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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 [[Derivative|derivative]] 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/p071620/p0716201.png" /> is defined in some neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071620/p0716202.png" />, then the partial derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071620/p0716203.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071620/p0716204.png" /> with respect to the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071620/p0716205.png" /> at that point is equal to the ordinary derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071620/p0716206.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071620/p0716207.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071620/p0716208.png" /> in the single variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071620/p0716209.png" />. In other words,
+
The [[Derivative|derivative]] 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 derivative $  ( \partial  f / \partial  x _ {1} ) ( x _ {1}  ^ {(0)}, \dots, x _ {n}  ^ {(0)} ) $
 +
of $  f $
 +
with respect to the variable $  x _ {1} $
 +
at that point is equal to the ordinary derivative $  ( d f /d x _ {1} ) ( x _ {1} , x _ {2}  ^ {(0)}, \dots, x _ {n}  ^ {(0)} ) $
 +
at the point $  x _ {1}  ^ {( 0)} $
 +
of the function $  f ( x _ {1} , x _ {2}  ^ {(0)}, \dots, x _ {n}  ^ {(0)} ) $
 +
in the single variable $  x _ {1} $.  
 +
In other words,
  
<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/p071620/p07162010.png" /></td> </tr></table>
+
$$
 +
\left .
 +
\frac{\partial  f }{\partial  x _ {1} }
 +
( x _ {1}  ^ {(0)},
 +
\dots, x _ {n}  ^ {(0)} )  =
 +
\frac{d f }{d x _ {1} }
 +
( x _ {1} , x _ {2}  ^ {(0)} \dots x _ {n}  ^ {(0)} )
 +
\right | _ {x _ {1}  = x _ {1}  ^ {(0)} } =
 +
$$
  
<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/p071620/p07162011.png" /></td> </tr></table>
+
$$
 +
= \
 +
\lim\limits _ {\Delta x _ {1} \rightarrow 0 } 
 +
\frac{\Delta _ {x _ {1}  } f ( x _ {1}  ^ {(0)}, \dots, x _ {n}  ^ {(0)} ) }{\Delta x _ {1} }
 +
,
 +
$$
  
 
where
 
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/p/p071/p071620/p07162012.png" /></td> </tr></table>
+
$$
 +
\Delta _ {x _ {1}  } f ( x _ {1}  ^ {(0)} , \dots,
 +
x _ {n}  ^ {(0)} ) =
 +
$$
  
<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/p071620/p07162013.png" /></td> </tr></table>
+
$$
 +
= \
 +
f ( x _ {1}  ^ {(0)} + \Delta x _ {1} , x _ {2}  ^ {(0)}, \dots, x _ {n}  ^ {(0)} ) - f ( x _ {1}  ^ {(0)}, \dots, x _ {n}  ^ {(0)} ) .
 +
$$
  
 
The partial derivatives
 
The partial derivatives
  
<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/p071620/p07162014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ \tag{* }
 +
 
 +
\frac{\partial  ^ {m} f }{\partial  x _ {1} ^ {m _ {1} } \cdots \partial  x _ {n} ^ {m _ {n} } }
 +
,\ \
 +
m _ {1} + \cdots + m _ {n} = m ,
 +
$$
 +
 
 +
of order  $  m > 1 $
 +
are defined by induction: If the partial derivative
  
of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071620/p07162015.png" /> are defined by induction: If the partial 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/p/p071/p071620/p07162016.png" /></td> </tr></table>
+
\frac{\partial  ^ {k- 1} f }{\partial  x _ {1} ^ {k _ {1} } \cdots \partial  x _ {i} ^ {k _ {i} } \cdots
 +
\partial  x _ {n} ^ {k _ {n} } }
 +
,\ \
 +
k _ {1} + \cdots + k _ {n} = k - 1 ,
 +
$$
  
 
has been defined, then by definition
 
has been defined, then by definition
  
<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/p071620/p07162017.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/p071620/p07162018.png" /></td> </tr></table>
+
\frac{\partial  ^ {k} f }{\partial  x _ {1} ^ {k _ {1} } \cdots \partial  x _ {i} ^ {k _ {i} + 1 } \cdots \partial  x _ {n} ^ {k _ {n} } }
 +
=
 +
$$
  
The partial derivative (*) is also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071620/p07162019.png" />. A partial derivative (*) in which at least two distinct indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071620/p07162020.png" /> are non-zero is called a mixed partial derivative; otherwise, that is, if the partial derivative has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071620/p07162021.png" />, it is called unmixed. Under fairly broad conditions, mixed partial derivatives do not depend on the order of differentiation with respect to the different variables. This holds, for example, if all the partial derivatives under consideration are continuous.
+
$$
 +
= \
  
