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''of the first order of a function in several variables''
 
''of the first order of a function in several variables''
  
The [[Differential|differential]] of the function with respect to one of the variables, keeping the remaining variables fixed. For example, if a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p0716301.png" /> is defined in some neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p0716302.png" />, then the partial differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p0716303.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p0716304.png" /> with respect to the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p0716305.png" /> at the given point is equal to the ordinary differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p0716306.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p0716307.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p0716308.png" /> in the single variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p0716309.png" />, i.e.
+
The [[Differential|differential]] of the function with respect to one of the variables, keeping the remaining variables fixed. For example, if a function $  f ( x _ {1} \dots x _ {n} ) $
 +
is defined in some neighbourhood of a point $  ( x _ {1}  ^ {(} 0) \dots x _ {n}  ^ {(} 0) ) $,  
 +
then the partial differential $  d _ {x _ {1}  } f ( x _ {1}  ^ {(} 0) \dots x _ {n}  ^ {(} 0) ) $
 +
of $  f $
 +
with respect to the variable $  x _ {1} $
 +
at the given point is equal to the ordinary differential $  d f ( x _ {1} , x _ {2}  ^ {(} 0) \dots x _ {n}  ^ {(} 0) ) $
 +
at $  x _ {1}  ^ {(} 0) $
 +
of the function $  f ( x _ {1} , x _ {2}  ^ {(} 0) \dots x _ {n}  ^ {(} 0) ) $
 +
in the single variable $  x _ {1} $,  
 +
i.e.
 +
 
 +
$$
 +
\left . d _ {x _ {1}  } f ( x _ {1}  ^ {(} 0) \dots
 +
x _ {n}  ^ {(} 0) )  =  d f ( x _ {1} , x _ {2}  ^ {(} 0) \dots
 +
x _ {n}  ^ {(} 0) ) \right | _ {x _ {1}  = x _ {1}  ^ {(} 0) } =
 +
$$
  
<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/p/p071/p071630/p07163010.png" /></td> </tr></table>
+
$$
 +
= \
  
<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/p/p071/p071630/p07163011.png" /></td> </tr></table>
+
\frac{\partial  f }{\partial  x _ {1} }
 +
( x _ {1}  ^ {(} 0) \dots x _ {n}  ^ {(} 0) )  d x _ {1} .
 +
$$
  
 
It follows that
 
It follows 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/p/p071/p071630/p07163012.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\partial  f }{\partial  x _ {1} }
 +
  = \
  
Partial differentials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p07163013.png" /> are defined analogously. For example, the partial differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p07163014.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p07163015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p07163016.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p07163017.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p07163018.png" /> is just the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p07163019.png" />-th order differential of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p07163020.png" /> in the single variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p07163021.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071630/p07163022.png" />. Hence,
+
\frac{d _ {x _ {1}  } f }{d x _ {1} }
 +
.
 +
$$
  
<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/p/p071/p071630/p07163023.png" /></td> </tr></table>
+
Partial differentials of order  $  k > 1 $
 +
are defined analogously. For example, the partial differential  $  d _ {x _ {1}  }  ^ {k} f ( x _ {1}  ^ {(} 0) \dots x _ {n}  ^ {(} 0) ) $
 +
of order  $  k $
 +
of  $  f ( x _ {1} \dots x _ {n} ) $
 +
with respect to  $  x _ {1} $
 +
at  $  ( x _ {1}  ^ {(} 0) \dots x _ {n}  ^ {(} 0) ) $
 +
is just the  $  k $-
 +
th order differential of the function  $  f ( x _ {1} , x _ {2}  ^ {(} 0) \dots x _ {n}  ^ {(} 0) ) $
 +
in the single variable  $  x _ {1} $
 +
at the point  $  x _ {1}  ^ {(} 0) $.  
 +
Hence,
  
<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/p/p071/p071630/p07163024.png" /></td> </tr></table>
+
$$
 +
d _ {x _ {i}  }  ^ {k} f ( x _ {1}  ^ {(} 0) \dots x _ {n}  ^ {(} 0) )
 +
=
 +
\frac{\partial  ^ {k} f }{\partial  x _ {i}  ^ {k} }
  
 +
( x _ {1}  ^ {(} 0) \dots x _ {n}  ^ {(} 0) )  d x _ {i}  ^ {k} ,
 +
$$
  
 +
$$
 +
i  =  1 \dots n ; \  k  =  1 , 2 , .  .  . .
 +
$$
  
 
====Comments====
 
====Comments====
 
For references see [[Differential calculus|Differential calculus]] and [[Differential|Differential]].
 
For references see [[Differential calculus|Differential calculus]] and [[Differential|Differential]].

Latest revision as of 08:05, 6 June 2020


of the first order of a function in several variables

The differential of the function with respect to one of the variables, keeping the remaining variables fixed. For example, if a function $ f ( x _ {1} \dots x _ {n} ) $ is defined in some neighbourhood of a point $ ( x _ {1} ^ {(} 0) \dots x _ {n} ^ {(} 0) ) $, then the partial differential $ d _ {x _ {1} } f ( x _ {1} ^ {(} 0) \dots x _ {n} ^ {(} 0) ) $ of $ f $ with respect to the variable $ x _ {1} $ at the given point is equal to the ordinary differential $ d f ( x _ {1} , x _ {2} ^ {(} 0) \dots x _ {n} ^ {(} 0) ) $ at $ x _ {1} ^ {(} 0) $ of the function $ f ( x _ {1} , x _ {2} ^ {(} 0) \dots x _ {n} ^ {(} 0) ) $ in the single variable $ x _ {1} $, i.e.

$$ \left . d _ {x _ {1} } f ( x _ {1} ^ {(} 0) \dots x _ {n} ^ {(} 0) ) = d f ( x _ {1} , x _ {2} ^ {(} 0) \dots x _ {n} ^ {(} 0) ) \right | _ {x _ {1} = x _ {1} ^ {(} 0) } = $$

$$ = \ \frac{\partial f }{\partial x _ {1} } ( x _ {1} ^ {(} 0) \dots x _ {n} ^ {(} 0) ) d x _ {1} . $$

It follows that

$$ \frac{\partial f }{\partial x _ {1} } = \ \frac{d _ {x _ {1} } f }{d x _ {1} } . $$

Partial differentials of order $ k > 1 $ are defined analogously. For example, the partial differential $ d _ {x _ {1} } ^ {k} f ( x _ {1} ^ {(} 0) \dots x _ {n} ^ {(} 0) ) $ of order $ k $ of $ f ( x _ {1} \dots x _ {n} ) $ with respect to $ x _ {1} $ at $ ( x _ {1} ^ {(} 0) \dots x _ {n} ^ {(} 0) ) $ is just the $ k $- th order differential of the function $ f ( x _ {1} , x _ {2} ^ {(} 0) \dots x _ {n} ^ {(} 0) ) $ in the single variable $ x _ {1} $ at the point $ x _ {1} ^ {(} 0) $. Hence,

$$ d _ {x _ {i} } ^ {k} f ( x _ {1} ^ {(} 0) \dots x _ {n} ^ {(} 0) ) = \frac{\partial ^ {k} f }{\partial x _ {i} ^ {k} } ( x _ {1} ^ {(} 0) \dots x _ {n} ^ {(} 0) ) d x _ {i} ^ {k} , $$

$$ i = 1 \dots n ; \ k = 1 , 2 , . . . . $$

Comments

For references see Differential calculus and Differential.

How to Cite This Entry:
Partial differential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partial_differential&oldid=13943
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article