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''of a function of several variables''
 
''of a function of several variables''
  
The increment acquired by the function when all the arguments undergo, in general non-zero, increments. More precisely, let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t0933801.png" /> be defined in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t0933802.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t0933803.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t0933804.png" /> of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t0933805.png" />. The increment
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The increment acquired by the function when all the arguments undergo, in general non-zero, increments. More precisely, let a function $  f $
 +
be defined in a neighbourhood of the point $  x  ^ {(} 0) = ( x _ {1}  ^ {(} 0) \dots x _ {n}  ^ {(} 0) ) $
 +
in the $  n $-
 +
dimensional space $  \mathbf R  ^ {n} $
 +
of the variables $  x _ {1} \dots x _ {n} $.  
 +
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/t/t093/t093380/t0933806.png" /></td> </tr></table>
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$$
 +
\Delta f  = f( x  ^ {(} 0) + \Delta x) - f( x  ^ {(} 0) )
 +
$$
  
of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t0933807.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t0933808.png" />, where
+
of the function $  f $
 +
at $  x  ^ {(} 0) $,  
 +
where
  
<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/t/t093/t093380/t0933809.png" /></td> </tr></table>
+
$$
 +
\Delta x  = ( \Delta x _ {1} \dots \Delta x _ {n} ),
 +
$$
  
<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/t/t093/t093380/t09338010.png" /></td> </tr></table>
+
$$
 +
x  ^ {(} 0) + \Delta x  = ( x _ {1}  ^ {(} 0) + \Delta x _ {1} \dots x _ {n}  ^ {(} 0) + \Delta x _ {n} ),
 +
$$
  
is called the total increment if it is considered as a function of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t09338011.png" /> possible increments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t09338012.png" /> of the arguments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t09338013.png" />, which are subject only to the condition that the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t09338014.png" /> belongs to the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t09338015.png" />. Along with the total increment of the function, one can consider the partial increments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t09338016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t09338017.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t09338018.png" /> with respect to the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t09338019.png" />, i.e. increments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t09338020.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t09338021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t09338022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t09338023.png" /> is fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093380/t09338024.png" />.
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is called the total increment if it is considered as a function of the $  n $
 +
possible increments $  \Delta x _ {1} \dots \Delta x _ {n} $
 +
of the arguments $  x _ {1} \dots x _ {n} $,  
 +
which are subject only to the condition that the point $  x  ^ {(} 0) + \Delta x $
 +
belongs to the domain of definition of $  f $.  
 +
Along with the total increment of the function, one can consider the partial increments $  \Delta _ {x _ {k}  } f $
 +
of $  f $
 +
at a point $  x  ^ {(} 0) $
 +
with respect to the variable $  x _ {k} $,  
 +
i.e. increments $  \Delta f $
 +
for which $  \Delta x _ {j} = 0 $,
 +
$  j = 1 \dots k- 1 , k+ 1 \dots n $,  
 +
and $  k $
 +
is fixed $  ( k = 1 \dots n) $.

Latest revision as of 08:26, 6 June 2020


of a function of several variables

The increment acquired by the function when all the arguments undergo, in general non-zero, increments. More precisely, let a function $ f $ be defined in a neighbourhood of the point $ x ^ {(} 0) = ( x _ {1} ^ {(} 0) \dots x _ {n} ^ {(} 0) ) $ in the $ n $- dimensional space $ \mathbf R ^ {n} $ of the variables $ x _ {1} \dots x _ {n} $. The increment

$$ \Delta f = f( x ^ {(} 0) + \Delta x) - f( x ^ {(} 0) ) $$

of the function $ f $ at $ x ^ {(} 0) $, where

$$ \Delta x = ( \Delta x _ {1} \dots \Delta x _ {n} ), $$

$$ x ^ {(} 0) + \Delta x = ( x _ {1} ^ {(} 0) + \Delta x _ {1} \dots x _ {n} ^ {(} 0) + \Delta x _ {n} ), $$

is called the total increment if it is considered as a function of the $ n $ possible increments $ \Delta x _ {1} \dots \Delta x _ {n} $ of the arguments $ x _ {1} \dots x _ {n} $, which are subject only to the condition that the point $ x ^ {(} 0) + \Delta x $ belongs to the domain of definition of $ f $. Along with the total increment of the function, one can consider the partial increments $ \Delta _ {x _ {k} } f $ of $ f $ at a point $ x ^ {(} 0) $ with respect to the variable $ x _ {k} $, i.e. increments $ \Delta f $ for which $ \Delta x _ {j} = 0 $, $ j = 1 \dots k- 1 , k+ 1 \dots n $, and $ k $ is fixed $ ( k = 1 \dots n) $.

How to Cite This Entry:
Total increment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Total_increment&oldid=13847
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article