Cauchy invariance rule

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 function

, then according to the

http://

planetmath

.org/node/2798

chain rule

, we have the

partial derivatives

{

f

¯

x

′

⁢

(

x

,

y

)

=

f

u

′

⁢

(

u

,

v

,

w

)

⁢

u

x

′

⁢

(

x

,

y

)

+

f

v

′

⁢

(

u

,

v

,

w

)

⁢

v

x

′

⁢

(

x

,

y

)

+

f

w

′

⁢

(

u

,

v

,

w

)

⁢

w

x

′

⁢

(

x

,

y

)

,

f

¯

y

′

⁢

(

x

,

y

)

=

f

u

′

⁢

(

u

,

v

,

w

)

⁢

u

y

′

⁢

(

x

,

y

)

+

f

v

′

⁢

(

u

,

v

,

w

)

⁢

v

y

′

⁢

(

x

,

y

)

+

f

w

′

⁢

(

u

,

v

,

w

)

⁢

w

y

′

⁢

(

x

,

y

)

.

(2)

Multiplying these two equations by

d

⁢

x

and

d

⁢

y

, respectively, and then adding them, we obtain for the

total differential

of the composite function the

expression

d

⁢

f

¯

⁢

(

x

,

y

)

=

f

¯

x

′

⁢

(

x

,

y

)

⁢

d

⁢

x

+

f

¯

y

′

⁢

(

x

,

y

)

⁢

d

⁢

y

=

(

f

u

′

⁢

u

x

′

+

f

v

′

⁢

v

x

′

+

f

w

′

⁢

w

x

′

)

⁢

d

⁢

x

+

(

f

u

′

⁢

u

y

′

+

f

v

′

⁢

v

y

′

+

f

w

′

⁢

w

y

′

)

⁢

d

⁢

y

=

f

u

′

⁢

[

u

x

′

⁢

d

⁢

x

+

u

y

′

⁢

d

⁢

y

]

+

f

v

′

⁢

[

v

x

′

⁢

d

⁢

x

+

v

y

′

⁢

d

⁢

y

]

+

f

w

′

⁢

[

w

x

′

⁢

d

⁢

x

+

w

y

′

⁢

d

⁢

y

]

.

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

d

⁢

f

¯

⁢

(

x

,

y

)

=

f

u

′

⁢

(

u

,

v

,

w

)

⁢

d

⁢

u

⁢

(

x

,

y

)

+

f

v

′

⁢

(

u

,

v

,

w

)

⁢

d

⁢

v

⁢

(

x

,

y

)

+

f

w

′

⁢

(

u

,

v

,

w

)

⁢

d

⁢

w

⁢

(

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

d

⁢

f

⁢

(

u

,

v

,

w

)

=

f

u

′

⁢

(

u

,

v

,

w

)

⁢

d

⁢

u

+

f

v

′

⁢

(

u

,

v

,

w

)

⁢

d

⁢

v

+

f

w

′

⁢

(

u

,

v

,

w

)

⁢

d

⁢

w

(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

d

⁢

u

,

d

⁢

v

,

d

⁢

w

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.

References

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

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