@The Pentacle Queen,
Here is an excerpt from Wikipedia about Dimensional Analysis. (It is beginning to dawn on me that maybe I don't need textbooks as long as I know what I want I can simply get it from the Internet.)
Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving certain physical quantities. It is routinely used by mathematicians, statisticians, physical scientists and engineers to check the plausibility of derived equations and computations. It is also used to form reasonable hypotheses about complex physical situations that can be tested by experiment or by more developed theories of the phenomena, and to categorise types of physical quantities and units based on their relations to or dependence on other units, or their "dimensions", or their lack thereof.
Dimensional analysis was developed by the 19th century French mathematician Joseph Fourier.[1], based on the idea that the physical laws like F = ma should be independent of the units employed to measure the physical variables. This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually formalized in the Buckingham π theorem. This theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n ??' m dimensionless parameters, where m is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.
A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization, which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or natural units of nature. This gives insight into the fundamental properties of the system, as illustrated in the examples below.
Definition
The dimensions of a physical quantity are associated with combinations of mass, length, time, electric charge, and temperature, represented by symbols M, L, T, Q, and Θ, respectively, each raised to rational powers.
Note that dimension is more abstract than scale unit: mass is a dimension, while kilograms are a scale unit (choice of standard) in the mass dimension.
As examples, the dimension of the physical quantity speed is "distance/time" (L/T or LT ??'1), and the dimension of the physical quantity force is "mass ?- acceleration" or "mass?-(distance/time)/time" (ML/T 2 or MLT ??'2). In principle, other dimensions of physical quantity could be defined as "fundamental" (such as momentum or energy or electric current) in lieu of some of those shown above. Most physicists do not recognize temperature, Θ, as a fundamental dimension of physical quantity since it essentially expresses the energy per particle per degree of freedom, which can be expressed in terms of energy (or mass, length, and time). Still others do not recognize electric charge, Q, as a separate fundamental dimension of physical quantity, since it has been expressed in terms of mass, length, and time in unit systems such as the electrostatic cgs system. There are also physicists that have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity.[2]
The unit of a physical quantity and its dimension are related, but not precisely identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g., length may have units of metres, feet, inches, miles or micrometres; but any length always has a dimension of L, independent of what units are arbitrarily chosen to measure it. Two different units of the same physical quantity have conversion factors between them. For example: 1 in = 2.54 cm; then (2.54 cm/in) is called a conversion factor (between two representations expressed in different units of a common quantity) and is itself dimensionless and equal to one. There are no conversion factors between dimensional symbols.
[edit] Mathematical properties
Dimensional symbols, such as L, form a group: There is an identity, L0 = 1; there is an inverse to L, which is 1/L or L??'1, and L raised to any rational power p is a member of the group, having an inverse of L??'p or 1/Lp. The operation of the group is multiplication, with the usual rules for handling exponents (Ln ?- Lm = Ln+m).
Indeed, dimensional symbols can be seen as a vector space over the rational numbers, with the coordinates of the vector being the powers of the exponents " expressing a dimensional symbol as MiLjTk corresponds to the vector (i,j,k). A basis for a given space of dimensional symbols is called a set of fundamental quantities or fundamental dimensions.
When quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by each other, their dimensional symbols are likewise multiplied or divided; this corresponds to vector addition or subtraction (on the exponents). When dimensioned quantities are raised to a rational power, the same is done to the dimensional symbols attached to those quantities; this corresponds to scalar multiplication on the exponents.
As in any vector space, one may choose different bases, which yield different physical interpretations.
[edit] Mechanics
In mechanics, the dimension of any physical quantity can be expressed in terms of the fundamental dimensions (or base dimensions) M, L and T " these form a 3-dimensional vector space. This is not the only possible choice, but it is the one most commonly used. For example, one might choose force, length and mass as the base dimensions (as some have done), with associated dimensions F, L, M; this corresponds to a different basis, and one may convert between these representations by a change of basis. The choice of the base set of dimensions is, thus, partly a convention, resulting in increased utility and familiarity. It is, however, important to note that the choice of the set of dimensions cannot be chosen arbitrarily " it is not just a convention " because the dimensions must form a basis: they must span the space, and be linearly independent.
For example, F, L, M form a set of fundamental dimensions because they form an equivalent basis to M, L, T: the former can be expressed as [F = ML / T2],L,M, while the latter can be expressed as M,L,[T = (ML / F)1 / 2].
On the other hand, using length, velocity and time (L, V, T) as base dimensions will not work well (they do not form a set of fundamental dimensions), for two reasons:
* Firstly, because there is no way to obtain mass " or anything derived from it, such as force " without introducing another base dimension (thus these do not span the space).
* Secondly, because velocity, being derived from length and time (V = L / T), is redundant (the set is not linearly independent).
http://en.wikipedia.org/wiki/Dimensional_analysis
