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The main linear part of increment of a function.
 
The main linear part of increment of a function.
  
1) A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d0318101.png" /> of a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d0318102.png" /> is said to be differentiable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d0318103.png" /> if it is defined in some neighbourhood of this point and if there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d0318104.png" /> such that the increment
+
1) A real-valued function $  f $
 +
of a real variable $  x $
 +
is said to be differentiable at a point $  x $
 +
if it is defined in some neighbourhood of this point and if there exists a number $  A $
 +
such that the increment
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d0318105.png" /></td> </tr></table>
+
$$
 +
\Delta y  = f ( x + \Delta x ) - f ( x)
 +
$$
  
may be written (if the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d0318106.png" /> lies in this neighbourhood) in the form
+
may be written (if the point $  x + \Delta x $
 +
lies in this neighbourhood) in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d0318107.png" /></td> </tr></table>
+
$$
 +
\Delta y  = A \Delta x + \omega ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d0318108.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d0318109.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181010.png" /> is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181011.png" /> and is called the differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181012.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181013.png" />. For a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181014.png" /> the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181015.png" /> is proportional to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181016.png" />, i.e. is a linear function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181017.png" />. By definition, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181018.png" /> the additional term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181019.png" /> is infinitely small of a higher order than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181020.png" /> (and also than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181021.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181022.png" />). This is why the differential is said to be the main part of increment of the function.
+
where $  \omega / \Delta x \rightarrow 0 $
 +
as $  \Delta x \rightarrow 0 $.  
 +
Here $  A \Delta x $
 +
is usually denoted by $  dy $
 +
and is called the differential of $  f $
 +
at $  x $.  
 +
For a given $  x $
 +
the differential $  dy $
 +
is proportional to $  \Delta x $,  
 +
i.e. is a linear function of $  \Delta x $.  
 +
By definition, as $  \Delta x \rightarrow 0 $
 +
the additional term $  \omega $
 +
is infinitely small of a higher order than $  \Delta x $(
 +
and also than $  dy $
 +
if $  A \neq 0 $).  
 +
This is why the differential is said to be the main part of increment of the function.
  
For a function which is differentiable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181024.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181025.png" />, i.e. a function which is differentiable at a point is continuous at that point. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181026.png" /> is differentiable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181027.png" /> if and only if it has, at that point, a finite derivative
+
For a function which is differentiable at a point $  x $,  
 +
$  \Delta y \rightarrow 0 $
 +
if $  \Delta x \rightarrow 0 $,  
 +
i.e. a function which is differentiable at a point is continuous at that point. A function $  f $
 +
is differentiable at a point $  x $
 +
if and only if it has, at that point, a finite derivative
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181028.png" /></td> </tr></table>
+
$$
 +
f ^ { \prime } ( x)  = \lim\limits _ {\Delta x \rightarrow 0 } \
 +
 
 +
\frac{\Delta y }{\Delta x }
 +
  = A ;
 +
$$
  
 
moreover,
 
moreover,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181029.png" /></td> </tr></table>
+
$$
 +
d y  = f ^ { \prime } ( x) \Delta x .
 +
$$
  
 
There exist continuous functions which are not differentiable.
 
There exist continuous functions which are not differentiable.
  
The designation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181030.png" /> may be used instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181031.png" />, and the above equation assumes the form
+
The designation $  df ( x) $
 +
may be used instead of $  dy $,  
 +
and the above equation assumes the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181032.png" /></td> </tr></table>
+
$$
 +
d f ( x)  = f ^ { \prime } ( x) \Delta x .
 +
$$
  
The increment of the argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181033.png" /> is then usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181034.png" />, and is said to be the differential of the independent variable. Accordingly, one may write
+
The increment of the argument $  \Delta x $
 +
is then usually denoted by $  dx $,  
 +
and is said to be the differential of the independent variable. Accordingly, one may write
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181035.png" /></td> </tr></table>
+
$$
 +
d y  = f ^ { \prime } ( x)  d x .
 +
$$
  
Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181036.png" />, i.e. the derivative is equal to the ratio of the differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181038.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181039.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181040.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181041.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181042.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181044.png" /> are infinitesimals of the same order as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181045.png" />; this fact, along with the simple structure of the differential (i.e. linearity with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181046.png" />), is often used in approximate computations, by assuming that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181047.png" /> for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181048.png" />. E.g., if it is desired to compute <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181049.png" /> from a known <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181050.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181051.png" /> is small, it is assumed that
+
Hence $  f ^ { \prime } ( x) = dy / dx $,  
 +
i.e. the derivative is equal to the ratio of the differentials $  dy $
 +
and $  dx $.  
 +
If $  A \neq 0 $,  
 +
then $  \Delta y / dy \rightarrow 1 $
 +
as $  \Delta x \rightarrow 0 $,  
 +
i.e. if $  A \neq 0 $,  
 +
then $  \Delta y $
 +
and $  dy $
 +
are infinitesimals of the same order as $  \Delta x \rightarrow 0 $;  
 +
this fact, along with the simple structure of the differential (i.e. linearity with respect to $  \Delta x $),  
 +
is often used in approximate computations, by assuming that $  \Delta y \approx dy $
 +
for small $  \Delta x $.  
 +
E.g., if it is desired to compute $  f ( x + \Delta x ) $
 +
from a known $  f ( x) $
 +
when $  \Delta x $
 +
is small, it is assumed that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181052.png" /></td> </tr></table>
+
$$
 +
f ( x + \Delta x )  \approx  f( x) + dy .
 +
$$
  
 
Obviously, such reasoning is useful only if it is possible to estimate the magnitude of the error involved.
 
