Sadri Hassani - Collected Quotes

"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)

"Differential equations have found their way into all areas of physics from the motion of planets around the Sun to standing waves on a rope or a drum, to electrical properties of conductors, and the behavior of electromagnetic fields and beyond. As is always the case, no mathematics can draw more attention than that which deals directly with Nature. The urgency of finding solutions to these differential equations prompted many mathematicians of the latter part of the eighteenth and the beginning of the nineteenth centuries to concentrate heavily on certain specific differential equations. It appeared that every differential equation dictated by Nature gave rise to a new function. The most common scheme for solving these differential equations was to assume a power series solution, substitute the assumed solution in the differential equation, and determine the (unknown) coefficients from the resulting equality of power series." (Sadri Hassani, "Mathematical Methods: For Students of Physics and Related Fields", 2009)

"How do we specify the “state” of a fluid? A fluid is an extended object, different parts of which behave differently. Attaching a vector to different points of the fluid to represent the velocity at that point, and taking snapshots at different times, we can get an idea of how the fluid behaves. [...] A complete determination of the fluid, therefore, entails a specification of the velocity vector at different points of the fluid for different times. A vector which varies from point to point is called a vector field. A problem involving a classical fluid is, therefore, solved vector field once we find its velocity field as a function of position and time. The concept of a field can be abstracted from the physical reality of the fluid. It then becomes a legitimate physical entity whose specification requires a position, a time, and a direction (if the field happens to be a vector field), just like the specification of the velocity field of a fluid." (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)

"Physics deals with both the large and the small. Its domain of study includes the interior of the nucleus of an atom as well as the exterior of a galaxy. It is, therefore, natural for the scope of physical theories to switch between global, or large-scale, and local, or small-scale regimes. Such an interplay between the local and the global has existed ever since Newton and others discovered the mathematical translation of this interplay: Derivatives are defined as local objects while integrals encompass global properties." (Sadri Hassani, "Mathematical Methods: For Students of Physics and Related Fields", 2009)

"The concept of a field was actually introduced first in the context of electricity and magnetism by Michael Faraday as a means of 'visualizing' electromagnetic effects to replace certain mathematical ideas for which he had little talent. However, in the hands of James Maxwell, fields were molded into a physical entity having an existence of their own in the form of electromagnetic waves to be produced in 1887 by Hertz and used in 1901 by Marconi in the development of radio." (Sadri Hassani, "Mathematical Methods: For Students of Physics and Related Fields", 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)