📚 Depiction of Functions of Two Variables | 二元函数的图形描绘
tions changes dramatically. In IB Mathematics, the depiction of functions of two variables – typically written as z = f(x, y) – ic rules with geometric shapes, and prepares you for applications in physics, economics, and data science.
当我们从单变量微积分进入多元函数的世界，函数可视化和解读的方式发生了巨大变化。在IB数学中，二元函数（通常写作 z = f(x, y)）的图形描绘为我们理解曲面、等高线图以及实际情境中的优化问题打开了大门。掌握这一主题能帮助你建立代数规则与几何形状之间的联系，并为物理、经济学和数据科学中的应用做好准备。
1. What is a Function of Two Variables? | 什么是二元函数？
x, y) and returns a real number as long as the input does not force a negative value under the square root.
二元函数为平面上某个集合 D 中的每一个有序对 (x, y) 指定一个唯一的实数 z。我们记作 z = f(x, y)。集合 D 称为定义域，所有可能的输出值构成的集合称为值域。例如，f(x, y) = √9 - x² - y² 会为任何 (x, y) 返回一个实数，只要输入不会导致平方根内出现负值。
Much of our work with two-variable functions involves moving between algebraic expressions, tables of values, and geometric depictions. The primary geometric representations are three-dimensional surface plots and two-dimensional contour diagrams. Both are essential tools in the IB syllabus.
我们处理二元函数时，经常要在代数表达式、数值表和几何图形之间切换。最主要的几何表示是三维曲面图和二维等高线图，两者都是IB大纲中不可或缺的工具。
2. Visualising with 3D Surfaces | 用三维曲面图可视化
unction f(x, y) = x² + y² produces a paraboloid opening upward, often called a bowl shape.
为了在三维空间中描绘 z = f(x, y)，我们建立一个互相垂直的 x 轴、y 轴和 z 轴构成的坐标系。定义域中的每一个点 (x, y) 被提升到 xy 平面上方（或下方）的高度 z 处。由此得到的点集 (x, y, f(x, y)) 形成一个曲面。例如，函数 f(x, y) = x² + y² 产生一个开口向上的抛物面，常被形容为碗形。
Plotting surfaces by hand can be challenging, so we often rely on traces and sections. By setting one variable constant, we obtain a curve that lies on the surface. These curves help us build a wireframe model of the function. Technology like graphing calculators or GeoGebra can generate the surface instantly, but understanding the underlying construction remains crucial for the exam.
手工绘制曲面可能很困难，因此我们通常借助迹线和截面。通过将一个变量固定为常数，我们得到一条位于曲面上的曲线。这些曲线能帮助我们构建函数的线框模型。像图形计算器或GeoGebra这样的技术工具可以瞬间生成曲面，但理解其背后的构造原理对应试仍然至关重要。
3. Domain and Range in Two Variables | 二元函数的定义域与值域
Identifying the domain of a two-variable function is a natural extension of the one-variable case, but now the domain is a subset of the xy-plane. Common restrictions arise from denominators (cannot be zero), square roots (radicand must be non-negative), and logarithmic functions (argument must be positive). The domain is often described by inequalities and sketched as a shaded region in the plane.
确定二元函数的定义域是单变量情况的自然推广，但此时定义域是 xy 平面的一个子集。常见的限制条件来自分母（不能为零）、平方根（被开方数必须非负）以及对数函数（真数必须为正）。定义域常用不等式描述，并在平面上画出阴影区域。
For example, the domain of f(x, y) = √9 - x² - y² t the origin. The range of this function is [0, 3] because the square root yields non-negative values up to 3. Understanding the domain and range sets the stage for contour plots and optimisation.
例如，函数 f(x, y) = √9 - x² - y² 的定义域是满足 x² + y² ≤ 9 的点集，即以原点为中心、半径为3的闭圆盘。该函数的值域是 [0, 3]，因为平方根给出的非负值最大为3。理解定义域和值域为等高线图和优化问题奠定了基础。
4. Contour Plots: The Key to Map Interpretation | 等高线图：解读地图的关键
es are projected onto the xy-plane, much like the contour lines on a topographic map showing different altitudes.
