In

physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

, scalars (or scalar quantities) are physical quantities
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 t ...

that are unaffected by changes to a vector space basis
In mathematics, a Set (mathematics), set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred ...

(i.e., a coordinate system
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

transformation). Scalars are often accompanied by units of measurement
A unit of measurement is a definite magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematic ...

, as in "10 cm". A change of a vector space basis changes the description of a vector in terms of the basis used but does not change the vector itself, while a scalar has nothing to do with this change. This physical definition of scalars, in classical theories, like Newtonian mechanics, means that rotations or reflections preserve scalars, while in relativistic theories, Lorentz transformation
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

s or space-time translations preserve scalars.
A scalar in physics is also a scalar in mathematics (as an element of a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

used to define a vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

). The magnitude (or length) of an electric is calculated as the square root
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of the inner product
In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...

of the electric field with itself and the outcome of the inner product is an element of the field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

for the vector space in which the electric field is described. As the field for the vector space in this example and usual cases in physics is the field of real numbers or complex numbers, the square root of the inner product is also an element of the field so it is a mathematically scalar. Since the inner product is independent of any vector space basis, the electric field magnitude is also a physically scalar. For a mass of an object that is unaffected by a change of a vector space basis so is a physically scalar, it is described by a real number as an element of the real number field. Since a field F is a vector space F over a field F, where addition defined on F is vector addition
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

and multiplication defined on F is scalar multiplication
250px, The scalar multiplications −a and 2a of a vector a
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

, the mass is also a mathematically scalar. Other quantities such as a distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

, charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* Charge (David Ford album), ''Charge'' (David Ford album)
* Charge (Machel Montano album), ''Charge'' (Mac ...

, volume
Volume is a scalar quantity expressing the amount
Quantity or amount is a property that can exist as a multitude
Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

, time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

, speed
In everyday use and in kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, bodies (objects), and systems of bodies (groups of objects) without considerin ...

(the magnitude of a velocity vector) are also mathematically and physically scalars in similar senses.
Scalar field

Since scalars mostly may be treated as special cases of multi-dimensional quantities such asvectors
Vector may refer to:
Biology
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector
*Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...

and tensor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

s, ''physical scalar fields'' might be regarded as a special case of more general fields, like vector field
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product ...

s, spinor fieldIn differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential g ...

s, and tensor field
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s.
Physical quantity

Like otherphysical quantities
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 t ...

, a physical quantity
Quantity 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 in terms of a unit of measu ...

of scalar is also typically expressed by a numerical value
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can b ...

and a physical unit
A unit of measurement is a definite magnitude (mathematics), magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can ...

, not merely a number, to provide its physical meaning. It may be regarded as the product of the number and the unit (e.g., 1 km as a physical distance is the same as 1,000 m). A physical distance does not depend on the length of each base vector of the coordinate system where the base vector length corresponds to the physical distance unit in use. (E.g., 1 m base vector length means the meter unit is used.) A physical distance differs from a metric
METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...

in the sense that it is not just a real number while the metric is calculated to a real number, but the metric can be converted to the physical distance by converting each base vector length to the corresponding physical unit.
Any change of a coordinate system may affect the formula for computing scalars (for example, the Euclidean formula for distance in terms of coordinates relies on the basis being orthonormalIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vector
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quanti ...

), but not the scalars themselves. Vectors themselves also do not change by a change of a coordinate system, but their descriptions changes (e.g., a change of numbers representing a position vector
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

by rotating a coordinate system in use).
Non-relativistic scalars

Temperature

An example of a scalar quantity istemperature
Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy
Thermal radiation in visible light can be seen on this hot metalwork.
Thermal energy refers to several distinct physical concept ...

: The temperature at a given point is a single number. Velocity, on the other hand, is a vector quantity.
Other examples

Some examples of scalar quantities in physics aremass
Mass is the quantity
Quantity 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 ...

, charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* Charge (David Ford album), ''Charge'' (David Ford album)
* Charge (Machel Montano album), ''Charge'' (Mac ...

, volume
Volume is a scalar quantity expressing the amount
Quantity or amount is a property that can exist as a multitude
Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

, time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

, speed
In everyday use and in kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, bodies (objects), and systems of bodies (groups of objects) without considerin ...

, and electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work
Work may refer to:
* Work (human activity), intentional activity people perform to support the ...

at a point inside a medium. The distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

between two points in three-dimensional space is a scalar, but the direction
Direction may refer to:
*Relative direction, for instance left, right, forward, backwards, up, and down
** Anatomical terms of location for those used in anatomy
*Cardinal direction
Mathematics and science
*Direction vector, a unit vector that ...

from one of those points to the other is not, since describing a direction requires two physical quantities such as the angle on the horizontal plane and the angle away from that plane. Force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

cannot be described using a scalar, since force has both direction and magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...

; however, the magnitude of a force alone can be described with a scalar, for instance the gravitation
Gravity (), or gravitation, is a natural phenomenon
Types of natural phenomena include:
Weather, fog, thunder, tornadoes; biological processes, decomposition, germination
seedlings, three days after germination.
Germination is t ...

al force acting on a particle is not a scalar, but its magnitude is. The speed of an object is a scalar (e.g. 180 km/h), while its velocity
The velocity of an object is the rate of change of its position with respect to a frame of reference
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical scie ...

is not (e.g. 108 km/h northward and 144 km/h westward).
Some other examples of scalar quantities in Newtonian mechanics are electric charge
Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like c ...

and charge density
In electromagnetism
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ...

.
Relativistic scalars

In thetheory of relativity
The theory of relativity usually encompasses two interrelated theories by Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born , widely acknowledged to be one of the greatest physicists of all time ...

, one considers changes of coordinate systems that trade space for time. As a consequence, several physical quantities that are scalars in "classical" (non-relativistic) physics need to be combined with other quantities and treated as four-vector
In special relativity
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in oth ...

s or tensors. For example, the charge density
In electromagnetism
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ...

at a point in a medium, which is a scalar in classical physics, must be combined with the local current density
In electromagnetism
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ...

(a 3-vector) to comprise a relativistic 4-vector. Similarly, energy density
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

must be combined with momentum density and pressure
Pressure (symbol: ''p'' or ''P'') is the force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

into the stress–energy tensor
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the Cauchy str ...

.
Examples of scalar quantities in relativity include electric charge
Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like c ...

, spacetime interval
In physics, spacetime is any mathematical model
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A sys ...

(e.g., proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural ...

and proper length
Proper length or rest length is the length of an object in the object's rest frame.
The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on t ...

), and invariant mass
Invariant and invariance may refer to: Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, invariants used to prove properties of loops
* A data type in method ove ...

.
See also

* Relative scalar *Pseudoscalar
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not.
Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. ...

**An example of a pseudoscalar is the scalar triple product
In geometry and algebra, the triple product is a product of three 3-dimension (vector space), dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, ...

(see vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

), and thus the signed volume. Another example is magnetic charge
In particle physics
Particle physics (also known as high energy physics) is a branch of physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is th ...

(as it is mathematically defined, regardless of whether it actually exists physically).
*Scalar (mathematics)
A scalar is an element of a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an ar ...

Notes

References

* Feynman, Leighton & Sands 1963. * * {{DEFAULTSORT:Scalar (Physics)