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Date: 21-8-2018
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Date: 21-9-2018
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Date: 12-10-2018
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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) |
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) |
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) |
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) |
If the continuity requirement for mixed partials is dropped, it is possible to construct functions for which mixed partials are notequal. An example is the function
(7) |
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.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 883-885, 1972.
Fischer, G. (Ed.). Plate 121 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 118, 1986.
Thomas, G. B. and Finney, R. L. §16.8 in Calculus and Analytic Geometry, 9th ed. Reading, MA: Addison-Wesley, 1996.
Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 83-85, 1991.
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