Partial differential

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.