͡agin |title=L.S. Pontryagin: Selected Works Vol. 2: Topological Groups |page= 429 |year= |url=http://books.google.com/books?id=JU0DT_wXu2oC&pg=PA429&dq=algebra+%22coordinate+system%22|isbn=2-88124-133-6 |publisher=Gordon and Breach|edition=3rd Edition }}. Less abstractly, a coordinate system in a space of n-dimensions is defined in terms of a basis set of vectors {'''e'''<sub>1</sub>, '''e'''<sub>2</sub>,… '''e'''<sub>n</sub>}; see {{cite book |title=Linear Algebra: A Geometric Approach |author= Edoardo Sernesi, J. Montaldi |page=95 |url=http://books.google.com/books?id=1dZOuFo1QYMC&pg=PA95&dq=algebra+%22coordinate+system%22|isbn=0-412-40680-2 |year=1993 |publisher=CRC Press }} As such, the coordinate system is a mathematical construct, a language, that may be related to motion, but has no necessary connection to motion. | |||
</ref> Consequently, an observer in an observational frame of reference can choose to employ any coordinate system (Cartesian, polar, curvilinear, generalized, …) to describe observations made from that frame of reference. A change in the choice of this coordinate system does not change an observer's state of motion, and so does not entail a change in the observer's ''observational'' frame of reference. This viewpoint can be found elsewhere as well.<ref name=Johansson> | |||
{{cite book |title=Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces |author=J X Zheng-Johansson and Per-Ivar Johansson | |||
|page=13 | |||
|url=http://books.google.com/books?id=I1FU37uru6QC&pg=PA13&dq=frame+coordinate+johansson|isbn=1-59454-260-0 | |||
|publisher=Nova Publishers | |||
|year=2006 }}</ref> Which is not to dispute that some coordinate systems may be a better choice for some observations than are others. | |||
*Choice of what to measure and with what observational apparatus is a matter separate from the observer's state of motion and choice of coordinate system. | |||
Here is a quotation applicable to moving observational frames <math>\mathfrak{R}</math> and various associated Euclidean three-space coordinate systems [''R'', ''R′'', ''etc.'']: | |||
<ref name=Lyle >{{cite book |title=Handbook of Continuum Mechanics: General Concepts, Thermoelasticity |page= 9 |author=Jean Salençon, Stephen Lyle |url=http://books.google.com/books?id=H3xIED8ctfUC&pg=PA9&dq=physical+%22frame+of+reference%22|isbn=3-540-41443-6 |year=2001 |publisher=Springer }}</ref> | |||
{{Cquote|We first introduce the notion of ''reference frame'', itself related to the idea of ''observer'': the reference frame is, in some sense, the "Euclidean space carried by the observer". Let us give a more mathematical definition:… the reference frame is... the set of all points in the Euclidean space with the rigid body motion of the observer. The frame, denoted <math>\mathfrak{R}</math>, is said to move with the observer.… The spatial positions of particles are labelled relative to a frame <math>\mathfrak{R}</math> by establishing a ''coordinate system'' ''R'' with origin ''O''. The corresponding set of axes, sharing the rigid body motion of the frame <math>\mathfrak{R}</math>, can be considered to give a physical realization of <math>\mathfrak{R}</math>. In a frame <math>\mathfrak{R}</math>, coordinates are changed from ''R'' to ''R′'' by carrying out, at each instant of time, the same coordinate transformation on the components of ''intrinsic'' objects (vectors and tensors) introduced to represent physical quantities ''in this frame''.| Jean Salençon, Stephen Lyle ''Handbook of Continuum Mechanics: General Concepts, Thermoelasticity'' p. 9}} | |||
and this on the utility of separating the notions of <math>\mathfrak{R}</math> and [''R'', ''R′'', ''etc.'']:<ref name= Lakhtakia>{{cite book |title=Essays on the Formal Aspects of Electromagnetic Theory |author=Patrick Cornille (Akhlesh Lakhtakia, editor) |page=149 |url=http://books.google.com/books?id=qsOBhKVM1qYC&pg=PA149&dq=coordinate+system+%22reference+frame%22|isbn=981-02-0854-5 |year=1993 |publisher=World Scientific }}</ref> | |||
{{Cquote|As noted by Brillouin, a distinction between mathematical sets of coordinates and physical frames of reference must be made. The ignorance of such distinction is the source of much confusion… the dependent functions such as velocity for example, are measured with respect to a physical reference frame, but one is free to choose any mathematical coordinate system in which the equations are specified.|L. Brillouin in ''Relativity Reexamined'' (as quoted by Patrick Cornille in '' Essays on the Formal Aspects of Electromagnetic Theory'' p. 149) }} | |||
and this, also on the distinction between <math>\mathfrak{R}</math> and [''R'', ''R′'', ''etc.'']:<ref name= Nerlich>{{cite book |title=What Spacetime Explains: Metaphysical essays on space and time |author=Graham Nerlich |page=64 |url=http://books.google.com/books?id=fKK7rKOpc7AC&pg=PA64&dq=%22idea+of+a+reference+frame%22|isbn=0-521-45261-9 |year=1994 |publisher=Cambridge University Press}}</ref> | |||
{{Cquote|The idea of a reference frame is really quite different from that of a coordinate system. Frames differ just when they define different ''spaces'' (sets of ''rest'' points) or times (sets of simultaneous events). So the ideas of a space, a time, of rest and simultaneity, go inextricably together with that of frame. However, a mere shift of origin, or a purely spatial rotation of space coordinates results in a new coordinate system. So frames correspond at best to ''classes'' of coordinate systems.|Graham Nerlich: ''What Spacetime Explains'', p. 64}} | |||
and from J. D. Norton:<ref name=Norton>John D. Norton (1993). [http://www.pitt.edu/~jdnorton/papers/decades.pdf ''General covariance and the foundations of general relativity: eight decades of dispute''], ''Rep. Prog. Phys.'', '''56''', pp. 835-7.</ref> | |||
{{Cquote|In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas. The first is the notion of a coordinate system, understood simply as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods. The second, the frame of reference, refers to an idealized system used to assign such numbers … To avoid unnecessary restrictions, we can divorce this arrangement from metrical notions. … Of special importance for our purposes is that each frame of reference has a definite state of motion at each event of spacetime.…Within the context of special relativity and as long as we restrict ourselves to frames of reference in inertial motion, then little of importance depends on the difference between an inertial frame of reference and the inertial coordinate system it induces. This comfortable circumstance ceases immediately once we begin to consider frames of reference in nonuniform motion even within special relativity.…More recently, to negotiate the obvious ambiguities of Einstein’s treatment, the notion of frame of reference has reappeared as a structure distinct from a coordinate system.|John D. Norton: ''General Covariance and the Foundations of General Relativity: eight decades of dispute'', ''Rep. Prog. Phys.'', '''56''', pp. 835-7.}} | |||