等高线图（或称等值线图）通过绘制使函数取常数值的曲线来表示曲面。对于给定的常数 k，等值线是满足 f(x, y) = k 的点 (x, y) 的集合。这些曲线被投影到 xy 平面上，很像地形图上显示不同海拔高度的等高线。
or to sketch level curves for a given function, such as f(x, y) = xy or f(x, y) = x² - y².
间距紧密的等高线表示坡度陡峭，间距稀疏的等高线表示坡度平缓。如果等高线图呈现同心圆，则曲面很可能是一个山丘或碗状。在IB考题中，经常要求你将曲面与其等高线图配对，或者为给定的函数绘制等值线，比如 f(x, y) = xy 或 f(x, y) = x² - y²。
5. Sections and Traces | 截面与迹线
plane. These traces are essentially the intersections of the surface with vertical planes.
取截面意味着固定其中一个变量，观察得到的单变量函数。通过设定 y = y₀ 得到的垂直迹线是 xz 平面上的曲线 z = f(x, y₀)。类似地，设定 x = x₀ 得到 yz 平面上的曲线 z = f(x₀, y)。这些迹线本质上是曲面与垂直平面的交线。
veal a family of ellipses and parabolas, confirming the surface is an elliptic paraboloid.
水平迹线通过设定 z = k 得到，这正是等值线 f(x, y) = k。结合多条垂直和水平迹线，你便能建立起清晰的曲面心理图像。例如，函数 f(x, y) = x² + 4y² 在平面 x = 0、y = 0 以及 z = k 上的迹线展示了一族椭圆和抛物线，由此可确认该曲面是一个椭圆抛物面。
6. Partial Derivatives and Tangent Planes | 偏导数与切平面
The partial derivative of f with respect to x, denoted ∂f/∂x or fx irection. These two slopes determine a tangent plane that locally approximates the surface.
函数 f 关于 x 的偏导数，记作 ∂f/∂x 或 fx，是通过将 y 视为常数并对 x 求导而得到的。从几何上看，(x₀, y₀) 处的 ∂f/∂x 给出了 x 方向迹线的切线斜率。类似地，∂f/∂y 给出了 y 方向的斜率。这两个斜率共同决定了一个局部近似曲面的切平面。
The equation of the tangent plane at (x₀, y₀, z₀) is:
z - z₀ = fx(x₀, y₀)(x - x₀) + fy(x₀, y₀)(y - y₀)
在 (x₀, y₀, z₀) 处的切平面方程为：
z - z₀ = fx(x₀, y₀)(x - x₀) + fy(x₀, y₀)(y - y₀)
This linearisation is used to approximate function values near a point and is fundamental to understanding differentiability in several variables. If the partial derivatives exist and are continuous near (x₀, y₀), the function is differentiable and the tangent plane provides a good linear approximation.
这一线性化近似用于估计某点附近的函数值，是理解多元函数可微性的基础。如果偏导数在 (x₀, y₀) 附近存在且连续，那么函数可微，且切平面能提供一个良好的线性近似。
7. Critical Points and Classification | 临界点与分类
the second derivative test based on the discriminant D = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)².
临界点出现在两个一阶偏导数同时为零，或者其中某个偏导数不存在的点处。在IB课程中，我们通常通过联立求解 ∂f/∂x = 0 和 ∂f/∂y = 0 来寻找驻点。为了对这些点进行分类，我们使用基于判别式 D = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)² 的二阶导数判别法。
wnward in another. If D = 0, the test is inconclusive and further investigation is needed.
如果 D > 0 且 ∂²f/∂x² > 0，则该临界点为局部极小值点。如果 D > 0 且 ∂²f/∂x² < 0，则为局部极大值点。如果 D < 0，该点为鞍点，此时曲面在某一方向上向上弯曲，在另一方向上向下弯曲。如果 D = 0，则二阶判别法无法确定，需要进一步分析。
| Discriminant D | ∂²f/∂x² | Classification |
|---|---|---|
| D > 0 | > 0 | Local minimum |
| D > 0 | < 0 | Local maximum |
| D < 0 | Any | Saddle point |
| D = 0 | -- | Inconclusive |
上表总结了利用判别式 D 和二阶偏导数 ∂²f/∂x² 对临界点进行分类的规则。鞍点是二元函数特有的现象，在单变量函数中不存在。
8. Gradient Vector and Directional Derivatives | 梯度向量与方向导数
The gradient vector ∇ e in understanding the behaviour of f. It points in the direction of the steepest ascent on the surface, and its magnitude gives the maximum rate of change. Conversely, the negative gradient points in the direction of steepest descent.
