Total increment

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) $.