Physical quantities

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A physical quantity is a
physical property A physical property is any property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have ...
of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For example, the physical quantity of mass can be quantified as '32.3 kg ', where '32.3' is the numerical value and 'kg' is the Unit. A physical quantity possesses at least two characteristics in common. # Numerical magnitude. # Units

# Symbols and nomenclature

International recommendations for the use of symbols for quantities are set out in ISO/IEC 80000, the IUPAP red book and the IUPAC green book. For example, the recommended symbol for the physical quantity ''mass'' is ''m'', and the recommended symbol for the quantity ''electric charge'' is ''Q''.

# Subscripts and indices

Subscripts are used for two reasons, to simply attach a name to the quantity or associate it with another quantity, or index a specific component (e.g., row or column). *Name reference: The quantity has a
subscript A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the Baseline (typog ...

ed or
superscript A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the Baseline (typog ...

ed single letter, group of letters, or complete word, to label what concept or entity they refer to, often to distinguish it from other quantities with the same main symbol. These subscripts or superscripts tend to be written in upright roman typeface rather than italic while the main symbol representing the quantity is in italic. For instance ''E''k or ''E''kinetic is usually used to denote
kinetic energy In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical sci ...
and ''E'' p or ''E'' potential is usually used to denote
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...

. *Quantity reference: The quantity has a
subscript A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the Baseline (typog ...

ed or
superscript A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the Baseline (typog ...

ed single letter, group of letters, or complete word, to parameterize what measurement/s they refer to. These subscripts or superscripts tend to be written in italic rather than upright roman typeface; the main symbol representing the quantity is in italic. For example ''cp'' or ''cpressure'' is
heat capacity Heat capacity or thermal capacity is a physical quantity, physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The International System of Units, SI unit of heat ca ...
at the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...

given by the quantity in the subscript. The type of subscript is expressed by its typeface: 'k' and 'p' are abbreviations of the words ''kinetic'' and ''potential'', whereas ''p'' (italic) is the symbol for the physical quantity ''pressure'' rather than an abbreviation of the word. *Indices: The use of indices is for mathematical formalism using
index notation In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to the ...
.

# Size

Physical quantities can have different "sizes", as a scalar, a vector, or a tensor.

## Scalars

A scalar is a physical quantity that has magnitude but no direction. Symbols for physical quantities are usually chosen to be a single letter of the
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...

or
Greek alphabet The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as we ...

, and are printed in italic type.

## Vectors

Vectors are physical quantities that possess both magnitude and direction and whose operations obey the
axioms An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
of a
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
. Symbols for physical quantities that are vectors are in bold type, underlined or with an arrow above. For example, if ''u'' is the speed of a particle, then the straightforward notations for its velocity are u, u, or $\vec\,\!$.

## Tensors

Scalars and vectors are the simplest
tensor In mathematics, a tensor is an mathematical object, algebraic object that describes a Multilinear map, multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as Vect ...
s, which can be used to describe more general physical quantities. For example, the
Cauchy stress tensor In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completel ...
possess magnitude, direction, and orientation qualities.

# Numbers and elementary functions

Numerical quantities, even those denoted by letters, are usually printed in roman (upright) type, though sometimes in italic. Symbols for elementary functions (circular trigonometric, hyperbolic, logarithmic etc.), changes in a quantity like Δ in Δ''y'' or operators like d in d''x'', are also recommended to be printed in roman type. Examples: *Real numbers, such as 1 or , *e, the base of
natural logarithms The natural logarithm of a number is its logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produc ...
, *i, the imaginary unit, *π for the ratio of a circle's circumference to its diameter, 3.14159265358979323846264338327950288... *δ''x'', Δ''y'', d''z'', representing differences (finite or otherwise) in the quantities ''x'', ''y'' and ''z'' *sin ''α'', sinh ''γ'', log ''x''

# Units and dimensions

## Units

There is often a choice of unit, though SI units (including submultiples and multiples of the basic unit) are usually used in scientific contexts due to their ease of use, international familiarity and prescription. For example, a quantity of mass might be represented by the symbol ''m'', and could be expressed in the units
kilogram The kilogram (also kilogramme) is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially. ...
s (kg), pounds (lb), or daltons (Da).

## Dimensions

The notion of ''dimension'' of a physical quantity was introduced by
Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French people, French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier an ...
in 1822.Fourier, Joseph. '' Théorie analytique de la chaleur'', Firmin Didot, Paris, 1822. (In this book, Fourier introduces the concept of ''physical dimensions'' for the physical quantities.) By convention, physical quantities are organized in a dimensional system built upon base quantities, each of which is regarded as having its own dimension.

# Base quantities

Base quantities are those quantities which are distinct in nature and in some cases have historically not been defined in terms of other quantities. Base quantities are those quantities on the basis of which other quantities can be expressed. The seven base quantities of the
International System of Quantities The International System of Quantities (ISQ) consists of the Quantity, quantities used in physics and in modern science in general, starting with basic quantities such as length and mass, and the relationships between those quantities. This syste ...
(ISQ) and their corresponding SI units and dimensions are listed in the following table. Other conventions may have a different number of base units (e.g. the CGS and MKS systems of units). The last two angular units,
plane angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
and
solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poi ...
, are subsidiary units used in the SI, but are treated as dimensionless. The subsidiary units are used for convenience to differentiate between a ''truly dimensionless'' quantity (pure number) and an ''angle'', which are different measurements.

# General derived quantities

Derived quantities are those whose definitions are based on other physical quantities (base quantities).

