Dimensional analysis: Difference between revisions

In [[engineering]] and [[science]], '''dimensional analysis''' is the analysis of the relationships between different [[Physical quantity|physical quantities]] by identifying their [[base quantity|base quantities]] (such as [[length]], [[mass]], [[time]], and [[electric current]]) and [[units of measurement]] (such as metersmetres and grams) and tracking these dimensions as calculations or comparisons are performed. The term dimensional analysis is also used to refer to [[conversion of units]] from one dimensional unit to another, which can be used to evaluate scientific formulae.

'''''Commensurable''''' physical quantities are of the same [[Kind of quantity|kind]] and have the same dimension, and can be directly compared to each other, even if they are expressed in differing units of measurement; e.g., metersmetres and feet, gallonsgrams and literspounds, seconds and years. ''Incommensurable'' physical [[Quantity|quantities]] are of different [[Kind of quantity|kinds]] and have different dimensions, and can not be directly compared to each other, no matter what [[units]] they are expressed in, e.g. metres and grams, seconds and grams, metres and seconds. For example, asking whether a gram is larger than an hour is meaningless.

Any physically meaningful [[equation]], or [[inequality (mathematics)|inequality]], ''must'' have the same dimensions on its left and right sides, a property known as ''dimensional homogeneity''. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on [[Formal proof|derived]] equations and [[Computation|computations]]. It also serves as a guide and constraint in deriving equations that may describe a physical [[system]] in the absence of a more rigorous derivation.

The dimension of the physical quantity [[electric potential differencevoltage]] {{math|''V''}} is

A related principle is that any physical law that accurately describes the real world must be independent of the units used to measure the physical variables.<ref>{{Cite book |last1=de Jong |first1=Frits J. |url=https://archive.org/details/dimensionalanaly0000jong |title=Dimensional analysis for economists |last2=Quade |first2=Wilhelm |publisher=North Holland |year=1967 |page=[https://archive.org/details/dimensionalanaly0000jong/page/28 28] |url-access=registration}}</ref> For example, [[Newton's laws of motion]] must hold true whether distance is measured in miles or kilometres. This principle gives rise to the form that a conversion factor between atwo unitunits that measuresmeasure the same dimension must take: multiplication by a simple constant. It also ensures equivalence; for example, if two buildings are the same height in feet, then they must be the same height in metres.

A simple application of dimensional analysis to mathematics is in computing the form of the [[N-sphere#Volume of the n-ball|volume of an {{math|''n''}}-ball]] (the solid ball in ''n'' dimensions), or the area of its surface, the [[n-sphere|{{math|''n''}}-sphere]]: being an {{math|''n''}}-dimensional figure, the volume scales as {{math|''x''{{sup|''n''}}}}, while the surface area, being {{math|(''n'' − 1)}}-dimensional, scales as {{math|''x''{{sup|''n''−1}}}}. Thus the volume of the {{math|''n''}}-ball in terms of the radius is {{math|''C''{{isupsub|''n''}}''r''{{isup|''n''}}}}, for some constant {{math|''C''{{isupsub|''n''}}}}. Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone.

Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. (In mathematics scalars are considered a special case of vectors;{{citation needed|date=September 2013}} vectors can be added to or subtracted from other vectors, and, inter alia, multiplied or divided by scalars. If a vector is used to define a position, this assumes an implicit point of reference: an [[origin (mathematics)|origin]]. While this is useful and often perfectly adequate, allowing many important errors to be caught, it can fail to model certain aspects of physics. A more rigorous approach requires distinguishing between position and displacement (or moment in time versus duration, or absolute temperature versus temperature change).