If in the definition of a partial derivative the usual notion of a derivative is replaced by that of a generalized derivative in some sense or another, then the definition of a generalized partial derivative is obtained.
+
\frac \partial {\partial x _ {i} }
 +
\left (
 +
\frac{\partial  ^ {k- 1} f }{\partial
 +
x _ {1} ^ {k _ {1} } \cdots \partial  x _ {i} ^ {k _ {i} } \cdots \partial  x _ {n} ^ {k _ {n} } }
 +
\right ) .
 +
$$
  
 +
The partial derivative (*) is also denoted by  $  D _ {m _ {1}  \dots m _ {n} }  ^ {m} f $.
 +
A partial derivative (*) in which at least two distinct indices  $  m _ {i} $
 +
are non-zero is called a mixed partial derivative; otherwise, that is, if the partial derivative has the form  $  \partial  ^ {m} f / \partial  x _ {i}  ^ {m} $,
 +
it is called unmixed. Under fairly broad conditions, mixed partial derivatives do not depend on the order of differentiation with respect to the different variables. This holds, for example, if all the partial derivatives under consideration are continuous.
  
 +
If in the definition of a partial derivative the usual notion of a derivative is replaced by that of a generalized derivative in some sense or another, then the definition of a generalized partial derivative is obtained.
  
 
====Comments====
 
====Comments====
 
For references see [[Differential calculus|Differential calculus]].
 
For references see [[Differential calculus|Differential calculus]].

Latest revision as of 07:57, 13 May 2022


of the first order of a function in several variables

The derivative 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 derivative $ ( \partial f / \partial x _ {1} ) ( x _ {1} ^ {(0)}, \dots, x _ {n} ^ {(0)} ) $ of $ f $ with respect to the variable $ x _ {1} $ at that point is equal to the ordinary derivative $ ( d f /d x _ {1} ) ( x _ {1} , x _ {2} ^ {(0)}, \dots, x _ {n} ^ {(0)} ) $ at the point $ x _ {1} ^ {( 0)} $ of the function $ f ( x _ {1} , x _ {2} ^ {(0)}, \dots, x _ {n} ^ {(0)} ) $ in the single variable $ x _ {1} $. In other words,

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

$$ = \ \lim\limits _ {\Delta x _ {1} \rightarrow 0 } \frac{\Delta _ {x _ {1} } f ( x _ {1} ^ {(0)}, \dots, x _ {n} ^ {(0)} ) }{\Delta x _ {1} } , $$

where

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

$$ = \ f ( x _ {1} ^ {(0)} + \Delta x _ {1} , x _ {2} ^ {(0)}, \dots, x _ {n} ^ {(0)} ) - f ( x _ {1} ^ {(0)}, \dots, x _ {n} ^ {(0)} ) . $$

The partial derivatives

$$ \tag{* } \frac{\partial ^ {m} f }{\partial x _ {1} ^ {m _ {1} } \cdots \partial x _ {n} ^ {m _ {n} } } ,\ \ m _ {1} + \cdots + m _ {n} = m , $$

of order $ m > 1 $ are defined by induction: If the partial derivative

$$ \frac{\partial ^ {k- 1} f }{\partial x _ {1} ^ {k _ {1} } \cdots \partial x _ {i} ^ {k _ {i} } \cdots \partial x _ {n} ^ {k _ {n} } } ,\ \ k _ {1} + \cdots + k _ {n} = k - 1 , $$

has been defined, then by definition

$$ \frac{\partial ^ {k} f }{\partial x _ {1} ^ {k _ {1} } \cdots \partial x _ {i} ^ {k _ {i} + 1 } \cdots \partial x _ {n} ^ {k _ {n} } } = $$

$$ = \ \frac \partial {\partial x _ {i} } \left ( \frac{\partial ^ {k- 1} f }{\partial x _ {1} ^ {k _ {1} } \cdots \partial x _ {i} ^ {k _ {i} } \cdots \partial x _ {n} ^ {k _ {n} } } \right ) . $$

The partial derivative (*) is also denoted by $ D _ {m _ {1} \dots m _ {n} } ^ {m} f $. A partial derivative (*) in which at least two distinct indices $ m _ {i} $ are non-zero is called a mixed partial derivative; otherwise, that is, if the partial derivative has the form $ \partial ^ {m} f / \partial x _ {i} ^ {m} $, it is called unmixed. Under fairly broad conditions, mixed partial derivatives do not depend on the order of differentiation with respect to the different variables. This holds, for example, if all the partial derivatives under consideration are continuous.

If in the definition of a partial derivative the usual notion of a derivative is replaced by that of a generalized derivative in some sense or another, then the definition of a generalized partial derivative is obtained.

Comments

For references see Differential calculus.

How to Cite This Entry:
Partial derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partial_derivative&oldid=17129
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article