Difference between revisions of "Partial differential"
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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 | + | 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. | ||
+ | |||
+ | $$ | ||
+ | \left . d _ {x _ {1} } f ( x _ {1} ^ {(} 0) \dots | ||
+ | x _ {n} ^ {(} 0) ) = d f ( x _ {1} , x _ {2} ^ {(} 0) \dots | ||
+ | x _ {n} ^ {(} 0) ) \right | _ {x _ {1} = x _ {1} ^ {(} 0) } = | ||
+ | $$ | ||
− | + | $$ | |
+ | = \ | ||
− | + | \frac{\partial f }{\partial x _ {1} } | |
+ | ( x _ {1} ^ {(} 0) \dots x _ {n} ^ {(} 0) ) d x _ {1} . | ||
+ | $$ | ||
It follows that | 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==== | ====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.
Partial differential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partial_differential&oldid=13943