Obviously, such reasoning is useful only if it is possible to estimate the magnitude of the error involved.
  
Geometric interpretation of the differential. The equation of the tangent to the graph of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181053.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181054.png" /> is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181055.png" />. If one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181056.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181057.png" />. The right-hand side represents the value of the differential of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181058.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181059.png" /> corresponding to the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181060.png" /> being considered. Thus, the differential is identical with the corresponding increment of the ordinate of the tangent to the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181061.png" /> (cf. the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181062.png" /> in Fig. a). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181063.png" />, i.e. the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181064.png" /> coincides with the length of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181065.png" />.
+
Geometric interpretation of the differential. The equation of the tangent to the graph of a function $  f $
 +
at a point $  ( x _ {0} , y _ {0} ) $
 +
is of the form $  y - y _ {0} = f ^ { \prime } ( x _ {0} ) ( x - x _ {0} ) $.  
 +
If one puts $  x = x _ {0} + \Delta x $,  
 +
then $  y - y _ {0} = f ^ { \prime } ( x _ {0} ) \Delta x $.  
 +
The right-hand side represents the value of the differential of the function $  f $
 +
at the point $  x _ {0} $
 +
corresponding to the value of $  \Delta x $
 +
being considered. Thus, the differential is identical with the corresponding increment of the ordinate of the tangent to the curve $  y = f ( x) $(
 +
cf. the segment $  NT $
 +
in Fig. a). Here $  \omega = \Delta y - dy $,  
 +
i.e. the value of $  | \omega | $
 +
coincides with the length of the segment $  TS $.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d031810a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d031810a.gif" />
Line 41: Line 127:
 
Figure: d031810a
 
Figure: d031810a
  
2) The definitions of differentiability and differential are readily extended to real-valued functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181066.png" /> real variables. Thus, in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181067.png" /> a real-valued function is said to be differentiable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181068.png" /> with respect to both variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181070.png" /> if it is defined in some neighbourhood of this point and if its total increment
+
2) The definitions of differentiability and differential are readily extended to real-valued functions of $  n $
 +
real variables. Thus, in the case $  n = 2 $
 +
a real-valued function is said to be differentiable at a point $  ( x , y ) $
 +
with respect to both variables $  x $
 +
and $  y $
 +
if it is defined in some neighbourhood of this point and if its total increment
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181071.png" /></td> </tr></table>
+
$$
 +
\Delta z  = f ( x + \Delta x , y + \Delta y ) - f ( x , y )
 +
$$
  
 
may be written as
 
may be written as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181072.png" /></td> </tr></table>
+
$$
 +
\Delta z  = A \Delta x + B \Delta y + \alpha ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181074.png" /> are real numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181075.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181077.png" />; it is assumed that the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181078.png" /> belongs to the neighbourhood mentioned above (Fig. b).
+
where $  A $
 +
and $  B $
 +
are real numbers, $  \alpha / \rho \rightarrow 0 $
 +
if $  \rho \rightarrow 0 $,  
 +
$  \rho = \sqrt {\Delta x  ^ {2} + \Delta y  ^ {2} } $;  
 +
it is assumed that the point $  ( x + \Delta x , y + \Delta y ) $
 +
belongs to the neighbourhood mentioned above (Fig. b).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d031810b.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d031810b.gif" />
Line 57: Line 158:
 
One introduces the notation
 
One introduces the notation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181079.png" /></td> </tr></table>
+
$$
 +
d z  = d f ( x , y )  = A \Delta x + B \Delta y;
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181080.png" /> is said to be the total differential, or simply the differential, of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181081.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181082.png" /> (the phrase  "with respect to both variables x and y"  is sometimes added). For a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181083.png" /> the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181084.png" /> is a linear function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181086.png" />; the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181087.png" /> is infinitely small of a higher order than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181088.png" />. In this sense <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181089.png" /> is the main linear part of the increment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181090.png" />.
+
$  dz $
 +
is said to be the total differential, or simply the differential, of the function $  f $
 +
at the point $  ( x , y ) $(
 +
the phrase  "with respect to both variables x and y"  is sometimes added). For a given point $  ( x , y ) $
 +
the differential $  dz $
 +
is a linear function of $  \Delta x $
 +
and $  \Delta y $;  
 +
the difference $  \alpha = \Delta z - dz $
 +
is infinitely small of a higher order than $  \rho $.  
 +
In this sense $  dz $
 +
is the main linear part of the increment $  \Delta z $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181091.png" /> is differentiable at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181092.png" />, then it is continuous at this point and has finite partial derivatives (cf. [[Derivative|Derivative]])
+
If $  f $
 +
is differentiable at the point $  ( x , y ) $,
 +
then it is continuous at this point and has finite partial derivatives (cf. [[Derivative|Derivative]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181093.png" /></td> </tr></table>
+
$$
 +
f _ {x} ^ { \prime } ( x , y )  = A ,\ \
 +
f _ {y} ^ { \prime } ( x , y )  = B
 +
$$
  
 
at this point. Thus
 
at this point. Thus
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181094.png" /></td> </tr></table>
+
$$
 +
d z  = d f ( x , y )  = f _ {x} ^ { \prime } ( x , y )
 +
\Delta x + f _ {y} ^ { \prime } ( x , y ) \Delta y .
 +
$$
  