The discussion is taken beyond simple space-time coordinate systems by Brading and Castellani.<ref name=Brading>{{cite book |title=Symmetries in Physics: Philosophical Reflections |author=Katherine Brading & Elena Castellani |page=417 |url=http://books.google.com/books?id=SnmBN64cAdYC&pg=PA417&dq=%22idea+of+a+reference+frame%22|isbn=0-521-82137-1 |year=2003 |publisher=Cambridge University Press}}</ref> Extension to coordinate systems using generalized coordinates underlies the [[Hamilton's principle|Hamiltonian]] and [[Lagrangian]] formulations<ref name=Johns>{{cite book |title=Analytical Mechanics for Relativity and Quantum Mechanics |page=Chapter 16 |author=Oliver Davis Johns |url=http://books.google.com/books?id=PNuM9YDN8CIC&pg=PA318&dq=coordinate+observer#PPA276,M1 | |||
|isbn=0-19-856726-X |year=2005 |publisher=Oxford University Press |nopp=true }}</ref> of [[quantum field theory]], [[classical mechanics|classical relativistic mechanics]], and [[quantum gravity]].<ref name=Greenwood>{{cite book |title=Classical dynamics |author=Donald T Greenwood |page=313 |year=1997 |edition=Reprint of 1977 edition by Prentice-Hall |publisher=Courier Dover Publications |url=http://books.google.com/books?id=x7rj83I98yMC&pg=RA2-PA314&dq=%22relativistic+%22+Lagrangian+OR+Hamiltonian#PRA2-PA313,M1 | |||
|isbn=0-486-69690-1 }}</ref><ref name=Trump>{{cite book |title=Classical Relativistic Many-Body Dynamics |author=Matthew A. Trump & W. C. Schieve |page= 99 |url=http://books.google.com/books?id=g2yfLOp0IzwC&pg=PA99&dq=relativity+%22generalized+coordinates%22#PPA99,M1 | |||
|year=1999 |publisher=Springer |isbn= 0-7923-5737-X }}</ref><ref name=Kompaneyets>{{cite book |author=A S Kompaneyets |title=Theoretical Physics |url=http://books.google.com/books?id=CQ2gBrL5T4YC&pg=PA118&dq=relativity+%22generalized+coordinates%22|page=118 |isbn=0-486-49532-9 |year=2003 |publisher=Courier Dover Publications |edition=Reprint of the 1962 2nd Edition }}</ref><ref name=Srednicki>{{cite book |title=Quantum Field Theory |page= Chapter 4|author=M Srednicki |publisher=Cambridge University Press |year=2007 |isbn=978-0-521-86449-7 |url=http://books.google.com/books?id=5OepxIG42B4C&pg=PA266&dq=isbn=9780521864497#PPA31,M1 |nopp=true }}</ref><ref name=Rovelli>{{cite book |title=Quantum Gravity |author=Carlo Rovelli |page= 98 ff |url=http://books.google.com/books?id=HrAzTmXdssQC&pg=PA179&dq=%22relativistic+%22+Lagrangian+OR+Hamiltonian#PPA98,M1 | |||
|isbn=0-521-83733-2 |year=2004 |publisher=Cambridge University Press }}</ref> | |||
===Coordinate systems=== | |||
{{main|Coordinate systems}} | |||
{{see also|Generalized coordinates|Axes conventions}} | |||
[[File:Reference frame and observer.svg|thumb|250px|"250px"|right| An observer O, situated at the origin of a local set of coordinates – a frame of reference '''F'''. The observer in this frame uses the coordinates (''x, y, z, t'') to describe a spacetime event, shown as a star.]] | |||
Although the term "coordinate system" is often used (particularly by physicists) in a nontechnical sense, the term "coordinate system" does have a precise meaning in mathematics, and sometimes that is what the physicist means as well. | |||
A coordinate system in mathematics is a facet of [[geometry]] or of [[algebra]],<ref name=Barker>{{cite book |title=Continuous symmetry: from Euclid to Klein |author=William Barker & Roger Howe |page=18 ff |url=http://books.google.com/books?id=NIxExnr2EjYC&pg=PA17&dq=geometry++axiom+%22coordinate+system%22#PPA18,M1 | |||
|isbn=0-8218-3900-4 |year=2008 |publisher=American Mathematical Society }}</ref><ref name=Ramsay>{{cite book |title=Introduction to Hyperbolic Geometry |author=Arlan Ramsay & Robert D. Richtmyer |page=11 |url=http://books.google.com/books?id=UVozmKVh7GsC&pg=PA202&dq=geometry++axiom+%22coordinate+system%22#PPA11,M1 | |||
|isbn=0-387-94339-0 |publisher=Springer |year=1995 }}</ref> in particular, a property of [[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=0-521-09906-4 |year=1973 |publisher=Cambridge University Press |page=11 |url=http://books.google.com/books?id=QagG_KI7Ll8C&pg=PA59&dq=manifold+%22The+Large+Scale+Structure+of+Space-Time%22#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=Morita>{{cite book |title=Geometry of Differential Forms |author=Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu |page=12 |url=http://books.google.com/books?id=5N33Of2RzjsC&pg=PA12&dq=geometry++axiom+%22coordinate+system%22#PPA12,M1 | |||
|isbn=0-8218-1045-6 |year=2001 |publisher=American Mathematical Society Bookstore }}</ref> The [[Cartesian coordinate system|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 |page=169 |url=http://books.google.com/books?id=xHNd5zCXt-EC&pg=PA169&dq=curvilinear+%22coordinate+system%22#PPA169,M1 | |||
|isbn=0-486-41147-8 |year=2000 |publisher=Courier Dover Publications}}</ref><ref name=encarta>See [http://encarta.msn.com/encyclopedia_761579532/Coordinate_System_(mathematics).html Encarta definition]. [http://www.webcitation.org/5kwcKb20f Archived] 2009-10-31.</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=3-540-20317-6 |year=2004 |publisher=Springer |pages=12–13 |url=http://books.google.com/books?id=tNrMiIx3fToC&pg=PA12&dq=generalized+coordinates+%22kinematic+chain%22#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>    <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>    <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:<ref name=Papapetrou>{{cite book |title=Lectures on General Relativity |author=Achilleus Papapetrou |page=5 |url=http://books.google.com/books?id=SWeOggyp1ZsC&pg=PA3&dq=relativistic++%22general+coordinates%22#PPA5,M1 | |||
|isbn=90-277-0540-2 |year=1974 |publisher=Springer }}</ref> | |||
:<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#Covariant_basis|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=0-521-00666-X |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}}</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 tensor]] ''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|isbn=0-486-63833-2 |publisher=Courier Dover Publications |year=1979 }}</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 [[Model theory|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 include time as a coordinate, and can be used to describe motion. Thus, [[Lorentz transformation]]s and [[Galilean transformation]]s may be viewed as [[Coordinate_system#Transformations|coordinate transformation]]s. | |||
General and specific topics of coordinate systems can be pursued following the [[#See also|See also]] links below. | |||
===Observational frames of reference=== | |||
{{main|Inertial frame of reference}} | |||
[[File:Minkowski diagram - 3 systems.svg|thumb|right|256px|Three frames of reference in special relativity. Black frame is at rest. Primed frame moves at 40% of light speed, double primed at 80%. Note scissors-like change as speed increases.]] | |||
