多元微分 Multivariable Differential

概念与定义 Concepts and Definitions

二元函数 Bivariate Function

一个二元函数是指有两个自变量的函数，形式为 $f (x, y)$ 。自变量 $x$ 和 $y$ 分别属于实数集 $R$ 的子集，函数值 $f (x, y)$ 属于实数集 $R$

A bivariate function refers to a function with two variables, in the form of $f (x, y)$ . The variables $x$ and $y$ belong to subsets of the real numbers $R$ , and the function value $f (x, y)$ belongs to the real numbers $R$

极限 Limits

设 $f (x, y)$ 为定义在区域 $D \subset R^{2}$ 上的函数。如果存在实数 $L$ 使得当 $(x, y)$ 充分接近点 $(a, b)$ 时， $f (x, y)$ 的值无限接近 $L$ ，则称 $L$ 为 $f (x, y)$ 在 $(a, b)$ 处的极限，记作：

(x, y) \to (a, b) lim f (x, y) = L

Let $f (x, y)$ be a function defined on a region $D \subset R^{2}$ . If there exists a real number $L$ such that as $(x, y)$ approaches the point $(a, b)$ , the value of $f (x, y)$ approaches $L$ , then $L$ is called the limit of $f (x, y)$ at $(a, b)$ , denoted as:

(x, y) \to (a, b) lim f (x, y) = L

连续性 Continuity

若 $f (x, y)$ 在点 $(a, b)$ 处有定义，且满足以下条件：

(x, y) \to (a, b) lim f (x, y) = f (a, b)

则称 $f (x, y)$ 在点 $(a, b)$ 处连续

If $f (x, y)$ is defined at the point $(a, b)$ and satisfies the following condition:

(x, y) \to (a, b) lim f (x, y) = f (a, b)

then $f (x, y)$ is continuous at the point $(a, b)$

偏导数 Partial Derivatives

二元函数 $f (x, y)$ 在点 $(a, b)$ 处对 $x$ 的偏导数定义为：

f_{x} (a, b) = h \to 0 lim \frac{f ( a + h , b ) - f ( a , b )}{h}

对 $y$ 的偏导数定义为：

f_{y} (a, b) = h \to 0 lim \frac{f ( a , b + h ) - f ( a , b )}{h}

The partial derivative of a bivariate function $f (x, y)$ at the point $(a, b)$ with respect to $x$ is defined as:

f_{x} (a, b) = h \to 0 lim \frac{f ( a + h , b ) - f ( a , b )}{h}

The partial derivative with respect to $y$ is defined as:

f_{y} (a, b) = h \to 0 lim \frac{f ( a , b + h ) - f ( a , b )}{h}

性质与定理 Properties and Theorems

可微性 Differentiability

若 $f (x, y)$ 在点 $(a, b)$ 处连续，并且存在常数 $A$ 和 $B$ 使得：

f (a + h, b + k) - f (a, b) = A h + B k + o (h^{2} + k^{2})

则称 $f (x, y)$ 在点 $(a, b)$ 处可微

If $f (x, y)$ is continuous at the point $(a, b)$ , and there exist constants $A$ and $B$ such that:

f (a + h, b + k) - f (a, b) = A h + B k + o (h^{2} + k^{2})

then $f (x, y)$ is differentiable at the point $(a, b)$

混合偏导数 Mixed Partial Derivatives

若函数 $f (x, y)$ 的二阶偏导数连续，即 $f_{x y} = f_{y x}$ ，则称 $f (x, y)$ 的混合偏导数连续

If the second-order partial derivatives of the function $f (x, y)$ are continuous, i.e., $f_{x y} = f_{y x}$ , then the mixed partial derivatives of $f (x, y)$ are continuous

泰勒展开 Taylor Expansion

若 $f (x, y)$ 在点 $(a, b)$ 处有二阶连续偏导数，则可以展开为泰勒级数形式：

f (x, y) \approx f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b) + \frac{1}{2} [f_{xx} (a, b) (x - a)^{2} + 2 f_{x y} (a, b) (x - a) (y - b) + f_{yy} (a, b) (y - b)^{2}]

