Functions of Two Variables: Geometry and Formulae | 二元函数的几何与公式

📚 Functions of Two Variables: Geometry and Formulae | 二元函数的几何与公式

, y), the geometry becomes richer: each input pair (x, y) produces an output z, and the graph becomes a surface in three-dimensional space. These functions arise naturally when modelling temperature distributions over a flat region, the height of terrain above a map, or economic quantities depending on two inputs. Understanding their geometry and the associated calculus – partial derivatives, tangent planes, stationary points – will guide you through the essential definitions, geometric interpretations and formulae.

在单变量微积分中，我们研究形如 y = f(x) 的函数，并将其解释为平面上的曲线。当我们转向两个自变量的函数 z = f(x, y) 时，几何变得更加丰富：每一对输入 (x, y) 产生一个输出 z，图形成为三维空间中的曲面。这些函数自然地出现在对平面上温度分布的建模、地图上方地形的高度或依赖两个输入的经济量描述中。理解它们的几何及相关微积分——偏导数、切平面、驻点——是 IB 数学“应用与解释”高级课程的一个重要主题。本文将带你梳理核心定义、几何解释及公式。

1. Definition and Domain | 定义与定义域

ables is a rule that assigns a unique real number z to each ordered pair (x, y) in a set D ⸤ ion; often it is the entire xy-plane or a region bounded by curves. For example, f(x, y) = √9 - x² - y² has domain {(x, y) | x² + y² ≤ 9}, a closed disk of radius 3.

二元函数是一种规则，它将一个唯一的实数 z 分配给集合 D ⸤ R² 中的每个有序对 (x, y)。我们记作 z = f(x, y)。集合 D 称为函数的定义域；它通常是整个 xy 平面或由曲线围成的区域。例如，f(x, y) = √9 - x² - y² 的定义域为 {(x, y) | x² + y² ≤ 9}，即半径为 3 的闭圆盘。

The notation (x, y) represents a point in the plane, and f(x, y) gives the 'height' above that point. If no domain is specified, we assume the natural domain is the set of all pairs (x, y) for which the expression is defined and yields a real number. Students must be able to identify and sketch domains, especially those involving inequalities like y > x² or disks and rectangles.

记号 (x, y) 表示平面上的一点，f(x, y) 给出该点上方的高度。如果没有指定定义域，我们假设自然定义域是使表达式有定义并产生实数的所有数对 (x, y)。学生必须能够识别并绘制定义域，特别是那些涉及不等式（如 y > x²）或圆盘、矩形的情况。

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2. Graphical Representation: Surfaces in 3D | 图形表示：三维空间中的曲面

e can recognise the shape: f(x, y) = x² + y² gives a paraboloid opening upwards; f(x, y) = √x² + y² ents a plane. Visualising these surfaces helps build intuition for partial derivatives and optimisation.

z = f(x, y) 的图形是三维空间中所有点 (x, y, f(x, y)) 的集合。这个图形是一个曲面。对于简单函数，我们可以识别其形状：f(x, y) = x² + y² 给出开口向上的抛物面；f(x, y) = √x² + y² 给出锥面；f(x, y) = 6 - 2x - 3y 表示一个平面。想象这些曲面有助于建立对偏导数和最优化的直观理解。

that behave like single-variable functions and are directly linked to partial derivatives.

由于手绘三维曲面较为困难，我们通常借助技术工具或使用等高线图。但是，绘制截痕是很有用的，即曲面与垂直平面 x = 常数 或 y = 常数 的交线。这些截痕是曲线，其行为类似于单变量函数，并且直接与偏导数相关。

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3. Level Curves and Contour Diagrams | 等高线与等高线图

evel curves for equally spaced values of k, we obtain a contour diagram, similar to a topographic map. On such a diagram, closely spaced contours indicate a steep surface, while widely spaced contours show a gentle slope.

对于常数 k，函数 f(x, y) 的等高线是 xy 平面上满足 f(x, y) = k 的点的集合。通过为等间距的 k 值绘制多条等高线，我们得到等高线图，这类似于地形图。在这种图上，密集的等高线表示陡峭的坡面，而稀疏的等高线则显示平缓的坡面。

. The surface gets steeper as we move away from the origin, which is reflected in the circles becoming more closely packed for larger k. Level curves are invaluable for understanding the behaviour of a function without a 3D view, and they appear frequently in IB exam questions.