An '''observational frame of reference''', often referred to as a ''physical frame of reference'', a ''frame of reference'', or simply a ''frame'', is a physical concept related to an [[Observer (special relativity)|observer]] and the observer's state of motion. Here we adopt the view expressed by Kumar and Barve: an observational frame of reference is characterized ''only by its state of motion''.<ref name=Kubar>See {{cite book |author=Arvind Kumar & Shrish Barve |page=115 |title=How and Why in Basic Mechanics |url=http://books.google.com/books?id=czlUPz38MOQC&pg=PA115&dq=%22characterized+only+by+its+state+of+motion%22+inauthor:Kumar|isbn=81-7371-420-7 |year= 2003 |publisher =Orient Longman}}</ref> However, there is lack of unanimity on this point. In special relativity, the distinction is sometimes made between an ''observer'' and a ''frame''. According to this view, a ''frame'' is an ''observer'' plus a coordinate lattice constructed to be an orthonormal right-handed set of spacelike vectors perpendicular to a timelike vector. See Doran.<ref name=Doran>{{cite book |url=http://www.worldcat.org/search?q=9780521715959&qt=owc_search |title=Geometric Algebra for Physicists |author= Chris Doran & Anthony Lasenby |page= §5.2.2, p. 133 |isbn=978-0-521-71595-9 |year=2003 |publisher=Cambridge University Press}}.</ref> This restricted view is not used here, and is not universally adopted even in discussions of relativity.<ref name=Moller>For example, Møller states: "Instead of Cartesian coordinates we can obviously just as well employ general curvilinear coordinates for the fixation of points in physical space.…we shall now introduce general "curvilinear" coordinates ''x''<sup>i</sup> in four-space…." {{cite book |author=C. Møller |title=The Theory of Relativity |page=222 and p. 233 |year=1952 |publisher=Oxford University Press}}</ref><ref name=Lightman>{{cite book |title=Problem Book in Relativity and Gravitation |author=A. P. Lightman, W. H. Press, R. H. Price & S. A. Teukolsky |page=15 |url=http://books.google.com/books?id=YtxGYnnP1PEC&pg=PA15&dq=relativistic++%22general+coordinates%22|isbn=0-691-08162-X |publisher=Princeton University Press |year=1975}}</ref> In [[general relativity]] the use of general coordinate systems is common (see, for example, the [[Karl Schwarzschild|Schwarzschild]] solution for the gravitational field outside an isolated sphere<ref name= Faber>{{cite book |title=Differential Geometry and Relativity Theory: an introduction |author=Richard L Faber |url=http://books.google.com/books?id=ctM3_afLuVEC&pg=PA149&dq=relativistic++%22general+coordinates%22#PPA211,M1 | |||
|page=211 |isbn=0-8247-1749-X |year=1983 |publisher=CRC Press }}</ref>). | |||
There are two types of observational reference frame: [[Inertial frame of reference|inertial]] and [[non-inertial reference frame|non-inertial]]. An inertial frame of reference is defined as one in which all laws of physics take on their simplest form. In [[special relativity]] these frames are related by [[Lorentz transformation]]s, which are parametrized by [[rapidity]]. In Newtonian mechanics, a more restricted definition requires only that [[Newton's first law]] holds true; that is, a Newtonian inertial frame is one in which a [[free particle]] travels in a [[straight line]] at constant [[speed]], or is at rest. These frames are related by [[Galilean transformation]]s. These relativistic and Newtonian transformations are expressed in spaces of general dimension in terms of [[Representation theory|representations]] of the [[Representation theory of the Poincaré group|Poincaré group]] and of the [[Representation theory of the Galilean group|Galilean group]]. | |||
In contrast to the inertial frame, a non-inertial frame of reference is one in which [[fictitious force]]s must be invoked to explain observations. An example is an observational frame of reference centered at a point on the Earth's surface. This frame of reference orbits around the center of the Earth, which introduces the fictitious forces known as the [[Coriolis force]], [[centrifugal force]], and [[gravitational force]]. (All of these forces including gravity disappear in a truly inertial reference frame, which is one of free-fall.) | |||
===Measurement apparatus=== | |||
A further aspect of a frame of reference is the role of the [[metrology|measurement apparatus]] (for example, clocks and rods) attached to the frame (see Norton quote above). This question is not addressed in this article, and is of particular interest in [[Measurement in quantum mechanics|quantum mechanics]], where the relation between observer and measurement is still under discussion (see [[measurement problem]]). | |||
In physics experiments, the frame of reference in which the laboratory measurement devices are at rest is usually referred to as the [[laboratory frame]] or simply "lab frame." An example would be the frame in which the detectors for a particle accelerator are at rest. The lab frame in some experiments is an inertial frame, but it is not required to be (for example the laboratory on the surface of the Earth in many physics experiments is not inertial). In particle physics experiments, it is often useful to transform energies and momenta of particles from the lab frame where they are measured, to the [[center of momentum frame]] "COM frame" in which calculations are sometimes simplified, since potentially all kinetic energy still present in the COM frame may be used for making new particles. | |||
In this connection it may be noted that the clocks and rods often used to describe observers' measurement equipment in thought, in practice are replaced by a much more complicated and indirect [[metrology]] that is connected to the nature of the [[vacuum]], and uses [[atomic clocks]] that operate according to the [[standard model]] and that must be corrected for [[gravitational time dilation]].<ref name= Wolfson>{{cite book |author= Richard Wolfson |title=Simply Einstein |url=http://books.google.com/books?id=OUJWKdlFKeQC&pg=PA216&dq=%22gravitational+time+dilation+%22|page=216 |isbn=0-393-05154-4 |publisher=W W Norton & Co. |year=2003 }}</ref> (See [[second]], [[meter]] and [[kilogram]]). | |||
In fact, Einstein felt that clocks and rods were merely expedient measuring devices and they should be replaced by more fundamental entities based upon, for example, atoms and molecules.<ref name=Rizzi>See {{cite book |title=Relativity in rotating frames |page=33 |url=http://books.google.com/books?id=_PGrlCLkkIgC&pg=PA226&dq=centrifugal+%22+%22+relativity+OR+relativistic#PPA33,M1 | |||
|isbn=1-4020-1805-3 |year=2003 |publisher=Springer |author=Guido Rizzi, Matteo Luca Ruggiero }}.</ref> | |||
==Examples of inertial frames of reference== | |||
{{unreferenced section|date=July 2013}} | |||
=== Simple example === | |||
[[File:Two reference frames.PNG|thumb|320px|Figure 1: Two cars moving at different but constant velocities observed from stationary inertial frame ''S'' attached to the road and moving inertial frame ''S′'' attached to the first car.]] | |||
Consider a situation common in everyday life. Two cars travel along a road, both moving at constant velocities. See Figure 1. At some particular moment, they are separated by 200 metres. The car in front is travelling at 22 metres per second and the car behind is travelling at 30 metres per second. If we want to find out how long it will take the second car to catch up with the first, there are three obvious "frames of reference" that we could choose. | |||
First, we could observe the two cars from the side of the road. We define our "frame of reference" ''S'' as follows. We stand on the side of the road and start a stop-clock at the exact moment that the second car passes us, which happens to be when they are a distance ''d'' = 200 m apart. Since neither of the cars is accelerating, we can determine their positions by the following formulas, where <math>x_1(t)</math> is the position in meters of car one after time ''t'' in seconds and <math>x_2(t)</math> is the position of car two after time ''t''. | |||
:<math> x_1(t)= d + v_1 t = 200\ + \ 22t\ ; \quad x_2(t)= v_2 t = 30t </math> | |||