The increments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181096.png" /> of the independent variables are usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181098.png" />, as in the case of a single variable. One may write, accordingly,
+
The increments $  \Delta x $
 +
and $  \Delta y $
 +
of the independent variables are usually denoted by $  dy $
 +
and $  dx $,  
 +
as in the case of a single variable. One may write, accordingly,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d03181099.png" /></td> </tr></table>
+
$$
 +
d z  = d f ( x , y )  = f _ {x} ^ { \prime } ( x , y )
 +
d x + f _ {y} ^ { \prime } ( x , y )  d y .
 +
$$
  
 
The existence of finite partial derivatives does not, in general, entail the differentiability of the function (even if it is assumed to be continuous).
 
The existence of finite partial derivatives does not, in general, entail the differentiability of the function (even if it is assumed to be continuous).
  
If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810100.png" /> has a partial derivative with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810101.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810102.png" />, the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810103.png" /> is said to be its partial differential with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810104.png" />; in the same manner, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810105.png" /> is the partial differential with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810106.png" />. If the function is differentiable, its total differential is equal to the sum of the partial differentials. Geometrically, the total differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810107.png" /> is the increment in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810108.png" />-direction in the tangent plane to the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810109.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810110.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810111.png" /> (Fig. c).
+
If a function $  f $
 +
has a partial derivative with respect to $  x $
 +
at a point $  ( x , y ) $,  
 +
the product $  f _ {x} ^ { \prime } ( x , y )  dx $
 +
is said to be its partial differential with respect to $  x $;  
 +
in the same manner, $  f _ {y} ^ { \prime } ( x , y )  dy $
 +
is the partial differential with respect to $  y $.  
 +
If the function is differentiable, its total differential is equal to the sum of the partial differentials. Geometrically, the total differential $  df ( x _ {0} , y _ {0} ) $
 +
is the increment in the $  z $-
 +
direction in the tangent plane to the surface $  z = f ( x , y ) $
 +
at the point $  ( x _ {0} , y _ {0} , z _ {0} ) $,  
 +
where $  z _ {0} = f ( x _ {0} , y _ {0} ) $(
 +
Fig. c).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d031810c.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d031810c.gif" />
Line 81: Line 221:
 
Figure: d031810c
 
Figure: d031810c
  
The following is a sufficient criterion for the differentiability of a function: If in a certain neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810112.png" /> a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810113.png" /> has a partial derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810114.png" /> which is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810115.png" /> and, in addition, has a partial derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810116.png" /> at that point, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810117.png" /> is differentiable at that point.
+
The following is a sufficient criterion for the differentiability of a function: If in a certain neighbourhood of a point $  ( x _ {0} , y _ {0} ) $
 +
a function $  f $
 +
has a partial derivative $  f _ {x} ^ { \prime } $
 +
which is continuous at $  ( x _ {0} , y _ {0} ) $
 +
and, in addition, has a partial derivative $  f _ {y} ^ { \prime } $
 +
at that point, then $  f $
 +
is differentiable at that point.
  
If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810118.png" /> is differentiable at all points of an open domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810119.png" />, then at any point of the domain
+
If a function $  f $
 +
is differentiable at all points of an open domain $  D $,  
 +
then at any point of the domain
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810120.png" /></td> </tr></table>
+
$$
 +
d z  = A ( x , y )  d x + B ( x , y )  d y ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810122.png" />. If, in addition, there exist continuous partial derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810123.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810124.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810125.png" />, then, everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810126.png" />,
+
where $  A ( x , y ) = f _ {x} ^ { \prime } ( x , y ) $,
 +
$  B ( x , y ) = f _ {y} ^ { \prime } ( x , y ) $.  
 +
If, in addition, there exist continuous partial derivatives $  A _ {y}  ^  \prime  $
 +
and $  B _ {x} ^ { \prime } $
 +
in $  D $,  
 +
then, everywhere in $  D $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810127.png" /></td> </tr></table>
+
$$
 +
A _ {y}  ^  \prime  = B _ {x} ^ { \prime } .
 +
$$
  
 
This proves, in particular, that not every expression
 
This proves, in particular, that not every expression
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810128.png" /></td> </tr></table>
+
$$
 +
A ( x , y )  d x + B ( x , y )  d y
 +
$$
  
with continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810129.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810130.png" /> (in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810131.png" />) is the total differential of some function of two variables. This is a difference from functions of one variable, where any expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810132.png" /> with a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810133.png" /> in some interval is the differential of some function.
+
with continuous $  A $
 +
and $  B $(
 +
in a domain $  D $)  
 +
is the total differential of some function of two variables. This is a difference from functions of one variable, where any expression $  A ( x)  d x $
 +
with a continuous function $  A $
 +
in some interval is the differential of some function.
  