例如，f(x, y) = x² + y² 的等高线是圆 x² + y² = k（k ≥ 0）。当我们远离原点时，曲面变得更陡，这反映在随着 k 增大，圆之间变得更为密集。等高线对于无需三维视图即可理解函数行为非常宝贵，并且它们经常出现在 IB 考题中。

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4. Partial Derivatives: Definition and Notation | 偏导数：定义与记号

The partial derivative of f with respect to x, denoted ∂f/∂x or f_x s a constant and differentiating with respect to x in the usual way. Similarly, ∂f/∂y or f_y is found by holding x constant. Formally:

f_x(x, y) = lim_h→0 [f(x+h, y) - f(x, y)] / h

函数 f 关于 x 的偏导数，记作 ∂f/∂x 或 f_x，是通过将 y 视为常数并对 x 进行常规求导得到的。类似地，∂f/∂y 或 f_y 是在固定 x 的情况下求得的。其形式化定义为上式。

Geometrically, f_x(a, b) is the slope of the tangent line to the surface at (a, b, f(a, b)) in the direction of the positive x-axis. In other words, it is the instantaneous rate of change of the function if we walk parallel to the x-axis. The partial derivative f_y n fy gives the slope in the y-direction. Computing partial derivatives is straightforward: for f(x, y) = x³y + e^x sin y, we have f_x = 3x²y + e^x sin y and f_y = x³ + e^x cos y.

几何上，f_x(a, b) 是曲面在点 (a, b, f(a, b)) 处沿正 x 轴方向的切线斜率。换句话说，它是如果我们沿着平行于 x 轴的方向移动时函数的瞬时变化率。偏导数 f_y 给出沿 y 方向的斜率。计算偏导数很直接：对于 f(x, y) = x³y + e^x sin y，我们有 f_x = 3x²y + e^x sin y 和 f_y = x³ + e^x cos y。

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

Just as with single-variable functions, we can differentiate partial derivatives again. The second-order partial derivatives are:

f_xx = ∂²f/∂x²,   f_yy = ∂²f/∂y²,   f_xy = ∂²f/∂y∂x,   f_yx = ∂²f/∂x∂y

与单变量函数一样，我们可以再次对偏导数求导。二阶偏导数如上所示。

Under the condition that the second partial derivatives are continuous, the mixed partials are equal: f_xy = f_yx. This is Clairaut' is extremely useful when classifying stationary points. For the function f(x, y) = x³y + e^x sin y, we obtain f_xx = 6xy + e^x sin y, f_yy = -e^x sin y, and f_xy = 3x² + e^x cos y.

在二阶偏导数连续的条件下，混合偏导数相等：f_xy = f_yx。这就是克莱罗定理，在对驻点进行分类时非常有用。对于函数 f(x, y) = x³y + e^x sin y，我们得到 f_xx = 6xy + e^x sin y, f_yy = -e^x sin y, f_xy = 3x² + e^x cos y。

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6. Tangent Planes and Linear Approximation | 切平面与线性近似

For a function of two variables, the tangent plane at (a, b) is the best linear approximation to the surface near that point. Its equation is:

z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)

对于二元函数，点 (a, b) 处的切平面是该点附近曲面最好的线性近似。其方程如上式。

he same partial slopes. The linear approximation of f near (a, b) is L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b). This formula is used to estimate function values for small changes in x and y. In IB problems, you may be asked to find the tangent plane or use linearisation to approximate a value like √4.1² + 2.9².

该平面在 (a, b, f(a, b)) 处与曲面相切，并具有相同的偏斜率。f 在 (a, b) 附近的线性近似为 L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)。该公式用于估计 x 和 y 发生微小变化时的函数值。在 IB 问题中，你可能需要求切平面或使用线性化来近似一个值，如 √4.1² + 2.9²。

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7. Stationary Points and Critical Points | 驻点与临界点

A point (a, b) is called a stationary point of f if both first-order partial derivatives vanish there:

f_x(a, b) = 0   and   f_y(a, b) = 0

如果函数 f 在点 (a, b) 处的两个一阶偏导数都为零，则该点称为驻点，如上式所示。

Geometrically, at a stationary point the tangent plane is horizontal. Stationary points are candidates for local maxima, local minima, or saddle points. To find them, we solve the system of equations f_x = 0 and f_y ves solving for one variable in terms of the other and substituting back. Critical points in the domain must also be examined if partial derivatives do not exist, but in IB the focus is on stationary points where partials exist.

