“Partial Derivative”的概念、定义、习题解题方法及翻译-数学辞典

单词

Partial Derivative

释义

Partial Derivative

Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interestare held fixed during the differentiation.

(1)

The above partial derivative is sometimes denoted

for brevity. For a ``nice'' 2-D function

(i.e., one forwhich

exist and are continuous in a Neighborhood

), then

. Partial derivatives involving more than one variable are called Mixed PartialDerivatives.

For nice functions, mixed partial derivatives must be equal regardless of the order in which the differentiation is performedso, for example,

(2)

(3)

For an Exact Differential,

(4)

(5)

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

(6)

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.

See also Ablowitz-Ramani-Segur Conjecture, Derivative, Mixed Partial Derivative, Monkey Saddle

References

Abramowitz, M. and Stegun, C. 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/Mathematical Models, Bildband/Photograph Volume. 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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