Notice that these formulas predict at ''t'' = 0 s the first car is 200 m down the road and the second car is right beside us, as expected. We want to find the time at which <math>x_1=x_2</math>. Therefore we set <math>x_1=x_2</math> and solve for <math>t</math>, that is: | |||
:<math>200 + 22 t = 30t \quad</math> | |||
:<math>8t = 200 \quad</math> | |||
:<math>t = 25 \quad \mathrm{seconds}</math> | |||
Alternatively, we could choose a frame of reference ''S′'' situated in the first car. In this case, the first car is stationary and the second car is approaching from behind at a speed of {{math|1=''v''<sub>2</sub> − ''v''<sub>1</sub> = 8 m / s}}. In order to catch up to the first car, it will take a time of {{math|1={{sfrac|''d''|''v''<sub>2</sub> − ''v''<sub>1</sub>}} = {{sfrac|200|8}} s}}, that is, 25 seconds, as before. Note how much easier the problem becomes by choosing a suitable frame of reference. The third possible frame of reference would be attached to the second car. That example resembles the case just discussed, except the second car is stationary and the first car moves backward towards it at 8 m / s. | |||
It would have been possible to choose a rotating, accelerating frame of reference, moving in a complicated manner, but this would have served to complicate the problem unnecessarily. It is also necessary to note that one is able to convert measurements made in one coordinate system to another. For example, suppose that your watch is running five minutes fast compared to the local standard time. If you know that this is the case, when somebody asks you what time it is, you are able to deduct five minutes from the time displayed on your watch in order to obtain the correct time. The measurements that an observer makes about a system depend therefore on the observer's frame of reference (you might say that the bus arrived at 5 past three, when in fact it arrived at three). | |||
===Additional example=== | |||
[[File:Wikipage pic.PNG|thumb|250px|Figure 2: Simple-minded frame-of-reference example]] | |||
{{Disputed-section|date=February 2012}} | |||
For a simple example involving only the orientation of two observers, consider two people standing, facing each other on either side of a north-south street. See Figure 2. A car drives past them heading south. For the person facing east, the car was moving toward the right. However, for the person facing west, the car was moving toward the left. This discrepancy is because the two people used two different frames of reference from which to investigate this system. | |||
For a more complex example involving observers in relative motion, consider Alfred, who is standing on the side of a road watching a car drive past him from left to right. In his frame of reference, Alfred defines the spot where he is standing as the origin, the road as the x-axis and the direction in front of him as the positive y-axis. To him, the car moves along the ''x'' axis with some [[velocity]] ''v'' in the positive x-direction. Alfred's frame of reference is considered an [[inertial frame of reference]] because he is not accelerating (ignoring effects such as Earth's rotation and gravity). | |||
Now consider Betsy, the person driving the car. Betsy, in choosing her frame of reference, defines her location as the origin, the direction to her right as the positive ''x''-axis, and the direction in front of her as the positive ''y''-axis. In this frame of reference, it is Betsy who is stationary and the world around her that is moving – for instance, as she drives past Alfred, she observes him moving with velocity ''v'' in the negative ''y''-direction. If she is driving north, then north is the positive ''y''-direction; if she turns east, east becomes the positive ''y''-direction. | |||
Finally, as an example of non-inertial observers, assume Candace is accelerating her car. As she passes by him, Alfred measures her [[acceleration]] and finds it to be ''a'' in the negative x-direction. Assuming Candace's acceleration is constant, what acceleration does Betsy measure? If Betsy's velocity ''v'' is constant, she is in an inertial frame of reference, and she will find the acceleration to be the same as Alfred in her frame of reference, ''a'' in the negative ''y''-direction. However, if she is accelerating at rate ''A'' in the negative ''y''-direction (in other words, slowing down), she will find Candace's acceleration to be ''a′'' = ''a'' − ''A'' in the negative ''y''-direction - a smaller value than Alfred has measured. Similarly, if she is accelerating at rate ''A'' in the positive y-direction (speeding up), she will observe Candace's acceleration as ''a′'' = ''a'' + ''A'' in the negative ''y''-direction – a larger value than Alfred's measurement. | |||
Frames of reference are especially important in [[special relativity]], because when a frame of reference is moving at some significant fraction of the speed of light, then the flow of time in that frame does not necessarily apply in another frame. The speed of light is considered to be the only true constant between moving frames of reference. | |||
===Remarks=== | |||
{{see also|Special theory of relativity|General theory of relativity}} | |||
It is important to note some assumptions made above about the various inertial frames of reference. Newton, for instance, employed universal time, as explained by the following example. Suppose that you own two clocks, which both tick at exactly the same rate. You synchronize them so that they both display exactly the same time. The two clocks are now separated and one clock is on a fast moving train, traveling at constant velocity towards the other. According to Newton, these two clocks will still tick at the same rate and will both show the same time. Newton says that the rate of time as measured in one frame of reference should be the same as the rate of time in another. That is, there exists a "universal" time and all other times in all other frames of reference will run at the same rate as this universal time irrespective of their position and velocity. This concept of time and simultaneity was later generalized by Einstein in his [[special theory of relativity]] (1905) where he developed transformations between inertial frames of reference based upon the universal nature of physical laws and their economy of expression ([[Lorentz transformations]]). | |||
It is also important to note that the definition of inertial reference frame can be extended beyond three dimensional Euclidean space. Newton's assumed a Euclidean space, but [[general relativity]] uses a more general geometry. As an example of why this is important, let us consider the [[geometry]] of an ellipsoid. In this geometry, a "free" particle is defined as one at rest or traveling at constant speed on a [[geodesic]] path. Two free particles may begin at the same point on the surface, traveling with the same constant speed in different directions. After a length of time, the two particles collide at the opposite side of the ellipsoid. Both "free" particles traveled with a constant speed, satisfying the definition that no forces were acting. No acceleration occurred and so Newton's first law held true. This means that the particles were in inertial frames of reference. Since no forces were acting, it was the geometry of the situation which caused the two particles to meet each other again. In a similar way, it is now common to describe<ref name="caveatondescription">That is, both descriptions are equivalent and can be used as needed. This equivalence does not hold outside of general relativity, e.g., in [[entropic gravity]].</ref> that we exist in a four dimensional geometry known as [[spacetime]]. In this picture, the curvature of this 4D space is responsible for the way in which two bodies with mass are drawn together even if no forces are acting. This curvature of spacetime replaces the force known as gravity in Newtonian mechanics and special relativity. | |||
