total differential

total differential

There is the generalisation of the theorem in the parent entry (

http://

planetmath

.org/

Differential

) concerning the

real functions

of several variables; here we formulate it for three variables:

Theorem.

Suppose that

S

is a ball in

ℝ

3

, the

function

f

:

S

→

ℝ

is

continuous

and has

partial derivatives

f

x

′

,

f

y

′

,

f

z

′

in

S

and the partial derivatives are continuous in a point

(

x

,

y

,

z

)

of

S

.  Then the increment

Δ

⁢

f

:=

f

⁢

(

x

+

Δ

⁢

x

,

y

+

Δ

⁢

y

,

z

+

Δ

⁢

z

)

-

f

⁢

(

x

,

y

,

z

)

,

which

f

gets when one moves from

(

x

,

y

,

z

)

to another point

(

x

+

Δ

⁢

x

,

y

+

Δ

⁢

y

,

z

+

Δ

⁢

z

)

of

S

, can be split into two parts as follows:

Δ

⁢

f

=

[

f

x

′

⁢

(

x

,

y

,

z

)

⁢

Δ

⁢

x

+

f

y

′

⁢

(

x

,

y

,

z

)

⁢

Δ

⁢

y

+

f

z

′

⁢

(

x

,

y

,

z

)

⁢

Δ

⁢

z

]

+

⟨

ϱ

⟩

⁢

ϱ

.

(1)

Here,

ϱ

:=

Δ

⁢

x

2

+

Δ

⁢

y

2

+

Δ

⁢

z

2

and

⟨

ϱ

⟩

is a quantity tending to 0 along with

ϱ

.

The former part of

Δ

⁢

x

is called the (

total

)

differential

or the

exact differential

of the function

f

in the point

(

x

,

y

,

z

)

and it is denoted by

d

⁢

f

⁢

(

x

,

y

,

z

)

of briefly

d

⁢

f

.  In the special case

f

⁢

(

x

,

y

,

z

)

≡

x

,  we see that

d

⁢

f

=

Δ

⁢

x

and thus

Δ

⁢

x

=

d

⁢

x

;  similarly

Δ

⁢

y

=

d

⁢

y

and

Δ

⁢

z

=

d

⁢

z

.  Accordingly, we obtain for the general case the more consistent notation

d

⁢

f

=

f

x

′

⁢

(

x

,

y

,

z

)

⁢

d

⁢

x

+

f

y

′

⁢

(

x

,

y

,

z

)

⁢

d

⁢

y

+

f

z

′

⁢

(

x

,

y

,

z

)

⁢

d

⁢

z

,

(2)

where

d

⁢

x

,

d

⁢

y

,

d

⁢

z

may be thought as

independent variables

.

We now assume conversely that the increment of a function

f

in

ℝ

3

can be split into two parts as follows:

f

⁢

(

x

+

Δ

⁢

x

,

y

+

Δ

⁢

y

,

z

+

Δ

⁢

z

)

-

f

⁢

(

x

,

y

,

z

)

=

[

A

⁢

Δ

⁢

x

+

B

⁢

Δ

⁢

y

+

C

⁢

Δ

⁢

z

]

+

⟨

ϱ

⟩

⁢

ϱ

(3)

where the

coefficients

A

,

B

,

C

are independent on the quantities

Δ

⁢

x

,

Δ

⁢

y

,

Δ

⁢

z

and

ϱ

,

⟨

ϱ

⟩

are as in the above theorem.  Then one can infer that the partial derivatives

f

x

′

,

f

y

′

,

f

z

′

exist in the point

(

x

,

y

,

z

)

and have the values

A

,

B

,

C

, respectively.  In fact, if we choose

Δ

⁢

y

=

Δ

⁢

z

=

0

, then

ϱ

=

|

Δ

⁢

x

|

whence (3) attains the form

f

⁢

(

x

+

Δ

⁢

x

,

y

+

Δ

⁢

y

,

z

+

Δ

⁢

z

)

-

f

⁢

(

x

,

y

,

z

)

=

A

⁢

Δ

⁢

x

+

⟨

Δ

⁢

x

⟩

⁢

Δ

⁢

x

and therefore

A

=

lim

Δ

⁢

x

→

0

⁡

f

⁢

(

x

+

Δ

⁢

x

,

y

+

Δ

⁢

y

,

z

+

Δ

⁢

z

)

-

f

⁢

(

x

,

y

,

z

)

Δ

⁢

x

=

f

x

′

⁢

(

x

,

y

,

z

)

.

Similarly we see the values of

f

y

′

and

f

z

′

.

The last consideration showed the uniqueness of the

total differential

.

Definition.

A function

f

in

ℝ

3

, satisfying the conditions of the above theorem is said to be

differentiable

in the point

(

x

,

y

,

z

)

.

Remark.

The differentiability of a function

f

of two variables in the point

(

x

,

y

)

means that the surface

z

=

f

⁢

(

x

,

y

)

has a

tangent plane

in this point.

References

1

Ernst Lindelöf

:

Differentiali- ja integralilasku
ja sen sovellutukset II

.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).

Title

total differential

Canonical name

TotalDifferential

Date of creation

2013-03-22 19:11:24

Last modified on

2013-03-22 19:11:24

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Definition

Classification

msc 53A04

Classification

msc 26B05

Classification

msc 01A45

Synonym

exact differential

Synonym

differential

Related topic

ExactDifferentialForm

Related topic

ExactDifferentialEquation

Related topic

Differential

Related topic

DifferntiableFunction

Defines

differentiable

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