Editing Pseudovector (section)

==Details==
{{See also|Covariance and contravariance of vectors|Euclidean vector}}

The definition of a "vector" in physics (including both polar vectors and pseudovectors) is more specific than the mathematical definition of "vector" (namely, any element of an abstract [[vector space]]). Under the physics definition, a "vector" is required to have [[tuple|components]] that "transform" in a certain way under a [[rotation (mathematics)|proper rotation]]: In particular, if everything in the universe were rotated, the vector would rotate in exactly the same way. (The coordinate system is fixed in this discussion; in other words this is the perspective of [[active and passive transformation|active transformations]].) Mathematically, if everything in the universe undergoes a rotation described by a [[rotation matrix]] ''R'', so that a [[displacement vector]] '''x''' is transformed to {{nowrap|1='''x'''{{prime}} = ''R'''''x'''}}, then any "vector" '''v''' must be similarly transformed to {{nowrap|1='''v'''{{prime}} = ''R'''''v'''}}. This important requirement is what distinguishes a ''vector'' (which might be composed of, for example, the ''x''-, ''y''-, and ''z''-components of [[velocity]]) from any other triplet of physical quantities (For example, the length, width, and height of a rectangular box ''cannot'' be considered the three components of a vector, since rotating the box does not appropriately transform these three components.)

(In the language of [[differential geometry]], this requirement is equivalent to defining a ''vector'' to be a [[tensor]] of [[Covariance and contravariance of vectors|contravariant]] rank one. In this more general framework, higher rank tensors can also have arbitrarily many and mixed covariant and contravariant ranks at the same time, denoted by raised and lowered indices within the [[Einstein summation convention]].)

A basic and rather concrete example is that of row and column vectors under the usual matrix multiplication operator: in one order they yield the dot product, which is just a scalar and as such a rank zero tensor, while in the other they yield the [[dyadic product]], which is a matrix representing a rank two mixed tensor, with one contravariant and one covariant index. As such, the noncommutativity of standard matrix algebra can be used to keep track of the distinction between covariant and contravariant vectors. This is in fact how the bookkeeping was done before the more formal and generalised tensor notation came to be. It still manifests itself in how the basis vectors of general tensor spaces are exhibited for practical manipulation.

The discussion so far only relates to proper rotations, i.e. rotations about an axis. However, one can also consider [[improper rotation]]s, i.e. a mirror-reflection possibly followed by a proper rotation. (One example of an improper rotation is [[inversion through a point]] in 3-dimensional space.) Suppose everything in the universe undergoes an improper rotation described by the improper rotation matrix ''R'', so that a position vector '''x''' is transformed to {{nowrap|1='''x'''{{prime}} = ''R'''''x'''}}. If the vector '''v''' is a polar vector, it will be transformed to {{nowrap|1='''v'''{{prime}} = ''R'''''v'''}}. If it is a pseudovector, it will be transformed to {{nowrap|1='''v'''{{prime}} = −''R'''''v'''}}.

The transformation rules for polar vectors and pseudovectors can be compactly stated as

: <math>
\begin{align}
\mathbf{v}' & = R\mathbf{v} & & \text{(polar vector)} \\
\mathbf{v}' & = (\det R)(R\mathbf{v}) & & \text{(pseudovector)}
\end{align}
</math>

where the symbols are as described above, and the rotation matrix ''R'' can be either proper or improper. The symbol det denotes [[determinant]]; this formula works because the determinant of proper and improper rotation matrices are +1 and −1, respectively.

===Behavior under addition, subtraction, scalar multiplication===

Suppose '''v'''<sub>1</sub> and '''v'''<sub>2</sub> are known pseudovectors, and '''v'''<sub>3</sub> is defined to be their sum, {{nowrap|1='''v'''<sub>3</sub> = '''v'''<sub>1</sub> + '''v'''<sub>2</sub>}}. If the universe is transformed by a rotation matrix ''R'', then '''v'''<sub>3</sub> is transformed to

: <math>
\begin{align}
\mathbf{v_3}' = \mathbf{v_1}'+\mathbf{v_2}' & = (\det R)(R\mathbf{v_1}) + (\det R)(R\mathbf{v_2}) \\
& = (\det R)(R(\mathbf{v_1}+\mathbf{v_2}))=(\det R)(R\mathbf{v_3}).
\end{align}
</math>

So '''v'''<sub>3</sub> is also a pseudovector. Similarly one can show that the difference between two pseudovectors is a pseudovector, that the sum or difference of two polar vectors is a polar vector, that multiplying a polar vector by any real number yields another polar vector, and that multiplying a pseudovector by any real number yields another pseudovector.

