Definition
Even and odd parity of a function
The transformation rules f(-x)=f(x) and f(-x)=-f(x) on a reflection-invariant domain.
On a domain invariant under , a scalar function has even parity if , and odd parity if . These are the two sign characters of reflection. Every function on such a domain splits uniquely as
an even part plus an odd part.
Calculus
Differentiation interchanges these parities. Products multiply their signs: odd times odd is even. If an odd function is integrable on a symmetric interval, its integral is zero. Parity in one coordinate means reflecting that coordinate while holding all others fixed; for vector fields, the transformation of the basis also matters.