梯度向量 ∇f = (∂f/∂x, ∂f/∂y) 在理解函数 f 的行为时扮演着核心角色。它指向曲面上最陡峭上升的方向，其模给出最大变化率。相反，负梯度则指向最陡下降的方向。
An important geometric fact is that the gradient vector at a point is always perpendicular to the level curve passing through that point. This property is used in optimisation with constraints and in interpreting contour maps. The directional derivative in the direction of a unit vector u is given by the dot product ∇f · u, which measures the rate of change of f in that specific direction.
一个重要的几何事实是，某一点的梯度向量总与经过该点的等值线垂直。这一性质可用于约束优化和等高线图的解读。函数沿单位向量 u 方向的方向导数由点积 ∇f · u 给出，它衡量了 f 在该特定方向上的变化率。
9. Optimisation in Real-World Contexts | 实际情境中的优化
Many IB questions frame functions of two variables within practical scenarios. A typical problem might ask you to maximise a profit function P(x, y) where x and y are quantities of two products, subject to a constraint like a budget or resource limit. Alternatively, you might minimise the surface area of a container with a fixed volume.
许多 IB 考题会将二元函数置于实际场景中。一个典型的问题是，给定两种产品的数量 x 和 y，要求最大化利润函数 P(x, y)，并可能附带预算或资源限制等约束条件。也可能要求你在容积固定的情况下最小化容器的表面积。
To solve such problems, you first eliminate one variable using the constraint, reducing the problem to a single-variable optimisation, or use Lagrange multipliers if studied. For unconstrained optimisation, you find critical points by setting partial derivatives to zero and then classify them. Always check the boundaries of the domain and the physical feasibility of your solutions.
求解此类问题时，你可以首先利用约束条件消去一个变量，从而将问题转化为单变量优化，或者若已学习拉格朗日乘数法则可使用该方法。对于无约束优化，通过令偏导数为零寻找临界点，然后进行分类。务必检查定义域的边界以及解的物理可行性。
10. Using Technology to Depict Functions | 利用技术描绘函数
Graphical display calculators (GDC) and software like GeoGebra or Desmos allow you to plot 3D surfaces and contour diagrams with ease. In the IB examination, you may be required to use your GDC to find numerical critical points, evaluate partial derivatives, or sketch level curves. However, the conceptual understanding must guide your interpretation of the displayed graphs.
图形显示计算器（GDC）以及 GeoGebra 或 Desmos 等软件能让你轻松绘制三维曲面和等高线图。在 IB 考试中，你可能需要使用计算器寻找数值临界点、计算偏导数值或绘制等值线草图。然而，对图形的解读必须立足于对概念的深入理解。
When using technology, always pay attention to the viewing window. A poorly chosen window can hide important features such as a saddle point or a sharp ridge. Experiment with different rotations and zoom levels to build intuition about how surfaces behave and how contour spacing reflects slope steepness.
在使用技术工具时，务必留意观察窗口的设置。选择不当的窗口可能会隐藏鞍点或锋锐棱线等重要特征。尝试不同的旋转角度和缩放水平，以建立关于曲面行为以及等高线间距如何反映坡度的直观认识。
11. Summary and Exam Tips | 总结与考试技巧
Depicting functions of two variables requires you to move flexibly between three representations: the algebraic formula, the 3D surface, and the 2D contour map. In the exam, be ready to sketch level curves for simple functions, interpret gradient vectors on contour plots, and classify critical points using the second derivative test.
描绘二元函数需要你在三种表示之间灵活转换：代数表达式、三维曲面图和二维等高线图。在考试中，应做好准备为简单函数绘制等值线草图、在等高线图上解读梯度向量，以及使用二阶导数判别法对临界点分类。
misidentifying saddle points as maxima, and ignoring domain boundary points in optimisation. Remember that technology can verify your results, but full marks go to those who can explain the geometry behind the numbers.
常见的失分点包括：在 D > 0 时忘记检查 ∂²f/∂x² 的符号、将鞍点误判为极大值、以及在优化中忽视定义域边界点。请记住，技术工具可以验证你的结果，但只有能够解释数字背后几何意义的学生才能获得满分。
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