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| In [[mathematics]], the '''Fox derivative''' is an [[algebra]]ic construction in the theory of [[free group]]s which bears many similarities to the conventional [[derivative]] of [[calculus]]. 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]] beginning in 1953.
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| ==Definition==
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| If ''G'' is a free group with identity element ''e'' and [[generating set of a group|generators]] ''g<sub>i</sub>'', then the Fox derivative with respect to ''g<sub>i</sub>'' is a function from ''G'' into the [[integral group ring]] '''''Z'''G'' which is denoted <math>\frac{\partial}{\partial g_i}</math>, and obeys the following [[axiom]]s:
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| * <math>\frac{\partial}{\partial g_i}(g_j) = \delta_{ij}</math>, where <math>\delta_{ij}</math> is the [[Kronecker delta]]
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| * <math>\frac{\partial}{\partial g_i}(e) = 0</math>
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| * <math>\frac{\partial}{\partial g_i}(uv) = \frac{\partial}{\partial g_i}(u) + u\frac{\partial}{\partial g_i}(v)</math> for any elements ''u'' and ''v'' of ''G''.
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| 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
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| * <math>\frac{\partial}{\partial g_i}(u^{-1}) = -u^{-1}\frac{\partial}{\partial g_i}(u)</math> for any element ''u'' of ''G''.
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| ==Applications==
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| The Fox derivative has applications in [[group cohomology]], [[knot theory]] and [[covering space]] theory, among other areas of mathematics.
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| ==See also==
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| * [[Alexander polynomial]]
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| * [[Alexander module]]
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| * [[Free group]]
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| * [[Ring (mathematics)]]
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| * [[Integral domain]]
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| ==References==
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| * {{Cite book | first=Kenneth S. | last1=Brown | authorlink = Kenneth Brown (mathematician) | title=Cohomology of Groups | publisher=[[Springer Verlag]] | year=1972 | isbn=0-387-90688-6 | series=Graduate Texts in Mathematics | mr=0672956 | volume=87 | postscript=<!--None--> }}
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| *{{cite journal |last=Fox |first=Ralph |authorlink=Ralph Fox |date=May 1953 |title=Free Differential Calculus, I: Derivation in the Free Group Ring |journal=[[Annals of Mathematics]] |volume=57 |issue=3 |pages=547–560 |doi=10.2307/1969736 |jstor=1969736 |publisher=Annals of Mathematics |mr=0053938}}
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| *{{cite journal |last=Fox |first=Ralph |date=March 1954 |title=Free Differential Calculus, II: The Isomorphism Problem of Groups |journal=Annals of Mathematics |volume=59 |issue=2 |pages=196–210 |doi=10.2307/1969686 |jstor=1969686 |publisher=Annals of Mathematics |mr=0062125}}
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| *{{cite journal |last=Fox |first=Ralph |date=November 1956 |title=Free Differential Calculus, III: Subgroups |journal=Annals of Mathematics |volume=64 |issue=2 |pages=407–419 |doi=10.2307/1969592 |jstor=1969592 |publisher=Annals of Mathematics |mr=0095876}}
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| *{{cite journal |last=Chen |first=Kuo-Tsai |authorlink=Kuo-Tsai Chen |coauthors=Ralph Fox, [[Roger Lyndon]] |date=July 1958 |title=Free Differential Calculus, IV: The Quotient Groups of the Lower Central Series |journal=Annals of Mathematics |volume=68 |issue=1 |pages=81–95 |doi=10.2307/1970044 |jstor=1970044 |publisher=Annals of Mathematics |mr=0102539}}
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| *{{cite journal |last=Fox |first=Ralph |date=May 1960 |title=Free Differential Calculus, V: The Alexander Matrices Re-Examined |journal=Annals of Mathematics |volume=71 |issue=3 |pages=408–422 |doi=10.2307/1969936 |jstor=1969936 |publisher=Annals of Mathematics |mr=0111781}}
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| [[Category:Geometric topology]]
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| [[Category:Combinatorial group theory]]
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| {{algebra-stub}}
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Hi there, I am Yoshiko Villareal but I by no means really favored that name. Some time in the past he selected to reside in Kansas. To perform handball is the thing she enjoys most of all. Interviewing is how he supports his family members but his promotion never arrives.
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