Pseudovector

In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation,^{[clarification needed]} but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation).

Physical examples

Physical examples of pseudovectors include the magnetic field, torque, vorticity, and the angular momentum.

References

↑ A simple example of an improper rotation in 3D (but not in 2D) is a coordinate inversion: x goes to −x, y to −y and z to −z. Under this transformation, a and b go to −a and −b (by the definition of a vector), but p clearly does not change. It follows that any improper rotation multiplies p by −1 compared to the rotation's effect on a true vector.
↑ Often, the distinction between vectors and pseudovectors is overlooked, but it becomes important in understanding and exploiting the effect of symmetry on the solution to physical systems.

George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (Harcourt: San Diego, 2001). (ISBN 0-12-059815-9)
John David Jackson, Classical Electrodynamics (Wiley: New York, 1999). (ISBN 0-471-30932-X)
Susan M. Lea, "Mathematics for Physicists" (Thompson: Belmont, 2004) (ISBN 0-534-37997-4)

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[1] A simple example of an improper rotation in 3D (but not in 2D) is a coordinate inversion: x goes to −x, y to −y and z to −z. Under this transformation, a and b go to −a and −b (by the definition of a vector), but p clearly does not change. It follows that any improper rotation multiplies p by −1 compared to the rotation's effect on a true vector.

[2] Often, the distinction between vectors and pseudovectors is overlooked, but it becomes important in understanding and exploiting the effect of symmetry on the solution to physical systems.