几何上，在驻点处切平面是水平的。驻点是局部极大值、局部极小值或鞍点的候选点。为了找到它们，我们联立求解方程组 f_x = 0 和 f_y = 0。这通常涉及将一个变量用另一个变量表示并代回。如果偏导数不存在，还须考察定义域内的临界点，但在 IB 课程中重点在于偏导数存在的驻点。

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8. Classifying Stationary Points: The Second Derivative Test | 驻点分类：二阶导数判别法

Once we have found a stationary point (a, b), we compute the discriminant (also called the Hessian determinant) defined by:

D = f_xx(a, b) · f_yy(a, b) - [f_xy(a, b)]²

找到驻点 (a, b) 后，我们计算判别式（也称为黑塞行列式），定义如上。

The second derivative test for a function of two variables classifies the stationary point as follows:

• If D > 0 and f_xx(a, b) > 0, then f has a local minimum at (a, b).
• 如果 D > 0 且 f_xx(a, b) > 0，则 f 在 (a, b) 处有局部极小值。
• If D > 0 and f_xx(a, b) < 0, then f has a local maximum at (a, b).
• 如果 D > 0 且 f_xx(a, b) < 0，则 f 在 (a, b) 处有局部极大值。
• If D < 0, then (a, b) is a saddle point (neither a maximum nor a minimum).
• 如果 D < 0，则 (a, b) 是一个鞍点（既非极大值也非极小值）。
• If D = 0, the test is inconclusive; we need further investigation.
• 如果 D = 0，该判别法无法确定；我们需要进一步研究。

It is important to note that the sign of f_xx (or f_yy) matters only when D > 0. When D > 0, f_xx and f_yy must have the same sign, so it suffices to check either one. The condition D > 0 ensures the surface curves in the same way in all directions around the point.

需要注意的是，仅当 D > 0 时 f_xx（或 f_yy）的符号才有决定性。当 D > 0 时，f_xx 和 f_yy 必然同号，因此只需检查其中一个。条件 D > 0 确保曲面在该点周围的所有方向上以相同方式弯曲。

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9. Local Maxima, Minima, and Saddle Points – Geometric View | 局部极值与鞍点 – 几何视角

A local minimum occurs at a point where the surface forms a bowl shape; both principal curvatures are positive, so the discriminant is positive and f_xx A familiar example is f(x, y) = x² + y², which has a minimum at (0,0) with D = 4 > 0 and f_xx = 2 > 0.

局部极小值出现在曲面呈碗状的点；两个主曲率均为正，因此判别式为正且 f_xx > 0。一个熟悉的例子是 f(x, y) = x² + y²，它在 (0,0) 处有极小值，D = 4 > 0，f_xx = 2 > 0。

A local maximum is like an upside-down bowl: both principal curvatures are negative, D > 0 and f_xx < 0. For g(x, y) = -x² - y², the point (0,0) is a maximum.

局部极大值就像倒扣的碗：两个主曲率均为负，D > 0 且 f_xx < 0。对于 g(x, y) = -x² - y²，点 (0,0) 是极大值。

n another. The classic example is h(x, y) = x² - y² (a hyperbolic paraboloid). At (0,0), f_xx = 2, f_yy = -2, f_xy = 0, so D = (2)(-2) - 0 = -4 < 0, confirming a saddle point.

鞍点既不是极大值也不是极小值；曲面在一个方向上向上弯曲，在另一个方向上向下弯曲。经典的例子是 h(x, y) = x² - y²（双曲抛物面）。在 (0,0) 处，f_xx = 2, f_yy = -2, f_xy = 0，因此 D = (2)(-2) - 0 = -4 < 0，确认为鞍点。

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10. Worked Example: Finding and Classifying Stationary Points | 例题：寻找并分类驻点

Consider the function f(x, y) = x³ + y³ - 3xy. First, find the partial derivatives:

f_x = 3x² - 3y,   f_y = 3y² - 3x

so x(x³ - 1) = 0, leading to x = 0 or x = 1. Hence stationary points are (0,0) and (1,1).

考虑函数 f(x, y) = x³ + y³ - 3xy。首先求偏导数：f_x = 3x² - 3y, f_y = 3y² - 3x。令其为零：3x² - 3y = 0 → y = x²，3y² - 3x = 0 → x = y²。将 y = x² 代入 x = y² 得 x = x⁴，即 x(x³ - 1) = 0，解得 x = 0 或 x = 1。因此驻点为 (0,0) 和 (1,1)。

Now compute second partial derivatives: f_xx = 6x, f_yy = 6y, f_xy = -3. The discriminant is D = (6x)(6y) - (-3)² = 36xy - 9.

现在计算二阶偏导数：f_xx = 6x, f_yy = 6y, f_xy = -3。判别式为 D = (6x)(6y) - (-3)² = 36xy - 9。

At (0,0): D = -9 < 0, so (0,0) is a saddle point. At (1,1): D = 36 - 9 = 27 > 0, and f_xx cal minimum with value f(1,1) = -1. This kind of problem is typical in IB HL examinations.

在 (0,0) 处：D = -9 < 0，因此 (0,0

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