View source for Partial derivative

''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,

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where

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The partial derivatives

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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

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has been defined, then by definition

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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.



====Comments====
For references see [[Differential calculus|Differential calculus]].