Partial Derivatives and Functions of Two Variables | 二元函数的定义与偏导数

📚 Partial Derivatives and Functions of Two Variables | 二元函数的定义与偏导数

1. Introduction | 引言

Single-variable calculus studies functions of the form y = f(x). In many real situations, however, the output depends on two or more inputs, such as temperature depending on longitude and latitude, or production output depending on labour and capital. The concept of a function of two variables extends the familiar idea of a function to a rule that accepts a pair of inputs.

一元微积分研究形如 y = f(x) 的函数。然而在许多实际问题中，输出往往依赖于两个或更多输入，例如温度依赖于经度和纬度，产量依赖于劳动与资本。二元函数的概念把熟悉的函数概念推广到接收两个输入并产生一个输出的规则。

Partial derivatives are the natural extension of the derivative. They measure how quickly the function changes when only one input changes and all other inputs are held fixed. This article defines functions of two variables, explains their domains and graphs, and develops the rules for computing partial derivatives and interpreting them.

偏导数是导数的自然推广。它度量的是：当只有一个输入变化、而其余输入保持不变时，函数变化的快慢。本文将定义二元函数，解释其定义域与图像，并系统讲解偏导数的计算方法及其几何意义。

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2. Definition of a Function of Two Variables | 二元函数的定义

A function of two variables assigns a single real output to each ordered pair of inputs. If the pair is written as (x, y) and the output as z, we write z = f(x, y).

二元函数将每一个有序实数对 (x, y) 对应到唯一的一个实数输出 z，记作 z = f(x, y)。

A function of two variables is a rule that assigns to each ordered pair (x, y) in a subset D of ℝ² exactly one real number z, written z = f(x, y).

二元函数是一个对应规则，它将定义域 D（ℝ² 的子集）中的每个有序对 (x, y) 对应到唯一实数 z，记作 z = f(x, y)。

Here x and y are called independent variables, and z is called the dependent variable. The set D is the domain of the function. For example, f(x, y) = x² + y² assigns to every pair (x, y) the square of its distance from the origin.

其中 x 与 y 称为自变量，z 称为因变量。集合 D 是函数的定义域。例如 f(x, y) = x² + y² 将每个点 (x, y) 对应到该点到原点距离的平方。

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3. Domain and Range | 定义域与值域

Finding the domain of a two-variable function follows the same restrictions used in single-variable calculus, but now the restrictions define a region in the xy-plane rather than an interval on the x-axis.

求二元函数的定义域遵循与一元微积分相同的限制条件，只是现在限制条件在 xy 平面中确定一个区域，而不是在 x 轴上确定一个区间。

• Denominators cannot be zero: for f(x, y) = 1/(x − y), the domain is all points with x ≠ y.

  分母不能为零：对于 f(x, y) = 1/(x − y)，定义域为满足 x ≠ y 的所有点。

• Even roots require a nonnegative radicand: for f(x, y) = √(1 − x² − y²), the domain is the closed unit disc x² + y² ≤ 1.

  偶次根号下必须非负：对于 f(x, y) = √(1 − x² − y²)，定义域为闭单位圆盘 x² + y² ≤ 1。

• Logarithms require a positive argument: for f(x, y) = ln(x − y), the domain is x − y > 0.

  对数函数的真数必须为正：对于 f(x, y) = ln(x − y)，定义域为 x − y > 0。

The range is the set of all possible output values. For f(x, y) = x² + y², the range is [0, ∞). For f(x, y) = sin(xy), the range is [−1, 1].

值域是所有可能输出值的集合。例如 f(x, y) = x² + y² 的值域为 [0, ∞)；f(x, y) = sin(xy) 的值域为 [−1, 1]。

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4. Graphs and Level Curves | 图像与等值线

The graph of z = f(x, y) is a surface in three-dimensional space. Every point on the surface has coordinates (x, y, f(x, y)).

z = f(x, y) 的图像是三维空间中的一张曲面。曲面上的每个点坐标为 (x, y, f(x, y))。

Level curves are curves in the xy-plane obtained by setting f(x, y) = c for a constant c. Drawing several level curves produces a contour map, which helps us visualise the surface without drawing it in three dimensions.

等值线是令 f(x, y) = c（c 为常数）后在 xy 平面中得到的一组曲线。画出多条等值线即可得到等高线图，这有助于我们在不绘制三维图像的情况下理解曲面。

For f(x, y) = x² + y², the level curves are circles x² + y² = c for c > 0. As c increases, the circles are spaced more closely near the origin? In fact for quadratic surface, spacing changes; closer spacing indicates a steeper surface.

对于 f(x, y) = x² + y²，等值线是圆 x² + y² = c（c > 0）。等值线越密集，表示曲面越陡峭；等值线越稀疏，表示曲面越平缓。

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5. Partial Derivatives: Definition | 偏导数的定义

The partial derivative of f with respect to x is obtained by holding y constant and differentiating f with respect to x. It is denoted by ∂f/∂x or f_x.

函数 f 对 x 的偏导数通过保持 y 不变、仅对 x 求导得到，记作 ∂f/∂x 或 fₓ。

∂f/∂x = lim_Δx→0 [ f(x + Δx, y) − f(x, y) ] / Δx

∂f/∂y = lim_Δy→0 [ f(x, y + Δy) − f(x, y) ] / Δy

When we need to indicate the point where the derivative is evaluated, we write ∂f/∂x at (a, b), or f_x(a, b). The same limit definition applies: fix y = b, then differentiate with respect to x at x = a.

当需要指明在某点求导时，可写成 ∂f/∂x 在 (a, b) 处的值，或 fₓ(a, b)。极限定义仍然适用：固定 y = b，再在 x = a 处对 x 求导。

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6. Notation and Basic Computation | 偏导数的记号与基本计算

Several equivalent notations appear in textbooks and examinations.

教材与考试中会出现多种等价记号。

Leibniz notation	Subscript notation	Meaning
∂z/∂x	z_x	Derivative with respect to x, y fixed
∂z/∂y	z_y	Derivative with respect to y, x fixed
∂f/∂x|_{(a,b)}	f_x(a,b)	Derivative evaluated at a specific point

To compute a partial derivative, treat every occurrence of the other variable as a constant, then apply ordinary differentiation rules.

计算偏导数时，将另一个变量视为常数，然后使用普通求导法则即可。

For example, if f(x, y) = x² sin(y), then ∂f/∂x = 2x sin(y) because sin(y) is constant with respect to x, and ∂f/∂y = x² cos(y) because x² is constant with respect to y.

例如，

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