==Non-inertial frames== | |||
{{main|Fictitious force|Non-inertial frame|Rotating frame of reference}} | |||
Here the relation between inertial and non-inertial observational frames of reference is considered. The basic difference between these frames is the need in non-inertial frames for fictitious forces, as described below. | |||
An accelerated frame of reference is often delineated as being the "primed" frame, and all variables that are dependent on that frame are notated with primes, e.g. ''x′'', ''y′'', ''a′''. | |||
The vector from the origin of an inertial reference frame to the origin of an accelerated reference frame is commonly notated as '''R'''. Given a point of interest that exists in both frames, the vector from the inertial origin to the point is called '''r''', and the vector from the accelerated origin to the point is called '''r′'''. | |||
From the geometry of the situation, we get | |||
: <math>\mathbf r = \mathbf R + \mathbf r'</math> | |||
Taking the first and second derivatives of this, we obtain | |||
: <math>\mathbf v = \mathbf V + \mathbf v'</math> | |||
: <math>\mathbf a = \mathbf A + \mathbf a'</math> | |||
where '''V''' and '''A''' are the velocity and acceleration of the accelerated system with respect to the inertial system and '''v''' and '''a''' are the velocity and acceleration of the point of interest with respect to the inertial frame. | |||
These equations allow transformations between the two coordinate systems; for example, we can now write [[Newton's laws of motion#Newton.27s second law|Newton's second law]] as | |||
: <math>\mathbf F = m\mathbf a = m\mathbf A + m\mathbf a'</math> | |||
When there is accelerated motion due to a force being exerted there is manifestation of inertia. If an electric car designed to recharge its battery system when decelerating is switched to braking, the batteries are recharged, illustrating the physical strength of manifestation of inertia. However, the manifestation of inertia does not prevent acceleration (or deceleration), for manifestation of inertia occurs in response to change in velocity due to a force. Seen from the perspective of a rotating frame of reference the manifestation of inertia appears to exert a force (either in [[centrifugal force (rotating reference frame)|centrifugal]] direction, or in a direction orthogonal to an object's motion, the [[Coriolis effect]]). | |||
A common sort of accelerated reference frame is a frame that is both rotating and translating (an example is a frame of reference attached to a CD which is playing while the player is carried). This arrangement leads to the equation (see [[Fictitious force]] for a derivation): | |||
: <math>\mathbf a = \mathbf a' + \dot{\boldsymbol\omega} \times \mathbf r' + 2\boldsymbol\omega \times \mathbf v' + \boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r') + \mathbf A_0</math> | |||
or, to solve for the acceleration in the accelerated frame, | |||
: <math>\mathbf a' = \mathbf a - \dot{\boldsymbol\omega} \times \mathbf r' - 2\boldsymbol\omega \times \mathbf v' - \boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r') - \mathbf A_0</math> | |||
Multiplying through by the mass ''m'' gives | |||
: <math>\mathbf F' = \mathbf F_\mathrm{physical} + \mathbf F'_\mathrm{Euler} + \mathbf F'_\mathrm{Coriolis} + \mathbf F'_\mathrm{centripetal} - m\mathbf A_0</math> | |||
where | |||
: <math>\mathbf F'_\mathrm{Euler} = -m\dot{\boldsymbol\omega} \times \mathbf r'</math> ([[Euler force]]) | |||
: <math>\mathbf F'_\mathrm{Coriolis} = -2m\boldsymbol\omega \times \mathbf v'</math> ([[Coriolis force]]) | |||
: <math>\mathbf F'_\mathrm{centrifugal} = -m\boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r')=m(\omega^2 \mathbf r'- (\boldsymbol\omega \cdot \mathbf r')\boldsymbol\omega) </math> ([[centrifugal force (rotating reference frame)|centrifugal force]]) | |||
== Particular frames of reference in common use == | |||
* [[International Terrestrial Reference Frame]] | |||
* [[International Celestial Reference Frame]] | |||
* In fluid mechanics, [[Lagrangian and Eulerian specification of the flow field]] | |||
===Other frames=== | |||
* [[Frame fields in general relativity]] | |||
* [[Linguistic frame of reference]] | |||
* [[Moving frame|Moving frame in Mathematics]] | |||
==See also== | |||
{{Col-begin}} | |||
{{Col-1-of-3}} | |||
* [[Analytical mechanics]] | |||
* [[Applied mechanics]] | |||
* [[Cartesian coordinate system]] | |||
* [[Center-of-momentum frame]] | |||
* [[Centrifugal force]] | |||
* [[Centripetal force]] | |||
* [[Classical mechanics]] | |||
* [[Coriolis force]] | |||
* [[Curvilinear coordinates]] | |||
{{Col-2-of-3}} | |||
* [[Dynamics (physics)]] | |||
* [[Fictitious force]] | |||
* [[Frenet–Serret formulas]] | |||
* [[Galilean invariance]] | |||
* [[General relativity]] | |||
* [[Generalized coordinates]] | |||
* [[Generalized forces]] | |||
* [[Inertial frame of reference]] | |||
{{Col-3-of-3}} | |||
* [[Kinematics]] | |||
* [[Laboratory frame of reference]] | |||
* [[Lorentz transformation]] | |||
* [[Mach's principle]] | |||
* [[Orthogonal coordinates]] | |||
* [[Principle of relativity]] | |||
* [[Special relativity]] | |||
* [[Theory of relativity]] | |||
{{col-end}} | |||
== Notes == | |||
{{reflist|2}} | |||
{{Relativity}} | |||
<!-- [[Category:Special relativity]] --> | |||
<!-- [[Category:Astronomy]] too general --> | |||
[[Category:Frames of reference| ]] | |||
[[Category:Theory of relativity]] | |||
[[Category:Geodesy]] | |||
[[Category:Navigation]] | |||
[[Category:Surveying]] |
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In physics, a frame of reference (or reference frame) may refer to a coordinate system used to represent and measure properties of objects, such as their position and orientation, at different moments of time. It may also refer to a set of axes used for such representation. In a weaker sense, a reference frame does not specify coordinates, but only defines the same 3-dimensional space for all moments of time such that the frame can distinguish objects at rest from those that are moving.
In Einsteinian relativity, reference frames are used to specify the relationship between a moving observer and the phenomenon or phenomena under observation. In this context, the phrase often becomes "observational frame of reference" (or "observational reference frame"), which implies that the observer is at rest in the frame, although not necessarily located at its origin. A relativistic reference frame includes (or implies) the coordinate time, which does not correspond across different frames moving relatively each other. The situation thus differs from Galilean relativity, where all possible coordinate times are essentially equivalent.