The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810134.png" /> is the total differential of some function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810135.png" /> in a simply-connected open domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810136.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810137.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810138.png" /> are continuous in this domain, meet the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810139.png" /> and, in addition: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810140.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810141.png" /> are continuous or b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810143.png" /> are everywhere differentiable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810144.png" /> with respect to both variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810145.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810146.png" /> [[#References|[7]]], [[#References|[8]]].
+
The expression $  A  dx + B  dy $
 +
is the total differential of some function $  z = f ( x , y ) $
 +
in a simply-connected open domain $  D $
 +
if $  A $
 +
and $  B $
 +
are continuous in this domain, meet the condition $  A _ {y}  ^  \prime  = B _ {x} ^ { \prime } $
 +
and, in addition: a) $  A _ {y}  ^  \prime  $
 +
and $  B _ {x} ^ { \prime } $
 +
are continuous or b) $  A $
 +
and $  B $
 +
are everywhere differentiable in $  D $
 +
with respect to both variables $  x $
 +
and $  y $[[#References|[7]]], [[#References|[8]]].
  
 
See also [[Differential calculus|Differential calculus]] for differentials of real-valued functions of one or more real variables and for differentials of higher orders.
 
See also [[Differential calculus|Differential calculus]] for differentials of real-valued functions of one or more real variables and for differentials of higher orders.
  
3) Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810147.png" /> be defined on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810148.png" /> of real numbers, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810149.png" /> be a limit point of this set, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810150.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810151.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810152.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810153.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810154.png" />; then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810155.png" /> is called differentiable with respect to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810156.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810157.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810158.png" /> is called its differential with respect to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810159.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810160.png" />. This is a generalization of the differential of a real-valued function of one real variable. Special kinds of this generalization include differentials at the end points of the interval within which the function is defined, and the approximate differential (cf. [[Approximate differentiability|Approximate differentiability]]).
+
3) Let a function $  f $
 +
be defined on some set $  E $
 +
of real numbers, let $  x $
 +
be a limit point of this set, let $  x \in E $,  
 +
$  x + \Delta x \in E $,  
 +
$  \Delta y = A \Delta x + \alpha $,  
 +
where $  \alpha / \Delta x \rightarrow 0 $
 +
if $  \Delta x \rightarrow 0 $;  
 +
then the function $  f $
 +
is called differentiable with respect to the set $  E $
 +
at $  x $,  
 +
while $  dy = A \Delta x $
 +
is called its differential with respect to the set $  E $
 +
at $  x $.  
 +
This is a generalization of the differential of a real-valued function of one real variable. Special kinds of this generalization include differentials at the end points of the interval within which the function is defined, and the approximate differential (cf. [[Approximate differentiability|Approximate differentiability]]).
  
 
Differentials with respect to a set for real-valued functions of several real variables are introduced in a similar manner.
 
Differentials with respect to a set for real-valued functions of several real variables are introduced in a similar manner.
Line 109: Line 299:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.P. Tolstov,  "Elements of mathematical analysis" , '''1–2''' , Moscow  (1974)  (In Russian)  {{MR|0357695}} {{MR|0354961}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Fichtenholz,  "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag Wissenschaft.  (1964)  {{MR|1191905}} {{MR|1056870}} {{MR|1056869}} {{MR|0887101}} {{MR|0845556}} {{MR|0845555}} {{MR|0524565}} {{MR|0473117}} {{MR|0344040}} {{MR|0344039}} {{MR|0238635}} {{MR|0238637}} {{MR|0238636}} {{ZBL|0143.27002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.D. Kudryavtsev,  "Mathematical analysis" , '''1''' , Moscow  (1973)  (In Russian)  {{MR|1617334}} {{MR|1070567}} {{MR|1070566}} {{MR|1070565}} {{MR|0866891}} {{MR|0767983}} {{MR|0767982}} {{MR|0628614}} {{MR|0619214}} {{ZBL|1080.00002}} {{ZBL|1080.00001}} {{ZBL|1060.26002}} {{ZBL|0869.00003}} {{ZBL|0696.26002}} {{ZBL|0703.26001}} {{ZBL|0609.00001}} {{ZBL|0632.26001}} {{ZBL|0485.26002}} {{ZBL|0485.26001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1''' , MIR  (1977)  (Translated from Russian)  {{MR|}} {{ZBL|0397.00003}} {{ZBL|0384.00004}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1953)  {{MR|0055409}} {{ZBL|0052.05301}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)  {{MR|1025126}} {{MR|0708717}} {{MR|0630899}} {{MR|0435771}} {{MR|0377444}} {{MR|0234241}} {{MR|0215962}} {{MR|0118796}} {{MR|1530727}} {{MR|0118795}} {{MR|0085462}} {{MR|0070045}} {{ZBL|0932.46001}} {{ZBL|0672.46001}} {{ZBL|0501.46001}} {{ZBL|0501.46002}} {{ZBL|0235.46001}} {{ZBL|0103.08801}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G.P. Tolstov,  "On curvilinear and iterated integrals"  ''Trudy Mat. Inst. Steklov.'' , '''35'''  (1950)  (In Russian)  {{MR|44612}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  G.P. Tolstov,  "On the total differential"  ''Uspekhi Mat. Nauk'' , '''3''' :  5  (1948)  pp. 167–170  {{MR|0027044}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.P. Tolstov,  "Elements of mathematical analysis" , '''1–2''' , Moscow  (1974)  (In Russian)  {{MR|0357695}} {{MR|0354961}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Fichtenholz,  "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag Wissenschaft.  (1964)  {{MR|1191905}} {{MR|1056870}} {{MR|1056869}} {{MR|0887101}} {{MR|0845556}} {{MR|0845555}} {{MR|0524565}} {{MR|0473117}} {{MR|0344040}} {{MR|0344039}} {{MR|0238635}} {{MR|0238637}} {{MR|0238636}} {{ZBL|0143.27002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.D. Kudryavtsev,  "Mathematical analysis" , '''1''' , Moscow  (1973)  (In Russian)  {{MR|1617334}} {{MR|1070567}} {{MR|1070566}} {{MR|1070565}} {{MR|0866891}} {{MR|0767983}} {{MR|0767982}} {{MR|0628614}} {{MR|0619214}} {{ZBL|1080.00002}} {{ZBL|1080.00001}} {{ZBL|1060.26002}} {{ZBL|0869.00003}} {{ZBL|0696.26002}} {{ZBL|0703.26001}} {{ZBL|0609.00001}} {{ZBL|0632.26001}} {{ZBL|0485.26002}} {{ZBL|0485.26001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1''' , MIR  (1977)  (Translated from Russian)  {{MR|}} {{ZBL|0397.00003}} {{ZBL|0384.00004}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1953)  {{MR|0055409}} {{ZBL|0052.05301}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)  {{MR|1025126}} {{MR|0708717}} {{MR|0630899}} {{MR|0435771}} {{MR|0377444}} {{MR|0234241}} {{MR|0215962}} {{MR|0118796}} {{MR|1530727}} {{MR|0118795}} {{MR|0085462}} {{MR|0070045}} {{ZBL|0932.46001}} {{ZBL|0672.46001}} {{ZBL|0501.46001}} {{ZBL|0501.46002}} {{ZBL|0235.46001}} {{ZBL|0103.08801}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G.P. Tolstov,  "On curvilinear and iterated integrals"  ''Trudy Mat. Inst. Steklov.'' , '''35'''  (1950)  (In Russian)  {{MR|44612}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  G.P. Tolstov,  "On the total differential"  ''Uspekhi Mat. Nauk'' , '''3''' :  5  (1948)  pp. 167–170  {{MR|0027044}} {{ZBL|}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Line 119: Line 307:
 
