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''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) $.