In
mathematics, the Fox derivative is an
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
ic construction in the theory of
free groups which bears many similarities to the conventional
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
. The Fox derivative and related concepts are often referred to as the Fox calculus, or (Fox's original term) the free differential calculus. The Fox derivative was developed in a series of five papers by mathematician
Ralph Fox, published in
Annals of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...
beginning in 1953.
Definition
If ''G'' is a free group with identity element ''e'' and
generators ''g
i'', then the Fox derivative with respect to ''g
i'' is a function from ''G'' into the
integral group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given ...
''ZG'' which is denoted
, and obeys the following
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy o ...
s:
*
, where
is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 ...
*
*
for any elements ''u'' and ''v'' of ''G''.
The first two axioms are identical to similar properties of the partial derivative of calculus, and the third is a modified version of the
product rule. As a consequence of the axioms, we have the following formula for inverses
*
for any element ''u'' of ''G''.
Applications
The Fox derivative has applications in
group cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomolog ...
,
knot theory and
covering space theory, among other areas of mathematics.
See also
*
Alexander polynomial
*
Free group
*
Ring (mathematics)
*
Integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
References
*
*
*
*
*
*
Geometric topology
Combinatorial group theory
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