Engineering Calculus Notes 357

Engineering Calculus Notes 357 - 3.7. HIGHER DERIVATIVES...

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Unformatted text preview: 3.7. HIGHER DERIVATIVES 345 and second-order partials f xx = 2 f 2 x = 2 f xy = 2 f yx = 2 + 3 y 2 f yx = 2 f xy = 2 + 3 y 2 f yy = 2 f 2 y = 6 xy. It is clear that the game of successive differentiation can be taken further; in general a sufficiently smooth function of two ( resp . three) variables will have 2 r ( resp . 3 r ) partial derivatives of order r . Recall that a function is called continuously differentiable , or C 1 , if its (first-order) partials exist and are continuous; Theorem 3.3.4 tells us that such functions are automatically differentiable. We shall extend this terminology to higher derivatives: a function is r times continuously differentiable or C r if all of its partial derivatives of order 1 , 2 ,...,r exist and are continuous. In practice, we shall seldom venture beyond the second-order partials. The alert reader will have noticed that the two mixed partials of the function above are equal. This is no accident; the phenomenon was firstfunction above are equal....
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