Different aspects of "frame of reference"
The need to distinguish between the various meanings of "frame of reference" has led to a variety of terms. For example, sometimes the type of coordinate system is attached as a modifier, as in Cartesian frame of reference. Sometimes the state of motion is emphasized, as in rotating frame of reference. Sometimes the way it transforms to frames considered as related is emphasized as in Galilean frame of reference. Sometimes frames are distinguished by the scale of their observations, as in macroscopic and microscopic frames of reference.[1]
In this article, the term observational frame of reference is used when emphasis is upon the state of motion rather than upon the coordinate choice or the character of the observations or observational apparatus. In this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. On the other hand, a coordinate system may be employed for many purposes where the state of motion is not the primary concern. For example, a coordinate system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, of course, the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors, which are only indirectly related to space and time. It seems useful to divorce the various aspects of a reference frame for the discussion below. We therefore take observational frames of reference, coordinate systems, and observational equipment as independent concepts, separated as below:
- An observational frame (such as an inertial frame or non-inertial frame of reference) is a physical concept related to state of motion.
- A coordinate system is a mathematical concept, amounting to a choice of language used to describe observations.[2] Consequently, an observer in an observational frame of reference can choose to employ any coordinate system (Cartesian, polar, curvilinear, generalized, …) to describe observations made from that frame of reference. A change in the choice of this coordinate system does not change an observer's state of motion, and so does not entail a change in the observer's observational frame of reference. This viewpoint can be found elsewhere as well.[3] Which is not to dispute that some coordinate systems may be a better choice for some observations than are others.
- Choice of what to measure and with what observational apparatus is a matter separate from the observer's state of motion and choice of coordinate system.
Here is a quotation applicable to moving observational frames and various associated Euclidean three-space coordinate systems [R, R′, etc.]: [4]
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and this on the utility of separating the notions of and [R, R′, etc.]:[5]
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and this, also on the distinction between and [R, R′, etc.]:[6]
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Banks usually engage skilled valuers to conduct valuations of properties. The valuers are required to abide by industry requirements and guidelines for property valuation prescribed by the Singapore Institute of Surveyors and Valuers. With Singapore conserving its greenery and with town sky view being the attraction to many, it is no wonder that increasingly condos are being developed on the island. Many future traders are placing extra value on units with a view than these with out one. Thus, this may be one of the reasons too as to why your unit of selection is either with a view or with out. How to Spot Abroad Property Scams (at Propwise.sg) New Condominium Eco Sanctuary Launch 01Dec 2012 Coming Quickly !!! New launch rental COCO Palm at Pasir Ris
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Dr Victor Lee is a surgeon gazetted to carry out multi-organ transplant surgical procedure. He underwent his fellowship coaching in transplant surgery on the Royal Infirmary of Edinburgh, United Kingdom in 2008 and 2009. He is presently a consultant surgeon at the Singapore Normal Hospital, and visiting guide to the National College Hospital. His scientific interest and skills are in the areas of transplant surgery, hepatobiliary pancreatic surgery and laparoscopic surgical procedure.
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and from J. D. Norton:[7]
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The simplest way is to contact a real estate agent who's a Advertising Workforce Member The agent will doubtless meet to show you details of the undertaking supplied by the developer. Attending a preview doesn't suggest you are obliged to purchase a unit. You can select not to and your cheque will likely be returned to you. No cost concerned. Your cheque will likely be returned to you in the event you decide to not proceed to purchase. Early hen low cost (up to 10%) at the preview earlier than the official launch. The purpose of that is to stimulate demand and to create a buying house in singapore momentum. As soon as this function is achieved, the developer would then take away the low cost so as to maximize his profitability. 2/3/4 bedrooms units obtainable ! GREATEST PURCHASE NOW !!! AVG $1900psf The Key Commercial Cluster in the North Area
The discussion is taken beyond simple space-time coordinate systems by Brading and Castellani.[8] Extension to coordinate systems using generalized coordinates underlies the Hamiltonian and Lagrangian formulations[9] of quantum field theory, classical relativistic mechanics, and quantum gravity.[10][11][12][13][14]
Coordinate systems
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
DTZ's public sale group in Singapore auctions all forms of residential, workplace and retail properties, outlets, homes, lodges, boarding homes, industrial buildings and development websites. Auctions are at present held as soon as a month.
We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.
The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.
Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.
is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease
In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value.
![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Reference_frame_and_observer.svg/250px-Reference_frame_and_observer.svg.png)
Although the term "coordinate system" is often used (particularly by physicists) in a nontechnical sense, the term "coordinate system" does have a precise meaning in mathematics, and sometimes that is what the physicist means as well.
A coordinate system in mathematics is a facet of geometry or of algebra,[15][16] in particular, a property of manifolds (for example, in physics, configuration spaces or phase spaces).[17][18] The coordinates of a point r in an n-dimensional space are simply an ordered set of n numbers:[19][20]
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.[21] Here we will suppose these coordinates can be related to a Cartesian coordinate system by a set of functions:
where x, y, z, etc. are the n Cartesian coordinates of the point. Given these functions, coordinate surfaces are defined by the relations:
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, …, en} at that point. That is:[22]
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.[23] 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 tensor gik, which determines the arc length ds in the coordinate system in terms of its coordinates:[24]
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 include time as a coordinate, and can be used to describe motion. Thus, Lorentz transformations and Galilean transformations may be viewed as coordinate transformations.
General and specific topics of coordinate systems can be pursued following the See also links below.
Observational frames of reference
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/44/Minkowski_diagram_-_3_systems.svg/256px-Minkowski_diagram_-_3_systems.svg.png)
An observational frame of reference, often referred to as a physical frame of reference, a frame of reference, or simply a frame, is a physical concept related to an observer and the observer's state of motion. Here we adopt the view expressed by Kumar and Barve: an observational frame of reference is characterized only by its state of motion.[25] However, there is lack of unanimity on this point. In special relativity, the distinction is sometimes made between an observer and a frame. According to this view, a frame is an observer plus a coordinate lattice constructed to be an orthonormal right-handed set of spacelike vectors perpendicular to a timelike vector. See Doran.[26] This restricted view is not used here, and is not universally adopted even in discussions of relativity.[27][28] In general relativity the use of general coordinate systems is common (see, for example, the Schwarzschild solution for the gravitational field outside an isolated sphere[29]).
There are two types of observational reference frame: inertial and non-inertial. An inertial frame of reference is defined as one in which all laws of physics take on their simplest form. In special relativity these frames are related by Lorentz transformations, which are parametrized by rapidity. In Newtonian mechanics, a more restricted definition requires only that Newton's first law holds true; that is, a Newtonian inertial frame is one in which a free particle travels in a straight line at constant speed, or is at rest. These frames are related by Galilean transformations. These relativistic and Newtonian transformations are expressed in spaces of general dimension in terms of representations of the Poincaré group and of the Galilean group.
