“higher‐order partial derivative”的概念、定义、翻译、参考文献-科学参考

higher‐order partial derivative

Mathematics

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

$\frac{\partial}{\partial x_{i}} (\frac{\partial f}{\partial x_{i}}) and \frac{\partial}{\partial x_{j}} (\frac{\partial f}{\partial x_{i}}) (for j \neq i)$

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

$\frac{\partial^{2} f}{\partial x_{i}^{2}} and \frac{\partial^{2} f}{\partial x_{j} \partial x_{i}} .$

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

$\frac{\partial^{2} f}{\partial x_{i} \partial x_{j}} and \frac{\partial^{2} f}{\partial x_{j} \partial x_{i}}$

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

$\frac{\partial^{3} f}{\partial x_{1}^{3}}, \frac{\partial^{3} f}{\partial x_{1} \partial x_{2}^{2}}, \frac{\partial^{3} f}{\partial x_{1} \partial x_{2} \partial x_{3}}, \frac{\partial^{3} f}{\partial x_{2} \partial x_{3} \partial x_{1}},$

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 f_x and f_y, then f_xx, f_xy, f_yx, f_yy are used to denote

$\frac{\partial^{2} f}{\partial x^{2}}, \frac{\partial^{2} f}{\partial y \partial x}, \frac{\partial^{2} f}{\partial x \partial y}, \frac{\partial^{2} f}{\partial y^{2}}$

respectively, noting particularly that f_xy means (f_x)_y and f_yx means (f_y)_x. Alternatively, these partial derivatives f_x and f_y of f(x,y) are denoted by f₁ and f₂, with similar notations for the higher-order partial derivatives.

单词	higher‐order partial derivative
释义	higher‐order partial derivative Mathematics Given a function f of n variables x₁, x₂,…, x_n, the partial derivative ∂f/∂x_i, where 1 ≤ i ≤ n, may also be reckoned to be a function of x₁, x₂,…, x_n. So the partial derivatives of ∂f/∂x_i can be considered. Thus $\frac{\partial}{\partial x_{i}} (\frac{\partial f}{\partial x_{i}}) and \frac{\partial}{\partial x_{j}} (\frac{\partial f}{\partial x_{i}}) (for j \neq i)$ can be formed, and these are denoted, respectively, by $\frac{\partial^{2} f}{\partial x_{i}^{2}} and \frac{\partial^{2} f}{\partial x_{j} \partial x_{i}} .$ These are the second‐order partial derivatives. When j ≠ i, $\frac{\partial^{2} f}{\partial x_{i} \partial x_{j}} and \frac{\partial^{2} f}{\partial x_{j} \partial x_{i}}$ 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 $\frac{\partial^{3} f}{\partial x_{1}^{3}}, \frac{\partial^{3} f}{\partial x_{1} \partial x_{2}^{2}}, \frac{\partial^{3} f}{\partial x_{1} \partial x_{2} \partial x_{3}}, \frac{\partial^{3} f}{\partial x_{2} \partial x_{3} \partial x_{1}},$ 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 f_x and f_y, then f_xx, f_xy, f_yx, f_yy are used to denote $\frac{\partial^{2} f}{\partial x^{2}}, \frac{\partial^{2} f}{\partial y \partial x}, \frac{\partial^{2} f}{\partial x \partial y}, \frac{\partial^{2} f}{\partial y^{2}}$ respectively, noting particularly that f_xy means (f_x)_y and f_yx means (f_y)_x. Alternatively, these partial derivatives f_x and f_y of f(x,y) are denoted by f₁ and f₂, with similar notations for the higher-order partial derivatives.
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