For generalizations to functions between abstract spaces see also [[Fréchet derivative|Fréchet derivative]]; [[Gâteaux derivative|Gâteaux derivative]].
 
For generalizations to functions between abstract spaces see also [[Fréchet derivative|Fréchet derivative]]; [[Gâteaux derivative|Gâteaux derivative]].
  
For the derivative of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031810/d031810161.png" /> see [[Analytic function|Analytic function]].
+
For the derivative of a function $  f : \mathbf C \rightarrow \mathbf C $
 +
see [[Analytic function|Analytic function]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Calculus" , '''1–2''' , Blaisdell  (1964)  {{MR|1908007}} {{MR|1182316}} {{MR|1182315}} {{MR|0595410}} {{MR|1536963}} {{MR|1535772}} {{MR|0271732}} {{MR|0248290}} {{MR|0247001}} {{MR|0261376}} {{MR|0250092}} {{MR|0236734}} {{MR|0236733}} {{MR|0214705}} {{MR|1532185}} {{MR|1531712}} {{MR|0087718}} {{ZBL|0123.25902}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.M. Apostol,  "Mathematical analysis" , Addison-Wesley  (1974)  {{MR|0344384}} {{ZBL|0309.26002}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Fleming,  "Functions of several variables" , Springer  (1977)  {{MR|0422527}} {{ZBL|0348.26002}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)  {{MR|0604364}} {{ZBL|0454.26001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Courant,  "Vorlesungen über Differential- und Integralrechnung" , '''1–2''' , Springer  (1971–1972)  {{MR|0190266}} {{MR|1521849}} {{MR|1521664}} {{ZBL|0224.26001}} {{ZBL|0217.37201}} {{ZBL|0121.28904}} {{ZBL|0066.30303}} {{ZBL|0064.04704}} {{ZBL|0003.05401}}  {{ZBL|57.0246.01}}  {{ZBL|56.0193.01}}  {{ZBL|53.0200.13}}  {{ZBL|55.0728.02}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I.P. Natanson,  "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M.  (1961)  (Translated from Russian)  {{MR|0640867}} {{MR|0409747}} {{MR|0259033}} {{MR|0063424}} {{ZBL|0097.26601}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G.E. Shilov,  B.L. Gurevich,  "Integral, measure, and derivative: a unified approach" , Dover, reprint  (1977)  (Translated from Russian)  {{MR|0466463}} {{ZBL|0391.28007}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Denjoy,  "Introduction à la théorie des fonctions des variables réelles" , Gauthier-Villars  (1937)  {{MR|}} {{ZBL|0017.10504}}  {{ZBL|63.0177.02}}  {{ZBL|63.0177.01}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Calculus" , '''1–2''' , Blaisdell  (1964)  {{MR|1908007}} {{MR|1182316}} {{MR|1182315}} {{MR|0595410}} {{MR|1536963}} {{MR|1535772}} {{MR|0271732}} {{MR|0248290}} {{MR|0247001}} {{MR|0261376}} {{MR|0250092}} {{MR|0236734}} {{MR|0236733}} {{MR|0214705}} {{MR|1532185}} {{MR|1531712}} {{MR|0087718}} {{ZBL|0123.25902}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.M. Apostol,  "Mathematical analysis" , Addison-Wesley  (1974)  {{MR|0344384}} {{ZBL|0309.26002}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Fleming,  "Functions of several variables" , Springer  (1977)  {{MR|0422527}} {{ZBL|0348.26002}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)  {{MR|0604364}} {{ZBL|0454.26001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Courant,  "Vorlesungen über Differential- und Integralrechnung" , '''1–2''' , Springer  (1971–1972)  {{MR|0190266}} {{MR|1521849}} {{MR|1521664}} {{ZBL|0224.26001}} {{ZBL|0217.37201}} {{ZBL|0121.28904}} {{ZBL|0066.30303}} {{ZBL|0064.04704}} {{ZBL|0003.05401}}  {{ZBL|57.0246.01}}  {{ZBL|56.0193.01}}  {{ZBL|53.0200.13}}  {{ZBL|55.0728.02}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I.P. Natanson,  "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M.  (1961)  (Translated from Russian)  {{MR|0640867}} {{MR|0409747}} {{MR|0259033}} {{MR|0063424}} {{ZBL|0097.26601}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G.E. Shilov,  B.L. Gurevich,  "Integral, measure, and derivative: a unified approach" , Dover, reprint  (1977)  (Translated from Russian)  {{MR|0466463}} {{ZBL|0391.28007}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Denjoy,  "Introduction à la théorie des fonctions des variables réelles" , Gauthier-Villars  (1937)  {{MR|}} {{ZBL|0017.10504}}  {{ZBL|63.0177.02}}  {{ZBL|63.0177.01}} </TD></TR></table>

