Cauchy invariance rule

If  f(u,v,w),  u(x,y),  v(x,y),  w(x,y)  are differentiable functions and

f¯(x,y):=f(u(x,y),v(x,y),w(x,y)) (1)

their composite functionMathworldPlanetmath, then according to the ruleMathworldPlanetmath, we have the partial derivativesMathworldPlanetmath

{f¯x(x,y)=fu(u,v,w)ux(x,y)+fv(u,v,w)vx(x,y)+fw(u,v,w)wx(x,y),f¯y(x,y)=fu(u,v,w)uy(x,y)+fv(u,v,w)vy(x,y)+fw(u,v,w)wy(x,y). (2)

Multiplying these two equations by dx and dy, respectively, and then adding them, we obtain for the total differentialMathworldPlanetmath of the composite function the expression

df¯(x,y) =f¯x(x,y)dx+f¯y(x,y)dy

But the sums in the brackets the total differentials of the inner functions, whence we may write

df¯(x,y)=fu(u,v,w)du(x,y)+fv(u,v,w)dv(x,y)+fw(u,v,w)dw(x,y) (3)

where one must still substitute  u:=u(x,y),v:=v(x,y),w:=w(x,y).  Comparing (3) with the expression of the total differential

df(u,v,w)=fu(u,v,w)du+fv(u,v,w)dv+fw(u,v,w)dw (4)

of the outer function, we infer the following

Rule.  The total differential of the composite function (1) is directly obtained from the expression of the total differential of the outer function, when one replaces in it the variables u,v,w with the corresponding inner functions and the differentials du,dv,dw with the total differentials of those inner functions.

This rule of Cauchy is analogical for any number of inner functions and their variables.  The rule also offers the simplest way to form the partial derivatives of the composite function.


  • 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset II.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).
Title Cauchy invariance rule
Canonical name CauchyInvarianceRule
Date of creation 2013-03-22 19:11:33
Last modified on 2013-03-22 19:11:33
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Derivation
Classification msc 53A04
Classification msc 01A45
Classification msc 26B05
Synonym total differential of composite function
Related topic ChainRuleSeveralVariables