Partial Differentiation for Functions of Two Variables | 二元函数的微分概念

📚 Partial Differentiation for Functions of Two Variables | 二元函数的微分概念

In single-variable calculus, the derivative measures how a function changes when its input changes. When we move to functions of two variables, the idea of “change” becomes richer: we can change one variable while holding the other fixed, or we can change both variables simultaneously. This article explores the core concepts of differentiation for functions of two variables, focusing on partial derivatives, total differentials, and their applications.

在单变量微积分中，导数衡量函数随输入变化的速率。当我们转向二元函数时，“变化”的含义变得更加丰富：我们可以在保持另一个变量不变的情况下改变一个变量，也可以同时改变两个变量。本文将深入探讨二元函数微分学的核心概念，重点讲解偏导数、全微分及其应用。

1. Functions of Two Variables | 二元函数

A function of two variables is a rule that assigns a unique real number to each ordered pair \((x,y)\) in a subset of the plane. We write \(z = f(x,y)\). The set of all possible input pairs is the domain, and the set of output values is the range. The graph of such a function is a surface in three-dimensional space.

二元函数是一个规则，它将平面某子集中的每个有序对 \((x,y)\) 对应到唯一实数。我们写作 \(z = f(x,y)\)。所有可能输入对的集合称为定义域，输出值的集合称为值域。二元函数的图像是三维空间中的一张曲面。

2. Partial Derivatives | 偏导数

The partial derivative of \(f\) with respect to \(x\), denoted as \(f_x\) or \(\frac{\partial f}{\partial x}\), is defined as the ordinary derivative of \(f\) treated as a function of \(x\) alone, while \(y\) is held constant. Formally:

函数 \(f\) 关于 \(x\) 的偏导数，记作 \(f_x\) 或 \(\frac{\partial f}{\partial x}\)，定义为将 \(f\) 视为仅关于 \(x\) 的函数、同时保持 \(y\) 不变时的普通导数。形式化定义如下：

\(f_x(x,y) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x, y) – f(x,y)}{\Delta x}\)

Similarly, the partial derivative with respect to \(y\) is:

类似地，关于 \(y\) 的偏导数为：

\(f_y(x,y) = \lim_{\Delta y \to 0} \frac{f(x, y+\Delta y) – f(x,y)}{\Delta y}\)

In practice, we compute \(f_x\) by using all the usual differentiation rules from single-variable calculus, treating \(y\) as a constant.

在实际计算中，我们使用单变量微积分中的所有常规求导法则来计算 \(f_x\)，将 \(y\) 视为常数。

3. Geometric Interpretation | 几何意义

The value of \(f_x(a,b)\) gives the slope of the tangent line to the surface \(z=f(x,y)\) at the point \((a,b,f(a,b))\) in the direction parallel to the \(x\)-axis. Similarly, \(f_y(a,b)\) gives the slope of the tangent line in the direction parallel to the \(y\)-axis. Thus partial derivatives describe the steepness of the surface along two perpendicular directions.

\(f_x(a,b)\) 的值给出了曲面 \(z=f(x,y)\) 在点 \((a,b,f(a,b))\) 处沿平行于 \(x\) 轴方向的切线斜率。类似地，\(f_y(a,b)\) 给出沿平行于 \(y\) 轴方向的切线斜率。因此，偏导数描述了曲面沿两个互相垂直方向的陡峭程度。

4. Higher-Order Partial Derivatives | 高阶偏导数

Just as we take derivatives of derivatives in one-variable calculus, we can take partial derivatives of partial derivatives. We denote the second partial derivatives as \(f_{xx}\), \(f_{xy}\), \(f_{yx}\), and \(f_{yy}\). For example,

正如单变量微积分中可以对导数再求导，我们也可以对偏导数再求偏导。二阶偏导数记作 \(f_{xx}\)、\(f_{xy}\)、\(f_{yx}\)、\(f_{yy}\)。例如，

\(f_{xy} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)\)

Under mild continuity conditions, the mixed partial derivatives \(f_{xy}\) and \(f_{yx}\) are equal. This is known as Clairaut’s Theorem, and it holds when both mixed partials are continuous on an open region.

在一定的连续性条件下，混合偏导数 \(f_{xy}\) 与 \(f_{yx}\) 相等。这称为克莱罗定理，当两个混合偏导数在开区域上连续时成立。

5. Total Differential | 全微分

For a function of two variables, if \(f\) has continuous partial derivatives, then the total differential \(dz\) approximates the change in \(z\) resulting from small changes \(dx\) and \(dy\). It is defined as:

对于二元函数，如果 \(f\) 具有连续的偏导数，则全微分 \(dz\) 近似表示由微小变化 \(dx\) 和 \(dy\) 引起的 \(z\) 的变化。其定义为：

\(dz = f_x(x,y)\,dx + f_y(x,y)\,dy\)

Here \(dx\) and \(dy\) are independent variables representing infinitesimal changes in \(x\) and \(y\). The total differential provides a linear approximation to the actual change \(\Delta z = f(x+\Delta x, y+\Delta y) – f(x,y)\).

