“higher‐order partial derivative”的概念、定义、习题解题方法及翻译-数学辞典

单词

higher‐order partial derivative

释义

higher‐order partial derivative

Given a function f of n variables x1, x2,…, xn, the partial derivative ∂f/∂xi, where 1 ≤ i ≤ n, may also be reckoned to be a function of x1, x2,…, xn. So the partial derivatives of ∂f/∂xi can be considered. Thus

can be formed, and these are denoted, respectively, by

These are the second‐order partial derivatives. When j ≠ i,

are different by definition, but the two are equal for most ‘straightforward' functions f—it is sufficient that either mixed derivative be continuous. Similarly, third‐order partial derivatives such as

can be defined, and so on. Then the nth‐order partial derivatives, where n≥2, are called the higher‐order partial derivatives.

When f is a function of two variables x and y, and the partial derivatives are denoted by fx and fy, then fxx, fxy, fyx, fyy are used to denote

respectively, noting particularly that fxy means (fx)y and fyx means (fy)x. Alternatively, these partial derivatives fx and fy of f(x,y) are denoted by f1 and f2, with similar notations for the higher-order partial derivatives.

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