Revision as of 17:33, 5 June 2020


The main linear part of increment of a function.

1) A real-valued function $ f $ of a real variable $ x $ is said to be differentiable at a point $ x $ if it is defined in some neighbourhood of this point and if there exists a number $ A $ such that the increment

$$ \Delta y = f ( x + \Delta x ) - f ( x) $$

may be written (if the point $ x + \Delta x $ lies in this neighbourhood) in the form

$$ \Delta y = A \Delta x + \omega , $$

where $ \omega / \Delta x \rightarrow 0 $ as $ \Delta x \rightarrow 0 $. Here $ A \Delta x $ is usually denoted by $ dy $ and is called the differential of $ f $ at $ x $. For a given $ x $ the differential $ dy $ is proportional to $ \Delta x $, i.e. is a linear function of $ \Delta x $. By definition, as $ \Delta x \rightarrow 0 $ the additional term $ \omega $ is infinitely small of a higher order than $ \Delta x $( and also than $ dy $ if $ A \neq 0 $). This is why the differential is said to be the main part of increment of the function.

For a function which is differentiable at a point $ x $, $ \Delta y \rightarrow 0 $ if $ \Delta x \rightarrow 0 $, i.e. a function which is differentiable at a point is continuous at that point. A function $ f $ is differentiable at a point $ x $ if and only if it has, at that point, a finite derivative

$$ f ^ { \prime } ( x) = \lim\limits _ {\Delta x \rightarrow 0 } \ \frac{\Delta y }{\Delta x } = A ; $$

moreover,

$$ d y = f ^ { \prime } ( x) \Delta x . $$

There exist continuous functions which are not differentiable.

The designation $ df ( x) $ may be used instead of $ dy $, and the above equation assumes the form

$$ d f ( x) = f ^ { \prime } ( x) \Delta x . $$

The increment of the argument $ \Delta x $ is then usually denoted by $ dx $, and is said to be the differential of the independent variable. Accordingly, one may write

$$ d y = f ^ { \prime } ( x) d x . $$

Hence $ f ^ { \prime } ( x) = dy / dx $, i.e. the derivative is equal to the ratio of the differentials $ dy $ and $ dx $. If $ A \neq 0 $, then $ \Delta y / dy \rightarrow 1 $ as $ \Delta x \rightarrow 0 $, i.e. if $ A \neq 0 $, then $ \Delta y $ and $ dy $ are infinitesimals of the same order as $ \Delta x \rightarrow 0 $; this fact, along with the simple structure of the differential (i.e. linearity with respect to $ \Delta x $), is often used in approximate computations, by assuming that $ \Delta y \approx dy $ for small $ \Delta x $. E.g., if it is desired to compute $ f ( x + \Delta x ) $ from a known $ f ( x) $ when $ \Delta x $ is small, it is assumed that

$$ f ( x + \Delta x ) \approx f( x) + dy . $$

Obviously, such reasoning is useful only if it is possible to estimate the magnitude of the error involved.