In contrast to the inertial frame, a non-inertial frame of reference is one in which fictitious forces must be invoked to explain observations. An example is an observational frame of reference centered at a point on the Earth's surface. This frame of reference orbits around the center of the Earth, which introduces the fictitious forces known as the Coriolis force, centrifugal force, and gravitational force. (All of these forces including gravity disappear in a truly inertial reference frame, which is one of free-fall.)
Measurement apparatus
A further aspect of a frame of reference is the role of the measurement apparatus (for example, clocks and rods) attached to the frame (see Norton quote above). This question is not addressed in this article, and is of particular interest in quantum mechanics, where the relation between observer and measurement is still under discussion (see measurement problem).
In physics experiments, the frame of reference in which the laboratory measurement devices are at rest is usually referred to as the laboratory frame or simply "lab frame." An example would be the frame in which the detectors for a particle accelerator are at rest. The lab frame in some experiments is an inertial frame, but it is not required to be (for example the laboratory on the surface of the Earth in many physics experiments is not inertial). In particle physics experiments, it is often useful to transform energies and momenta of particles from the lab frame where they are measured, to the center of momentum frame "COM frame" in which calculations are sometimes simplified, since potentially all kinetic energy still present in the COM frame may be used for making new particles.
In this connection it may be noted that the clocks and rods often used to describe observers' measurement equipment in thought, in practice are replaced by a much more complicated and indirect metrology that is connected to the nature of the vacuum, and uses atomic clocks that operate according to the standard model and that must be corrected for gravitational time dilation.[30] (See second, meter and kilogram).
In fact, Einstein felt that clocks and rods were merely expedient measuring devices and they should be replaced by more fundamental entities based upon, for example, atoms and molecules.[31]
Examples of inertial frames of reference
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Simple example
Consider a situation common in everyday life. Two cars travel along a road, both moving at constant velocities. See Figure 1. At some particular moment, they are separated by 200 metres. The car in front is travelling at 22 metres per second and the car behind is travelling at 30 metres per second. If we want to find out how long it will take the second car to catch up with the first, there are three obvious "frames of reference" that we could choose.
First, we could observe the two cars from the side of the road. We define our "frame of reference" S as follows. We stand on the side of the road and start a stop-clock at the exact moment that the second car passes us, which happens to be when they are a distance d = 200 m apart. Since neither of the cars is accelerating, we can determine their positions by the following formulas, where is the position in meters of car one after time t in seconds and is the position of car two after time t.
Notice that these formulas predict at t = 0 s the first car is 200 m down the road and the second car is right beside us, as expected. We want to find the time at which . Therefore we set and solve for , that is:
Alternatively, we could choose a frame of reference S′ situated in the first car. In this case, the first car is stationary and the second car is approaching from behind at a speed of Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
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If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.. In order to catch up to the first car, it will take a time of Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer., that is, 25 seconds, as before. Note how much easier the problem becomes by choosing a suitable frame of reference. The third possible frame of reference would be attached to the second car. That example resembles the case just discussed, except the second car is stationary and the first car moves backward towards it at 8 m / s.
It would have been possible to choose a rotating, accelerating frame of reference, moving in a complicated manner, but this would have served to complicate the problem unnecessarily. It is also necessary to note that one is able to convert measurements made in one coordinate system to another. For example, suppose that your watch is running five minutes fast compared to the local standard time. If you know that this is the case, when somebody asks you what time it is, you are able to deduct five minutes from the time displayed on your watch in order to obtain the correct time. The measurements that an observer makes about a system depend therefore on the observer's frame of reference (you might say that the bus arrived at 5 past three, when in fact it arrived at three).
Additional example
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You'll need to arrange the remainder of the security deposit and advance rental upon signing of the Tenancy Settlement. For 1 12 months lease - 1 month's deposit and 1 month's advance rental. For 2 years lease - 2 month's deposit and 1 month's advance rental. Minus the goodfaith deposit (if relevant) that you've got paid when signing the Letter of Intent. The explanation behind this clause is that the landlord had paid the total one month's agent fee for a 2 years lease however in the event you terminate the lease by exercising the diplomatic clause, hence unable to complete the total 2 years, you will have to refund the professional-rata commission. Since landlord grants the diplomatic clause, they'll usually demand reimbursement clause to be included within the tenancy settlement.
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An entirely owned subsidiary of SLP Worldwide, SLP Realty homes 900 associates to service SLP's fast growing portfolio of residential projects. Actual estate is a human-centric trade. Moreover offering complete training packages for our associates, SLP Realty places equal emphasis on growing human capabilities and creating strong teamwork all through all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of pros is pushed to ensure our shoppers meet their enterprise and investment goals. Under is an inventory of some notable purchasers from completely different industries and markets, who have entrusted their actual estate needs to the experience of SLP International.
Most agents that I've seen are afraid to invest. They try to find methods to save lots of as a lot as they will. Effectively, I've nothing towards discovering methods to avoid wasting on promoting cost as a result of I do it too. But these agents are so afraid to the extent that they provide up advertising the property after just 1 attempt. Most agents are simply focusing more on methods to cut value when they need to be thinking where can I make investments my money to usher in more clients. With a imaginative and prescient for making optimistic waves within the trade, each males set out with a single purpose in mind – to contribute to the native property scene. They've gone to the sting to make sure that this goal of theirs is met, and can cease at nothing till it's so. In the true estate industry, only the fittest survive.
Soufan.com owned by Nasdaq-listed Sina Corp, Meiaoju.com and Auproperty.com.au are other marketplaces that work on a similar business model, appearing as intermediaries for Chinese language patrons and overseas agents. Nowadays consumers look at critiques before they make a purchase. Social media is sweet for this. They've more belief in me than those that discovered me by ads, as a result of they have been following my Weibo, and that hurries up the shopping for course of." The shift comes because the rental market holds its personal in the housing sector, which has been hit by slowing demand following several rounds of cooling measures and lending curbs, industry practitioners told RIGHT NOW. a. is resident in Singapore; a. he is resident in Singapore; Association of Singapore Patent Agents
For a simple example involving only the orientation of two observers, consider two people standing, facing each other on either side of a north-south street. See Figure 2. A car drives past them heading south. For the person facing east, the car was moving toward the right. However, for the person facing west, the car was moving toward the left. This discrepancy is because the two people used two different frames of reference from which to investigate this system.
For a more complex example involving observers in relative motion, consider Alfred, who is standing on the side of a road watching a car drive past him from left to right. In his frame of reference, Alfred defines the spot where he is standing as the origin, the road as the x-axis and the direction in front of him as the positive y-axis. To him, the car moves along the x axis with some velocity v in the positive x-direction. Alfred's frame of reference is considered an inertial frame of reference because he is not accelerating (ignoring effects such as Earth's rotation and gravity).
Now consider Betsy, the person driving the car. Betsy, in choosing her frame of reference, defines her location as the origin, the direction to her right as the positive x-axis, and the direction in front of her as the positive y-axis. In this frame of reference, it is Betsy who is stationary and the world around her that is moving – for instance, as she drives past Alfred, she observes him moving with velocity v in the negative y-direction. If she is driving north, then north is the positive y-direction; if she turns east, east becomes the positive y-direction.