## Space

Important applied base units for space and time are below.
Area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an o ...
and
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
are thus, of course, derived from the length, but included for completeness as they occur frequently in many derived quantities, in particular densities.

## Densities, flows, gradients, and moments

Important and convenient derived quantities such as densities,
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
es, flows,
current Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...
s are associated with many quantities. Sometimes different terms such as ''current density'' and ''flux density'', ''rate'', ''frequency'' and ''current'', are used interchangeably in the same context, sometimes they are used uniquely. To clarify these effective template derived quantities, we let ''q'' be ''any'' quantity within some scope of context (not necessarily base quantities) and present in the table below some of the most commonly used symbols where applicable, their definitions, usage, SI units and SI dimensions – where 'q''denotes the dimension of ''q''. For time derivatives, specific, molar, and flux densities of quantities there is no one symbol, nomenclature depends on the subject, though time derivatives can be generally written using overdot notation. For generality we use ''qm'', ''qn'', and F respectively. No symbol is necessarily required for the gradient of a scalar field, since only the nabla/del operator ∇ or grad needs to be written. For spatial density, current, current density and flux, the notations are common from one context to another, differing only by a change in subscripts. For current density, $\mathbf$ is a unit vector in the direction of flow, i.e. tangent to a flowline. Notice the dot product with the unit normal for a surface, since the amount of current passing through the surface is reduced when the current is not normal to the area. Only the current passing perpendicular to the surface contributes to the current passing ''through'' the surface, no current passes ''in'' the (tangential) plane of the surface. The calculus notations below can be used synonymously. If ''X'' is a ''n''-variable function $X \equiv X \left \left( x_1, x_2 \cdots x_n \right \right)$, then ''Differential'' The differential ''n''-space
volume element In mathematics, a volume element provides a means for Integral, integrating a function (mathematics), function with respect to volume in various coordinate systems such as Spherical coordinate system, spherical coordinates and Cylindrical coordinate ...
is $\mathrm^n x \equiv \mathrm V_n \equiv \mathrm x_1 \mathrm x_2 \cdots \mathrm x_n$, :''Integral'': The ''multiple'' integral of ''X'' over the ''n''-space volume is $\int X \mathrm^n x \equiv \int X \mathrm V_n \equiv \int \cdots \int \int X \mathrm x_1 \mathrm x_2 \cdots \mathrm x_n \,\!$. The meaning of the term physical ''quantity'' is generally well understood (everyone understands what is meant by ''the frequency of a periodic phenomenon'', or ''the resistance of an electric wire''). The term ''physical quantity'' does not imply a physically ''invariant quantity''. ''Length'' for example is a ''physical quantity'', yet it is variant under coordinate change in special and general relativity. The notion of physical quantities is so basic and intuitive in the realm of science, that it does not need to be explicitly ''spelled out'' or even ''mentioned''. It is universally understood that scientists will (more often than not) deal with quantitative data, as opposed to qualitative data. Explicit mention and discussion of ''physical quantities'' is not part of any standard science program, and is more suited for a ''philosophy of science'' or ''philosophy'' program. The notion of ''physical quantities'' is seldom used in physics, nor is it part of the standard physics vernacular. The idea is often misleading, as its name implies "a quantity that can be physically measured", yet is often incorrectly used to mean a physical invariant. Due to the rich complexity of physics, many different fields possess different physical invariants. There is no known physical invariant sacred in all possible fields of physics. Energy, space, momentum, torque, position, and length (just to name a few) are all found to be experimentally variant in some particular scale and system. Additionally, the notion that it is possible to measure "physical quantities" comes into question, particularly in quantum field theory and normalization techniques. As infinities are produced by the theory, the actual “measurements” made are not really those of the physical universe (as we cannot measure infinities), they are those of the renormalization scheme which is expressly dependent on our measurement scheme, coordinate system and metric system.

# See also

* List of physical quantities *
Philosophy of science Philosophy of science is a branch of philosophy concerned with the foundations, methodology, methods, and implications of science. The central questions of this study concern Demarcation problem, what qualifies as science, the reliability of s ...
*
Quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a u ...
** Observable quantity **
Specific quantity In the natural sciences, including physiology and engineering, a specific quantity generally refers to an intensive quantity obtained by dividing an extensive quantity of interest by mass. For example, specific leaf area is leaf area divided by le ...

# References

## Computer implementations

DEVLIB
project in C#
Language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
and
Delphi Delphi (; ), in legend previously called Pytho (Πυθώ), in ancient times was a sacred precinct that served as the seat of Pythia, the major oracle who was consulted about important decisions throughout the ancient classical world. The oracle ...
Language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...

Physical Quantities
project in C#
Language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
at Code Plex
Physical Measure C# library
project in C#
Language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
at Code Plex
Ethical Measures
project in C#
Language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
at Code Plex
Engineer JS
online calculation and scripting tool supporting physical quantities.

# Sources

* Cook, Alan H. ''The observational foundations of physics'', Cambridge, 1994. * Essential Principles of Physics, P.M. Whelan, M.J. Hodgson, 2nd Edition, 1978, John Murray, * Encyclopedia of Physics, R.G. Lerner, G.L. Trigg, 2nd Edition, VHC Publishers, Hans Warlimont, Springer, 2005, pp 12–13 * Physics for Scientists and Engineers: With Modern Physics (6th Edition), P.A. Tipler, G. Mosca, W.H. Freeman and Co, 2008, 9-781429-202657 {{DEFAULTSORT:Physical Quantity