Geometric interpretation of the differential. The equation of the tangent to the graph of a function $ f $ at a point $ ( x _ {0} , y _ {0} ) $ is of the form $ y - y _ {0} = f ^ { \prime } ( x _ {0} ) ( x - x _ {0} ) $. If one puts $ x = x _ {0} + \Delta x $, then $ y - y _ {0} = f ^ { \prime } ( x _ {0} ) \Delta x $. The right-hand side represents the value of the differential of the function $ f $ at the point $ x _ {0} $ corresponding to the value of $ \Delta x $ being considered. Thus, the differential is identical with the corresponding increment of the ordinate of the tangent to the curve $ y = f ( x) $( cf. the segment $ NT $ in Fig. a). Here $ \omega = \Delta y - dy $, i.e. the value of $ | \omega | $ coincides with the length of the segment $ TS $.

Figure: d031810a

2) The definitions of differentiability and differential are readily extended to real-valued functions of $ n $ real variables. Thus, in the case $ n = 2 $ a real-valued function is said to be differentiable at a point $ ( x , y ) $ with respect to both variables $ x $ and $ y $ if it is defined in some neighbourhood of this point and if its total increment

$$ \Delta z = f ( x + \Delta x , y + \Delta y ) - f ( x , y ) $$

may be written as

$$ \Delta z = A \Delta x + B \Delta y + \alpha , $$

where $ A $ and $ B $ are real numbers, $ \alpha / \rho \rightarrow 0 $ if $ \rho \rightarrow 0 $, $ \rho = \sqrt {\Delta x ^ {2} + \Delta y ^ {2} } $; it is assumed that the point $ ( x + \Delta x , y + \Delta y ) $ belongs to the neighbourhood mentioned above (Fig. b).

Figure: d031810b

One introduces the notation

$$ d z = d f ( x , y ) = A \Delta x + B \Delta y; $$

$ dz $ is said to be the total differential, or simply the differential, of the function $ f $ at the point $ ( x , y ) $( the phrase "with respect to both variables x and y" is sometimes added). For a given point $ ( x , y ) $ the differential $ dz $ is a linear function of $ \Delta x $ and $ \Delta y $; the difference $ \alpha = \Delta z - dz $ is infinitely small of a higher order than $ \rho $. In this sense $ dz $ is the main linear part of the increment $ \Delta z $.

If $ f $ is differentiable at the point $ ( x , y ) $, then it is continuous at this point and has finite partial derivatives (cf. Derivative)

$$ f _ {x} ^ { \prime } ( x , y ) = A ,\ \ f _ {y} ^ { \prime } ( x , y ) = B $$

at this point. Thus

$$ d z = d f ( x , y ) = f _ {x} ^ { \prime } ( x , y ) \Delta x + f _ {y} ^ { \prime } ( x , y ) \Delta y . $$

The increments $ \Delta x $ and $ \Delta y $ of the independent variables are usually denoted by $ dy $ and $ dx $, as in the case of a single variable. One may write, accordingly,

$$ d z = d f ( x , y ) = f _ {x} ^ { \prime } ( x , y ) d x + f _ {y} ^ { \prime } ( x , y ) d y . $$

The existence of finite partial derivatives does not, in general, entail the differentiability of the function (even if it is assumed to be continuous).

If a function $ f $ has a partial derivative with respect to $ x $ at a point $ ( x , y ) $, the product $ f _ {x} ^ { \prime } ( x , y ) dx $ is said to be its partial differential with respect to $ x $; in the same manner, $ f _ {y} ^ { \prime } ( x , y ) dy $ is the partial differential with respect to $ y $. If the function is differentiable, its total differential is equal to the sum of the partial differentials. Geometrically, the total differential $ df ( x _ {0} , y _ {0} ) $ is the increment in the $ z $- direction in the tangent plane to the surface $ z = f ( x , y ) $ at the point $ ( x _ {0} , y _ {0} , z _ {0} ) $, where $ z _ {0} = f ( x _ {0} , y _ {0} ) $( Fig. c).

Figure: d031810c

The following is a sufficient criterion for the differentiability of a function: If in a certain neighbourhood of a point $ ( x _ {0} , y _ {0} ) $ a function $ f $ has a partial derivative $ f _ {x} ^ { \prime } $ which is continuous at $ ( x _ {0} , y _ {0} ) $ and, in addition, has a partial derivative $ f _ {y} ^ { \prime } $ at that point, then $ f $ is differentiable at that point.

If a function $ f $ is differentiable at all points of an open domain $ D $, then at any point of the domain

$$ d z = A ( x , y ) d x + B ( x , y ) d y , $$

where $ A ( x , y ) = f _ {x} ^ { \prime } ( x , y ) $, $ B ( x , y ) = f _ {y} ^ { \prime } ( x , y ) $. If, in addition, there exist continuous partial derivatives $ A _ {y} ^ \prime $ and $ B _ {x} ^ { \prime } $ in $ D $, then, everywhere in $ D $,

$$ A _ {y} ^ \prime = B _ {x} ^ { \prime } . $$

This proves, in particular, that not every expression

$$ A ( x , y ) d x + B ( x , y ) d y $$

with continuous $ A $ and $ B $( in a domain $ D $) is the total differential of some function of two variables. This is a difference from functions of one variable, where any expression $ A ( x) d x $ with a continuous function $ A $ in some interval is the differential of some function.

The expression $ A dx + B dy $ is the total differential of some function $ z = f ( x , y ) $ in a simply-connected open domain $ D $ if $ A $ and $ B $ are continuous in this domain, meet the condition $ A _ {y} ^ \prime = B _ {x} ^ { \prime } $ and, in addition: a) $ A _ {y} ^ \prime $ and $ B _ {x} ^ { \prime } $ are continuous or b) $ A $ and $ B $ are everywhere differentiable in $ D $ with respect to both variables $ x $ and $ y $[7], [8].