On the other hand, suppose '''v'''<sub>1</sub> is known to be a polar vector, '''v'''<sub>2</sub> is known to be a pseudovector, and '''v'''<sub>3</sub> is defined to be their sum, {{nowrap|1='''v'''<sub>3</sub> = '''v'''<sub>1</sub> + '''v'''<sub>2</sub>}}. If the universe is transformed by an improper rotation matrix ''R'', then '''v'''<sub>3</sub> is transformed to

: <math>
\mathbf{v_3}' = \mathbf{v_1}'+\mathbf{v_2}' = (R\mathbf{v_1}) + (\det R)(R\mathbf{v_2}) = R(\mathbf{v_1}+(\det R) \mathbf{v_2}).
</math>

Therefore, '''v'''<sub>3</sub> is neither a polar vector nor a pseudovector (although it is still a vector, by the physics definition). For an improper rotation, '''v'''<sub>3</sub> does not in general even keep the same magnitude:

: <math>|\mathbf{v_3}| = |\mathbf{v_1}+\mathbf{v_2}|, \text{ but } \left|\mathbf{v_3}'\right| = \left|\mathbf{v_1}'-\mathbf{v_2}'\right|</math>.

If the magnitude of '''v'''<sub>3</sub> were to describe a measurable physical quantity, that would mean that the laws of physics would not appear the same if the universe was viewed in a mirror. In fact, this is exactly what happens in the [[weak interaction]]: Certain radioactive decays treat "left" and "right" differently, a phenomenon which can be traced to the summation of a polar vector with a pseudovector in the underlying theory. (See [[parity violation]].)

===Behavior under cross products and curls===

[[Image:Uitwendig product onder inversie.svg|thumb|Under inversion the two vectors change sign, but their cross product is invariant [black are the two original vectors, grey are the inverted vectors, and red is their mutual cross product].]]
For a rotation matrix ''R'', either proper or improper, the following mathematical equation is always true:
:<math>(R\mathbf{v_1})\times(R\mathbf{v_2}) = (\det R)(R(\mathbf{v_1}\times\mathbf{v_2}))</math>,
where '''v'''<sub>1</sub> and '''v'''<sub>2</sub> are any three-dimensional vectors. (This equation can be proven either through a geometric argument or through an algebraic calculation.) Similarly, if '''v''' is any vector field, the following equation is always true:
:<math>\nabla \times (R\mathbf{v}) = (\det R)(R(\nabla \times \mathbf{v}))</math>
where {{nowrap|1=∇ ×}} denotes the [[curl (mathematics)|curl]] operation from vector calculus.

Suppose '''v'''<sub>1</sub> and '''v'''<sub>2</sub> are known polar vectors, and '''v'''<sub>3</sub> is defined to be their cross product, {{nowrap|1='''v'''<sub>3</sub> = '''v'''<sub>1</sub> × '''v'''<sub>2</sub>}}. If the universe is transformed by a rotation matrix ''R'', then '''v'''<sub>3</sub> is transformed to
:<math>\mathbf{v_3}' = \mathbf{v_1}' \times \mathbf{v_2}' = (R\mathbf{v_1}) \times (R\mathbf{v_2}) = (\det R)(R(\mathbf{v_1} \times \mathbf{v_2})) = (\det R)(R\mathbf{v_3}).</math>
So '''v'''<sub>3</sub> is a pseudovector. Likewise, one can show that the cross product of two pseudovectors is a pseudovector and the cross product of a polar vector with a pseudovector is a polar vector. In conclusion, we have:
*polar vector × polar vector = pseudovector
*pseudovector × pseudovector = pseudovector
*polar vector × pseudovector = polar vector
*pseudovector × polar vector = polar vector
This is isomorphic to addition modulo&nbsp;2, where "polar" corresponds to 1 and "pseudo" to&nbsp;0.

Similarly, if '''v'''<sub>1</sub> is any known polar vector field and '''v'''<sub>2</sub> is defined to be its curl {{nowrap|1='''v'''<sub>2</sub> = ∇ × '''v'''<sub>1</sub>}}, then if the universe is transformed by the rotation matrix ''R'', '''v'''<sub>2</sub> is transformed to
:<math>\mathbf{v_2}' = \nabla \times \mathbf{v_1}' = \nabla \times (R\mathbf{v_1}) = (\det R)(R(\nabla \times \mathbf{v_1})) = (\det R)(R\mathbf{v_2}).</math>
So '''v'''<sub>2</sub> is a pseudovector field. Likewise, one can show that the curl of a pseudovector field is a polar vector field. In conclusion, we have:
*∇ × polar vector field = pseudovector field
*∇ × pseudovector field = polar vector field
This is like the above rule for cross-products if one interprets the del operator ∇ as a polar vector.

===Examples===

From the definition, it is clear that linear displacement is a polar vector. Linear velocity is linear displacement (a polar vector) divided by time (a scalar), so is also a polar vector. Linear momentum is linear velocity (a polar vector) times mass (a scalar), so is a polar vector. Angular momentum (in a point object) is the [[cross product]] of linear displacement (a polar vector) and linear momentum (a polar vector), and is therefore a pseudovector. Torque is angular momentum (a pseudovector) divided by time (a scalar), so is also a pseudovector. Angular velocity (in a rotating body or fluid) is one-half times the [[curl (mathematics)|curl]] of linear velocity (a polar vector field), and thus is a pseudovector. Continuing this way, it is straightforward to classify any of the common vectors in physics as either a pseudovector or a polar vector. (There are the parity-violating vectors in the theory of weak-interactions, which are neither polar vectors nor pseudovectors. However, these occur very rarely in physics.)