View source for Scalar (physics)

= Scalar (Physics): Definition and Mathematical Representation =

== Introduction ==
In physics, a '''scalar''' is a quantity that is fully described by a single numerical value (magnitude) and has no direction. Scalars are used to measure and represent physical properties that do not depend on orientation in space.

Scalars contrast with [[Vector (Physics)|vectors]], which require both magnitude and direction for complete description.

== Definition ==

A scalar quantity is defined as:

<math>
S \in \mathbb{R}
</math>

Where:
* <math>S</math> is a scalar quantity,
* <math>\mathbb{R}</math> denotes the set of real numbers.

Scalars remain unchanged under rotation or transformation of coordinate systems, making them frame-independent.

== Examples of Scalar Quantities ==

* [[Mass]] (e.g., 5 kg)
* [[Temperature]] (e.g., 300 K)
* [[Time]] (e.g., 10 s)
* [[Energy]] (e.g., 50 J)
* [[Speed]] (e.g., 20 m/s)
* [[Work (Physics)|Work]] (e.g., 12 J)
* [[Power]] (e.g., 100 W)
* [[Electric charge]] (e.g., 2 C)
* [[Volume]] (e.g., 3.5 m³)

== SI Units of Scalars ==

Each scalar quantity has an associated SI unit:

{| class="wikitable"
! Quantity !! SI Unit !! Symbol
|-
| Mass || kilogram || kg
|-
| Time || second || s
|-
| Temperature || kelvin || K
|-
| Energy || joule || J
|-
| Speed || meter per second || m/s
|}

== Mathematical Operations with Scalars ==

* Scalars can be added, subtracted, multiplied, or divided using standard arithmetic.
* Scalars can multiply vectors (scalar multiplication):

<math>
k\vec{A} = (k A_x)\hat{i} + (k A_y)\hat{j} + (k A_z)\hat{k}
</math>

Where <math>k</math> is a scalar, and <math>\vec{A}</math> is a vector.

== Scalars vs. Vectors ==

{| class="wikitable"
! Property !! Scalars !! Vectors
|-
| Definition || Only magnitude || Magnitude and direction
|-
| Direction || Not applicable || Required
|-
| Examples || Time, mass, energy || Force, velocity, displacement
|-
| Notation || Plain letters (e.g., <math>t</math>, <math>m</math>) || Bold or arrowed letters (e.g., <math>\vec{v}</math>)
|}

== Use in Physics ==
Scalars are essential in:
* Thermodynamics (temperature, entropy)
* Mechanics (mass, energy, work)
* Electromagnetism (potential, charge)
* Kinematics (speed)

== See Also ==
* [[Vector]]
* [[Speed]]
* [[Time]]
* [[Mass]]
* [[Work]]
* [[Energy]]
* [[Power]]
* [[Scalar Field]]