See also Differential calculus for differentials of real-valued functions of one or more real variables and for differentials of higher orders.

3) Let a function $ f $ be defined on some set $ E $ of real numbers, let $ x $ be a limit point of this set, let $ x \in E $, $ x + \Delta x \in E $, $ \Delta y = A \Delta x + \alpha $, where $ \alpha / \Delta x \rightarrow 0 $ if $ \Delta x \rightarrow 0 $; then the function $ f $ is called differentiable with respect to the set $ E $ at $ x $, while $ dy = A \Delta x $ is called its differential with respect to the set $ E $ at $ x $. This is a generalization of the differential of a real-valued function of one real variable. Special kinds of this generalization include differentials at the end points of the interval within which the function is defined, and the approximate differential (cf. Approximate differentiability).

Differentials with respect to a set for real-valued functions of several real variables are introduced in a similar manner.

4) All definitions of differentiability and a differential given above can be extended, almost unchanged, to complex-valued functions of one or more real variables; to real-valued and complex-valued vector-functions of one or more real variables; and to complex functions and vector-functions of one or more complex variables. In functional analysis they are extended to functions of the points of an abstract space. One may speak of differentiability and of the differential of a set function with respect to some measure.

References

[1] G.P. Tolstov, "Elements of mathematical analysis" , 1–2 , Moscow (1974) (In Russian) MR0357695 MR0354961
[2] G.M. Fichtenholz, "Differential und Integralrechnung" , 1 , Deutsch. Verlag Wissenschaft. (1964) MR1191905 MR1056870 MR1056869 MR0887101 MR0845556 MR0845555 MR0524565 MR0473117 MR0344040 MR0344039 MR0238635 MR0238637 MR0238636 Zbl 0143.27002
[3] L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) (In Russian) MR1617334 MR1070567 MR1070566 MR1070565 MR0866891 MR0767983 MR0767982 MR0628614 MR0619214 Zbl 1080.00002 Zbl 1080.00001 Zbl 1060.26002 Zbl 0869.00003 Zbl 0696.26002 Zbl 0703.26001 Zbl 0609.00001 Zbl 0632.26001 Zbl 0485.26002 Zbl 0485.26001
[4] S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian) Zbl 0397.00003 Zbl 0384.00004
[5] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953) MR0055409 Zbl 0052.05301
[6] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) MR1025126 MR0708717 MR0630899 MR0435771 MR0377444 MR0234241 MR0215962 MR0118796 MR1530727 MR0118795 MR0085462 MR0070045 Zbl 0932.46001 Zbl 0672.46001 Zbl 0501.46001 Zbl 0501.46002 Zbl 0235.46001 Zbl 0103.08801
[7] G.P. Tolstov, "On curvilinear and iterated integrals" Trudy Mat. Inst. Steklov. , 35 (1950) (In Russian) MR44612
[8] G.P. Tolstov, "On the total differential" Uspekhi Mat. Nauk , 3 : 5 (1948) pp. 167–170 MR0027044

Comments

See also Differentiation; Differentiation of a mapping.

For differentiation of set functions cf. Set function; Radon–Nikodým theorem, [a7].

For generalizations to functions between abstract spaces see also Fréchet derivative; Gâteaux derivative.

For the derivative of a function $ f : \mathbf C \rightarrow \mathbf C $ see Analytic function.

References

[a1] T.M. Apostol, "Calculus" , 1–2 , Blaisdell (1964) MR1908007 MR1182316 MR1182315 MR0595410 MR1536963 MR1535772 MR0271732 MR0248290 MR0247001 MR0261376 MR0250092 MR0236734 MR0236733 MR0214705 MR1532185 MR1531712 MR0087718 Zbl 0123.25902
[a2] T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1974) MR0344384 Zbl 0309.26002
[a3] W. Fleming, "Functions of several variables" , Springer (1977) MR0422527 Zbl 0348.26002
[a4] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) MR0604364 Zbl 0454.26001
[a5] R. Courant, "Vorlesungen über Differential- und Integralrechnung" , 1–2 , Springer (1971–1972) MR0190266 MR1521849 MR1521664 Zbl 0224.26001 Zbl 0217.37201 Zbl 0121.28904 Zbl 0066.30303 Zbl 0064.04704 Zbl 0003.05401 Zbl 57.0246.01 Zbl 56.0193.01 Zbl 53.0200.13 Zbl 55.0728.02
[a6] I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian) MR0640867 MR0409747 MR0259033 MR0063424 Zbl 0097.26601
[a7] G.E. Shilov, B.L. Gurevich, "Integral, measure, and derivative: a unified approach" , Dover, reprint (1977) (Translated from Russian) MR0466463 Zbl 0391.28007
[a8] A. Denjoy, "Introduction à la théorie des fonctions des variables réelles" , Gauthier-Villars (1937) Zbl 0017.10504 Zbl 63.0177.02 Zbl 63.0177.01
How to Cite This Entry:
Differential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential&oldid=28172
This article was adapted from an original article by G.P. Tolstov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article