On Coordinates (2000-2009)

"By definition, a Kähler manifold is one with a complex structure (this means in particular that the coordinates changes are holomorphic for the complex coordinates) together with a Riemannian metric which has with this complex structure the best possible link, namely that multiplication of tangent vectors by unit complex numbers preserves the metric, but moreover the complex structure is invariant under parallel transport. This is equivalent to the condition that the holonomy group be included in the unitary group, hence equivalent also to ask for the existence of a 2-form of maximal rank and of zero covariant derivative."(Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"Descartes’ idea to use numbers to describe points in space involves the choice of a coordinate system or coordinate frame: an origin, together with axes and units of length along the axes. A recurring theme of all the different geometries [...] is the question of what a coordinate frame is, and what I can get out of it. While coordinates provide a convenient framework to discuss points, lines, and so on, it is a basic requirement that any meaningful statement in geometry is independent of the choice of coordinates. That is, coordinate frames are a humble technical aid in determining the truth, and are not allowed the dignity of having their own meaning." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"Descartes’ invention of coordinate geometry is another key ingredient in modern science. It is scarcely an accident that calculus was discovered by Leibnitz and Newton (independently, alphabetical order) in the fifty years following the dissemination of Descartes’ ideas. Interactions between the axiomatic and the coordinate-based points of view go in both ways: coordinate geometry gives models of axiomatic geometries, and conversely, axiomatic geometries allow the introduction of number systems and coordinates." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"Minkowski calls a spatial point existing at a temporal point a world point. These coordinates are now called 'space-time coordinates'. The collection of all imaginable value systems or the set of space-time coordinates Minkowski called the world. This is now called the manifold. The manifold is four-dimensional and each of its space-time points represents an event." (Friedel Weinert," The Scientist as Philosopher: Philosophical Consequences of Great Scientific Discoveries", 2005)

"Roughly speaking, a manifold is essentially a space that is locally similar to the Euclidean space. This resemblance permits differentiation to be defined. On a manifold, we do not distinguish between two different local coordinate systems. Thus, the concepts considered are just those independent of the coordinates chosen. This makes more sense if we consider the situation from the physics point of view. In this interpretation, the systems of coordinates are systems of reference." (Ovidiu Calin & Der-Chen Chang,  "Geometric Mechanics on Riemannian Manifolds : Applications to partial differential equations", 2005)

"When real numbers are used as coordinates, the number of coordinates is the dimension of the geometry. This is why we call the plane two-dimensional and space three-dimensional. However, one can also expect complex numbers to be useful, knowing their geometric properties […] What is remarkable is that complex numbers are if anything more appropriate for spherical and hyperbolic geometry than for Euclidean geometry. With hindsight, it is even possible to see hyperbolic geometry in properties of complex numbers that were studied as early as 1800, long before hyperbolic geometry was discussed by anyone." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"Coordinates and vectors - in one form or another - are two of the most fundamental concepts in any discussion of mathematics as applied to physical problems. So, it is beneficial to start our study with these two concepts. Both vectors and coordinates have generalizations that cover a wide variety of physical situations including not only ordinary three-dimensional space with its ordinary vectors, but also the four-dimensional spacetime of relativity with its so-called four vectors, and even the infinite-dimensional spaces used in quantum physics with their vectors of infinite components." (Sadri Hassani, "Mathematical Methods: For Students of Physics and Related Fields", 2009)

"Coordinates are 'functions' that specify points of a space. The smallest number of these functions necessary to specify a point is called the dimension of that space. For instance, a point of a plane is specified by two numbers, and as the point moves in the plane the two numbers change, i.e., the coordinates are functions of the position of the point." (Sadri Hassani, "Mathematical Methods: For Students of Physics and Related Fields", 2009)

"One of the great advantages of vectors is their ability to express results independent of any specific coordinate systems. Physical laws are always coordinate-independent. For example, when we write F = ma both F and a could be expressed in terms of Cartesian, spherical, cylindrical, or any other convenient coordinate system. This independence allows us the freedom to choose the coordinate systems most convenient for the problem at hand. For example, it is extremely difficult to solve the planetary motions in Cartesian coordinates, while the use of spherical coordinates facilitates the solution of the problem tremendously." (Sadri Hassani, "Mathematical Methods: For Students of Physics and Related Fields", 2009)

"One of the greatest achievements in the development of mathematics since Euclid was the introduction of coordinates. Two men take credit for this development: Fermat and Descartes. These two great French mathematicians were interested in the unification of geometry and algebra, which resulted in the creation of a most fruitful branch of mathematics now called analytic geometry. Fermat and Descartes who were heavily involved in physics, were keenly aware of both the need for quantitative methods and the capacity of algebra to deliver that method." (Sadri Hassani, "Mathematical Methods: For Students of Physics and Related Fields", 2009)

"The concept of symmetry is used widely in physics. If the laws that determine relations between physical magnitudes and a change of these magnitudes in the course of time do not vary at the definite operations (transformations), they say, that these laws have symmetry (or they are invariant) with respect to the given transformations. For example, the law of gravitation is valid for any points of space, that is, this law is in variant with respect to the system of coordinates." (Alexey Stakhov et al, "The Mathematics of Harmony", 2009)

"Which one of the three systems of coordinates to use in a given physical problem is dictated mainly by the geometry of that problem. As a rule, spherical coordinates are best suited for spheres and spherically symmetric problems. Spherical symmetry describes situations in which quantities of interest are functions only of the distance from a fixed point and not on the orientation of that distance. Similarly, cylindrical coordinates ease calculations when cylinders or cylindrical symmetries are involved. Finally, Cartesian coordinates are used in rectangular geometries." (Sadri Hassani, "Mathematical Methods: For Students of Physics and Related Fields", 2009)