Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation.
(1)
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The above partial derivative is sometimes denoted for brevity.
Partial derivatives can also be taken with respect to multiple variables, as denoted for examples
(2)
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(3)
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(4)
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Such partial derivatives involving more than one variable are called mixed partial derivatives.
For a "nice" two-dimensional function (i.e., one for which , , , , exist and are continuous in a neighborhood ), then
(5)
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More generally, for "nice" functions, mixed partial derivatives must be equal regardless of the order in which the differentiation is performed, so it also is true that
(6)
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If the continuity requirement for mixed partials is dropped, it is possible to construct functions for which mixed partials are not equal. An example is the function
(7)
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which has and (Wagon 1991). This function is depicted above and by Fischer (1986).
Abramowitz and Stegun (1972) give finite difference versions for partial derivatives.
A differential equation expressing one or more quantities in terms of partial derivatives is called a partial differential equation. Partial differential equations are extremely important in physics and engineering, and are in general difficult to solve.