If a multivariate function has continuous partial derivatives, the order in which the derivatives are calculated does not matter. Thus, for example, the function f(x, y) of two variables has equal third partials 03 dx² dy f(x, y) = 2³ əyəx² f(x, y). (a) How many fourth partial derivatives does a function of three variables have? (b) Prove that a function of n variables has ("+-¹) rth partial derivatives.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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If a multivariate function has continuous partial derivatives, the order in which the
derivatives are calculated does not matter. Thus, for example, the function f(x, y) of
two variables has equal third partials
8³
da ² dy f(x, y) =
2³
əyəx² f (x, y).
(a) How many fourth partial derivatives does a function of three variables have?
(b) Prove that a function of n variables has ("+-¹) rth partial derivatives.
Transcribed Image Text:If a multivariate function has continuous partial derivatives, the order in which the derivatives are calculated does not matter. Thus, for example, the function f(x, y) of two variables has equal third partials 8³ da ² dy f(x, y) = 2³ əyəx² f (x, y). (a) How many fourth partial derivatives does a function of three variables have? (b) Prove that a function of n variables has ("+-¹) rth partial derivatives.
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