其中 \(dx\) 和 \(dy\) 是表示 \(x\) 和 \(y\) 微小变化的独立变量。全微分提供了实际变化 \(\Delta z = f(x+\Delta x, y+\Delta y) – f(x,y)\) 的线性近似。

6. Differentiability of a Two-Variable Function | 二元函数的可微性

A function \(f(x,y)\) is differentiable at a point \((a,b)\) if it can be well approximated by a linear function near that point. More precisely, \(f\) is differentiable at \((a,b)\) if the increment can be written as:

函数 \(f(x,y)\) 在点 \((a,b)\) 处可微，是指它在该点附近可以被一个线性函数很好地近似。更精确地说，如果增量可以写成：

\(\Delta z = f_x(a,b)\Delta x + f_y(a,b)\Delta y + \varepsilon_1 \Delta x + \varepsilon_2 \Delta y\)

where \(\varepsilon_1 \to 0\) and \(\varepsilon_2 \to 0\) as \((\Delta x, \Delta y) \to (0,0)\). If both partial derivatives exist and are continuous in a neighbourhood of \((a,b)\), then \(f\) is differentiable there. However, existence of partial derivatives alone does not guarantee differentiability.

其中当 \((\Delta x, \Delta y) \to (0,0)\) 时，\(\varepsilon_1 \to 0\)，\(\varepsilon_2 \to 0\)。如果两个偏导数在 \((a,b)\) 的邻域内存在且连续，那么 \(f\) 在该点可微。然而，仅存在偏导数并不能保证可微性。

7. The Chain Rule | 链式法则

When a two-variable function depends on variables that themselves change with time or with other parameters, we use the chain rule. If \(z = f(x,y)\), and \(x = g(t)\), \(y = h(t)\), then the derivative of \(z\) with respect to \(t\) is:

当二元函数所依赖的变量本身随时间或其他参数变化时，我们使用链式法则。若 \(z = f(x,y)\)，且 \(x = g(t)\)，\(y = h(t)\)，则 \(z\) 关于 \(t\) 的导数为：

\(\frac{dz}{dt} = f_x(x,y)\frac{dx}{dt} + f_y(x,y)\frac{dy}{dt}\)

If \(x\) and \(y\) depend on two parameters \(s\) and \(t\), we get partial versions of the chain rule for \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t}\). The chain rule is essential in converting between coordinate systems and in implicit differentiation.

如果 \(x\) 和 \(y\) 依赖于两个参数 \(s\) 和 \(t\)，则得到 \(\frac{\partial z}{\partial s}\) 和 \(\frac{\partial z}{\partial t}\) 的偏链式法则。链式法则在坐标系转换和隐函数求导中至关重要。

8. Gradient and Directional Derivatives | 梯度与方向导数

The gradient of \(f\), denoted \(\nabla f\), is the vector of partial derivatives:

函数 \(f\) 的梯度，记作 \(\nabla f\)，是由偏导数组成的向量：

\(\nabla f(x,y) = \left( f_x(x,y),\ f_y(x,y) \right)\)

The directional derivative of \(f\) at \((x,y)\) in the direction of a unit vector \(\mathbf{u} = (a,b)\) is:

\(f\) 在 \((x,y)\) 处沿单位向量 \(\mathbf{u} = (a,b)\) 方向的方向导数为：

\(D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u} = f_x a + f_y b\)

The gradient points in the direction of the steepest increase of \(f\), and its magnitude gives the rate of increase in that direction.

梯度指向 \(f\) 增长最陡的方向，其大小给出了该方向上的增长率。

9. Linear Approximation and Differentials | 线性近似与微分

The total differential \(dz\) is the basis for linear approximation. Near a point \((a,b)\), the value of \(f\) can be approximated by:

全微分 \(dz\) 是线性近似的基础。在点 \((a,b)\) 附近，\(f\) 的值可以近似为：

\(f(x,y) \approx f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)\)

This linear approximation is often used in error estimation and numerical methods. For instance, if \(x\) and \(y\) are measured with small errors \(dx\) and \(dy\), the resulting error in \(z\) is approximately \(|dz|\).

这种线性近似常用于误差估计和数值方法。例如，若 \(x\) 和 \(y\) 的测量带有小误差 \(dx\) 和 \(dy\)，则 \(z\) 中的相应误差近似为 \(|dz|\)。

10. Application: Critical Points and Extrema | 应用：临界点与极值

To find local extrema of a differentiable function of two variables, we first locate critical points where both partial derivatives are zero:

为求二元可微函数的局部极值，我们首先寻找两个偏导数均为零的临界点：

\(f_x(a,b) = 0,\quad f_y(a,b) = 0\)

Then we use the second derivative test involving \(D = f_{xx}f_{yy} – (f_{xy})^2\). If \(D>0\) and \(f_{xx}>0\), the point is a local minimum; if \(D>0\) and \(f_{xx}<0\), it is a local maximum; if \(D<0\), it is a saddle point. This test draws on the ideas of differentiability and second-order approximation.

然后我们使用涉及 \(D = f_{xx}f_{yy} – (f_{xy})^2\) 的二阶导数判别法。若 \(D>0\) 且 \(f_{xx}>0\)，该点为局部极小值；若 \(D>0\) 且 \(f_{xx}<0\)，为局部极大值；若 \(D<0\)，则为鞍点。该判别法基于可微性和二阶近似的思想。

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Partial Differentiation for Functions of Two Variables | 二元函数的微分概念