Cauchy invariance rule

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

$\overline{f}(x,y):=f(u(x,y),v(x,y),w(x,y))$ (1)

their composite function, then according to the http://planetmath.org/node/2798chain rule, we have the partial derivatives

$\{\begin{array}{cc}{\overline{f}}_x^{\prime }(x,y)=f_u^{\prime }(u,v,w)u_x^{\prime }(x,y)+f_v^{\prime }(u,v,w)v_x^{\prime }(x,y)+f_w^{\prime }(u,v,w)w_x^{\prime }(x,y),\hfill & \\ {\overline{f}}_y^{\prime }(x,y)=f_u^{\prime }(u,v,w)u_y^{\prime }(x,y)+f_v^{\prime }(u,v,w)v_y^{\prime }(x,y)+f_w^{\prime }(u,v,w)w_y^{\prime }(x,y).\hfill & \end{array}$ (2)

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

$d\overline{f}(x,y)$ $\mathrm{\hspace{0.33em}}={\overline{f}}_x^{\prime }(x,y)dx+{\overline{f}}_y^{\prime }(x,y)dy$

$\mathrm{\hspace{0.33em}}=(f_u^{\prime }{u}_x^{\prime }+f_v^{\prime }{v}_x^{\prime }+f_w^{\prime }{w}_x^{\prime })dx+(f_u^{\prime }{u}_y^{\prime }+f_v^{\prime }{v}_y^{\prime }+f_w^{\prime }{w}_y^{\prime })dy$

$\mathrm{\hspace{0.33em}}=f_u^{\prime }[u_x^{\prime }dx+u_y^{\prime }dy]+f_v^{\prime }[v_x^{\prime }dx+v_y^{\prime }dy]+f_w^{\prime }[w_x^{\prime }dx+w_y^{\prime }dy].$

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

$d\overline{f}(x,y)=f_u^{\prime }(u,v,w)du(x,y)+f_v^{\prime }(u,v,w)dv(x,y)+f_w^{\prime }(u,v,w)dw(x,y)$ (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

$df(u,v,w)=f_u^{\prime }(u,v,w)du+f_v^{\prime }(u,v,w)dv+f_w^{\prime }(u,v,w)dw$ (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.

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