If $f (x, y)$ has second-order continuous partial derivatives at the point $(a, b)$ , it can be expanded in the form of a Taylor series:

f (x, y) \approx f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b) + \frac{1}{2} [f_{xx} (a, b) (x - a)^{2} + 2 f_{x y} (a, b) (x - a) (y - b) + f_{yy} (a, b) (y - b)^{2}]

计算技巧 Calculation Techniques

链式法则 Chain Rule

如果 $z = f (x, y)$ ，且 $x = g (t), y = h (t)$ ，则 $z$ 对 $t$ 的导数为：

\frac{d z}{d t} = \frac{\partial f}{\partial x} \frac{d x}{d t} + \frac{\partial f}{\partial y} \frac{d y}{d t}

If $z = f (x, y)$ , and $x = g (t), y = h (t)$ , then the derivative of $z$ with respect to $t$ is:

\frac{d z}{d t} = \frac{\partial f}{\partial x} \frac{d x}{d t} + \frac{\partial f}{\partial y} \frac{d y}{d t}

方向导数 Directional Derivative

在方向 $u = (u_{1}, u_{2})$ 上，函数 $f (x, y)$ 在点 $(a, b)$ 处的方向导数定义为：

D_{u} f (a, b) = f_{x} (a, b) u_{1} + f_{y} (a, b) u_{2}

In the direction $u = (u_{1}, u_{2})$ , the directional derivative of the function $f (x, y)$ at the point $(a, b)$ is defined as:

D_{u} f (a, b) = f_{x} (a, b) u_{1} + f_{y} (a, b) u_{2}

梯度 Gradient

函数 $f (x, y)$ 在点 $(a, b)$ 处的梯度为：

\nabla f (a, b) = (f_{x} (a, b), f_{y} (a, b))

The gradient of the function $f (x, y)$ at the point $(a, b)$ is:

\nabla f (a, b) = (f_{x} (a, b), f_{y} (a, b))

拉普拉斯算子 Laplacian

拉普拉斯算子用于描述函数的二阶导数和，定义为：

Δ f = f_{xx} + f_{yy}

The Laplacian operator is used to describe the sum of the second-order derivatives of a function, defined as:

Δ f = f_{xx} + f_{yy}

例题 Examples

例 1 Example 1

求函数 $f (x, y) = x^{2} + y^{2}$ 在点 $(1, 2)$ 处的梯度

Calculate the gradient of the function $f (x, y) = x^{2} + y^{2}$ at the point $(1, 2)$

解：

f_{x} (x, y) = 2 x f_{y} (x, y) = 2 y \nabla f (1, 2) = (2 \cdot 1, 2 \cd o t 2) = (2, 4)

Solution:

f_{x} (x, y) = 2 x f_{y} (x, y) = 2 y \nabla f (1, 2) = (2 \cdot 1, 2 \cdot 2) = (2, 4)

例 2 Example 2

计算 $z = x^{2} y + y^{3}$ 当 $x = cos (t)$ 且 $y = sin (t)$ 时的 $\frac{d z}{d t}$

Calculate $\frac{d z}{d t}$ for $z = x^{2} y + y^{3}$ when $x = cos (t)$ and $y = sin (t)$

解：

\frac{\partial z}{\partial x} = 2 x y \frac{\partial z}{\partial y} = x^{2} + 3 y^{2} \frac{d x}{d t} = - sin (t) \frac{d y}{d t} = cos (t) \frac{d z}{d t} = \frac{\partial z}{\partial x} \frac{d x}{d t} + \frac{\partial z}{\partial y} \frac{d y}{d t} = (2 x y) (- sin (t)) + (x^{2} + 3 y^{2}) (cos (t)) = (2 cos (t) sin (t)) (- sin (t)) + (cos^{2} (t) + 3 sin^{2} (t)) (cos (t))