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

partial derivative

Physics

The infinitesimal change in a function consisting of two or more variables when one of the variable changes and the others remain constant. If z=f(x,y), ∂z/∂x is the partial derivative of z with respect to x, while y remains unchanged. A partial differential equation, such as the Laplace equation, is an equation containing partial derivatives of a function.

Mathematics

Suppose that f(x₁, x₂,…, x_n) is a real function of n variables. If

$\frac{f (x_{1} + h, x_{2}, \dots, x_{n}) - f (x_{1}, x_{2}, \dots, x_{n})}{h}$

tends to a limit as h → 0, this limit is the partial derivative of f, at the point (x₁, x₂,…, x_n), with respect to x₁; it is denoted by f₁(x₁, x₂,…, x_n) or ∂f/∂x₁ (read as ‘partial d f by d x₁’). This partial derivative may be found using the normal rules of differentiation, by differentiating as though the function were a function of x₁ only and treating x₂,…, x_n as constants. The other partial derivatives,

$\frac{\partial f}{\partial x_{2}}, \dots, \frac{\partial f}{\partial x_{n}},$

are defined similarly. The partial derivatives may also be denoted by $f_{x_{1}}, f_{x_{2}}, \dots, f_{x_{n}} .$ For example, if f(x,y) = xy³, then the partial derivatives are f_x = y³ and f_y = 3xy². See also chain rule (multivariable), differential (multivariate case), directional derivative, higher-order partial derivative, Jacobian matrix.

Chemical Engineering

The derivative of a function of two or more variables with respect to one of the variables and with the others remaining constant. For example, for the function $z = f (x, y) = x^{3} y^{2} + x^{2} y + y^{3} + x$ then the partial derivative of z with respect to x, is:
$\frac{\partial z}{\partial x} = 3 x^{2} y^{2} + 2 x y + 1$

(p. 273) where y is unchanged; and;
$\frac{\partial z}{\partial y} = 2 x^{3} y + x^{2} + 3 y^{2}$

where x is fixed, y is the independent variable, and z is the dependent variable. There are many other examples that use various mathematical rules in their solution such as the chain rule and the quotient rule. They are widely used in chemical engineering, such as in applications involving the movement of fluids or solving energy balances within a space that does not have any defined boundary conditions.

单词	partial derivative
释义	partial derivative Physics The infinitesimal change in a function consisting of two or more variables when one of the variable changes and the others remain constant. If z=f(x,y), ∂z/∂x is the partial derivative of z with respect to x, while y remains unchanged. A partial differential equation, such as the Laplace equation, is an equation containing partial derivatives of a function. Mathematics Suppose that f(x₁, x₂,…, x_n) is a real function of n variables. If $\frac{f (x_{1} + h, x_{2}, \dots, x_{n}) - f (x_{1}, x_{2}, \dots, x_{n})}{h}$ tends to a limit as h → 0, this limit is the partial derivative of f, at the point (x₁, x₂,…, x_n), with respect to x₁; it is denoted by f₁(x₁, x₂,…, x_n) or ∂f/∂x₁ (read as ‘partial d f by d x₁’). This partial derivative may be found using the normal rules of differentiation, by differentiating as though the function were a function of x₁ only and treating x₂,…, x_n as constants. The other partial derivatives, $\frac{\partial f}{\partial x_{2}}, \dots, \frac{\partial f}{\partial x_{n}},$ are defined similarly. The partial derivatives may also be denoted by $f_{x_{1}}, f_{x_{2}}, \dots, f_{x_{n}} .$ For example, if f(x,y) = xy³, then the partial derivatives are f_x = y³ and f_y = 3xy². See also chain rule (multivariable), differential (multivariate case), directional derivative, higher-order partial derivative, Jacobian matrix. Chemical Engineering The derivative of a function of two or more variables with respect to one of the variables and with the others remaining constant. For example, for the function $z = f (x, y) = x^{3} y^{2} + x^{2} y + y^{3} + x$ then the partial derivative of z with respect to x, is: $\frac{\partial z}{\partial x} = 3 x^{2} y^{2} + 2 x y + 1$ (p. 273) where y is unchanged; and; $\frac{\partial z}{\partial y} = 2 x^{3} y + x^{2} + 3 y^{2}$ where x is fixed, y is the independent variable, and z is the dependent variable. There are many other examples that use various mathematical rules in their solution such as the chain rule and the quotient rule. They are widely used in chemical engineering, such as in applications involving the movement of fluids or solving energy balances within a space that does not have any defined boundary conditions.
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