Finally, as an example of non-inertial observers, assume Candace is accelerating her car. As she passes by him, Alfred measures her acceleration and finds it to be a in the negative x-direction. Assuming Candace's acceleration is constant, what acceleration does Betsy measure? If Betsy's velocity v is constant, she is in an inertial frame of reference, and she will find the acceleration to be the same as Alfred in her frame of reference, a in the negative y-direction. However, if she is accelerating at rate A in the negative y-direction (in other words, slowing down), she will find Candace's acceleration to be a′ = a − A in the negative y-direction - a smaller value than Alfred has measured. Similarly, if she is accelerating at rate A in the positive y-direction (speeding up), she will observe Candace's acceleration as a′ = a + A in the negative y-direction – a larger value than Alfred's measurement.
Frames of reference are especially important in special relativity, because when a frame of reference is moving at some significant fraction of the speed of light, then the flow of time in that frame does not necessarily apply in another frame. The speed of light is considered to be the only true constant between moving frames of reference.
Remarks
DTZ's public sale group in Singapore auctions all forms of residential, workplace and retail properties, outlets, homes, lodges, boarding homes, industrial buildings and development websites. Auctions are at present held as soon as a month.
We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.
The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.
Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.
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It is important to note some assumptions made above about the various inertial frames of reference. Newton, for instance, employed universal time, as explained by the following example. Suppose that you own two clocks, which both tick at exactly the same rate. You synchronize them so that they both display exactly the same time. The two clocks are now separated and one clock is on a fast moving train, traveling at constant velocity towards the other. According to Newton, these two clocks will still tick at the same rate and will both show the same time. Newton says that the rate of time as measured in one frame of reference should be the same as the rate of time in another. That is, there exists a "universal" time and all other times in all other frames of reference will run at the same rate as this universal time irrespective of their position and velocity. This concept of time and simultaneity was later generalized by Einstein in his special theory of relativity (1905) where he developed transformations between inertial frames of reference based upon the universal nature of physical laws and their economy of expression (Lorentz transformations).
It is also important to note that the definition of inertial reference frame can be extended beyond three dimensional Euclidean space. Newton's assumed a Euclidean space, but general relativity uses a more general geometry. As an example of why this is important, let us consider the geometry of an ellipsoid. In this geometry, a "free" particle is defined as one at rest or traveling at constant speed on a geodesic path. Two free particles may begin at the same point on the surface, traveling with the same constant speed in different directions. After a length of time, the two particles collide at the opposite side of the ellipsoid. Both "free" particles traveled with a constant speed, satisfying the definition that no forces were acting. No acceleration occurred and so Newton's first law held true. This means that the particles were in inertial frames of reference. Since no forces were acting, it was the geometry of the situation which caused the two particles to meet each other again. In a similar way, it is now common to describe[32] that we exist in a four dimensional geometry known as spacetime. In this picture, the curvature of this 4D space is responsible for the way in which two bodies with mass are drawn together even if no forces are acting. This curvature of spacetime replaces the force known as gravity in Newtonian mechanics and special relativity.
Non-inertial frames
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Here the relation between inertial and non-inertial observational frames of reference is considered. The basic difference between these frames is the need in non-inertial frames for fictitious forces, as described below.
An accelerated frame of reference is often delineated as being the "primed" frame, and all variables that are dependent on that frame are notated with primes, e.g. x′, y′, a′.
The vector from the origin of an inertial reference frame to the origin of an accelerated reference frame is commonly notated as R. Given a point of interest that exists in both frames, the vector from the inertial origin to the point is called r, and the vector from the accelerated origin to the point is called r′. From the geometry of the situation, we get
Taking the first and second derivatives of this, we obtain
where V and A are the velocity and acceleration of the accelerated system with respect to the inertial system and v and a are the velocity and acceleration of the point of interest with respect to the inertial frame.
These equations allow transformations between the two coordinate systems; for example, we can now write Newton's second law as
When there is accelerated motion due to a force being exerted there is manifestation of inertia. If an electric car designed to recharge its battery system when decelerating is switched to braking, the batteries are recharged, illustrating the physical strength of manifestation of inertia. However, the manifestation of inertia does not prevent acceleration (or deceleration), for manifestation of inertia occurs in response to change in velocity due to a force. Seen from the perspective of a rotating frame of reference the manifestation of inertia appears to exert a force (either in centrifugal direction, or in a direction orthogonal to an object's motion, the Coriolis effect).
A common sort of accelerated reference frame is a frame that is both rotating and translating (an example is a frame of reference attached to a CD which is playing while the player is carried). This arrangement leads to the equation (see Fictitious force for a derivation):
or, to solve for the acceleration in the accelerated frame,
Multiplying through by the mass m gives
where
Particular frames of reference in common use
- International Terrestrial Reference Frame
- International Celestial Reference Frame
- In fluid mechanics, Lagrangian and Eulerian specification of the flow field
Other frames
See also
Template:Col-begin Template:Col-1-of-3
- Analytical mechanics
- Applied mechanics
- Cartesian coordinate system
- Center-of-momentum frame
- Centrifugal force
- Centripetal force
- Classical mechanics
- Coriolis force
- Curvilinear coordinates
- Dynamics (physics)
- Fictitious force
- Frenet–Serret formulas
- Galilean invariance
- General relativity
- Generalized coordinates
- Generalized forces
- Inertial frame of reference
- Kinematics
- Laboratory frame of reference
- Lorentz transformation
- Mach's principle
- Orthogonal coordinates
- Principle of relativity
- Special relativity
- Theory of relativity
Notes
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- ↑ The distinction between macroscopic and microscopic frames shows up, for example, in electromagnetism where constitutive relations of various time and length scales are used to determine the current and charge densities entering Maxwell's equations. See, for example, 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534. These distinctions also appear in thermodynamics. See 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534. - ↑
In very general terms, a coordinate system is a set of arcs xi = xi (t) in a complex Lie group; see 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534. Less abstractly, a coordinate system in a space of n-dimensions is defined in terms of a basis set of vectors {e1, e2,… en}; see 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 As such, the coordinate system is a mathematical construct, a language, that may be related to motion, but has no necessary connection to motion. - ↑
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ John D. Norton (1993). General covariance and the foundations of general relativity: eight decades of dispute, Rep. Prog. Phys., 56, pp. 835-7.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 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." 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 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. - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ See Encarta definition. Archived 2009-10-31.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ See 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534. - ↑ For example, Møller states: "Instead of Cartesian coordinates we can obviously just as well employ general curvilinear coordinates for the fixation of points in physical space.…we shall now introduce general "curvilinear" coordinates xi in four-space…." 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ See 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534. - ↑ That is, both descriptions are equivalent and can be used as needed. This equivalence does not hold outside of general relativity, e.g., in entropic gravity.