Graphing Functions of Two Variables | 二元函数的图像表示

📚 Graphing Functions of Two Variables | 二元函数的图像表示

A function of two variables, typically written as \(z = f(x, y)\), assigns a real number \(z\) to each ordered pair \((x, y)\) in its domain. Its graph is the set of all points \((x, y, f(x, y))\) in three-dimensional space, forming a surface that encodes the function’s behaviour.

二元函数通常写作 \(z = f(x, y)\)，它将每一个有序数对 \((x, y)\) 对应到一个实数 \(z\)。它的图像是三维空间中所有点 \((x, y, f(x, y))\) 的集合，形成一个曲面，这个曲面承载了函数的全部行为信息。

1. Domain and Range | 定义域与值域

Before graphing, we must determine the domain of \(f(x, y)\). The domain is the set of all \((x, y)\) for which \(f\) is defined, often constrained by square roots, logarithms, or denominators. The range is the set of all output values \(z\) that the function can produce.

在绘制图像之前，我们必须先确定 \(f(x, y)\) 的定义域。定义域是使 \(f\) 有意义的全体 \((x, y)\) 的集合，常受到根号、对数或分母的限制。值域是函数可能产生的全部输出值 \(z\) 的集合。

For example, \(f(x, y) = \sqrt{1 – x^2 – y^2}\) has domain \(x^2 + y^2 \le 1\), a closed unit disk, and range \([0, 1]\).

例如，\(f(x, y) = \sqrt{1 – x^2 – y^2}\) 的定义域为 \(x^2 + y^2 \le 1\)，即单位闭圆盘，值域为 \([0, 1]\)。

2. The Graph as a Surface | 曲面图像

The graph of \(z = f(x, y)\) is a surface in \(\mathbb{R}^3\). Each point \((x, y)\) in the domain is lifted vertically to height \(z = f(x, y)\). This surface can be visualised using perspective drawings, contour maps, or computer software.

\(z = f(x, y)\) 的图像是 \(\mathbb{R}^3\) 中的一个曲面。定义域中的每个点 \((x, y)\) 都被垂直抬升到高度 \(z = f(x, y)\)。我们可以通过透视图、等高线图或计算机软件来可视化这个曲面。

For instance, \(f(x, y) = x^2 + y^2\) produces a circular paraboloid opening upward, while \(f(x, y) = -x^2 – y^2\) produces a downward-opening paraboloid.

例如，\(f(x, y) = x^2 + y^2\) 生成一个开口向上的圆形抛物面，而 \(f(x, y) = -x^2 – y^2\) 则生成开口向下的抛物面。

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

Level curves are sets of points in the \(xy\)-plane where \(f(x, y) = c\) for a constant \(c\). Plotting several level curves for different \(c\) values gives a contour map, which is a two-dimensional representation of the three-dimensional surface.

等高线是 \(xy\) 平面中满足 \(f(x, y) = c\)（\(c\) 为常数）的点的集合。对不同 \(c\) 值绘制多条等高线，就得到等高线图，它是三维曲面的二维表示。

Closely spaced level curves indicate a steep surface; widely spaced curves indicate a gentle slope. For \(f(x, y) = x^2 + y^2\), the level curves are concentric circles \(x^2 + y^2 = c\).

等高线密集表示曲面陡峭；等高线稀疏表示坡度平缓。对于 \(f(x, y) = x^2 + y^2\)，其等高线是同心圆 \(x^2 + y^2 = c\)。

4. Vertical Traces and Cross-Sections | 垂直截痕与横截面

By fixing one variable, say \(y = y_0\), we obtain a cross-section \(z = f(x, y_0)\) in the plane \(y = y_0\). Similarly, fixing \(x = x_0\) gives another cross-section. These traces help us understand the shape of the surface slice by slice.

固定一个变量，例如令 \(y = y_0\)，就得到平面 \(y = y_0\) 上的截面 \(z = f(x, y_0)\)。类似地，固定 \(x = x_0\) 可得到另一个截面。这些截痕帮助我们逐片理解曲面的形状。

For \(f(x, y) = x^2 – y^2\), the trace \(y = 0\) is \(z = x^2\) (a parabola opening up), and the trace \(x = 0\) is \(z = -y^2\) (a parabola opening down). The resulting surface is a hyperbolic paraboloid, often called a saddle.

对于 \(f(x, y) = x^2 – y^2\)，截面 \(y = 0\) 是 \(z = x^2\)（开口向上的抛物线），截面 \(x = 0\) 是 \(z = -y^2\)（开口向下的抛物线）。所得曲面是双曲抛物面，常称为马鞍面。

5. Partial Derivatives and Tangent Planes | 偏导数与切平面

The partial derivatives \(f_x(x_0, y_0)\) and \(f_y(x_0, y_0)\) represent the slopes of the surface in the \(x\)- and \(y\)-directions, respectively, at the point \((x_0, y_0, z_0)\). These slopes determine the tangent plane, which is the best linear approximation to the surface near that point.

偏导数 \(f_x(x_0, y_0)\) 和 \(f_y(x_0, y_0)\) 分别表示曲面在点 \((x_0, y_0, z_0)\) 处沿 \(x\) 方向和 \(y\) 方向的斜率。这些斜率决定了切平面，它是曲面在该点附近的最佳线性逼近。

z – z₀ = fₓ(x₀, y₀)(x – x₀) + fᵧ(x₀, y₀)(y – y₀)

This formula is essential for local linearisation and for approximating values of \(f\) near a known point.

这个公式对于局部线性化以及在已知点附近近似计算 \(f\) 的值至关重要。

6. Maxima, Minima, and Saddle Points | 极大值、极小值与鞍点

A point \((a, b)\) is a local maximum if \(f(x, y) \le f(a, b)\) for all points near \((a, b)\), and a local minimum if the inequality is reversed. A saddle point is a point where the surface is neither a maximum nor a minimum, but the partial derivatives are both zero.

若在 \((a, b)\) 附近的所有点都满足 \(f(x, y) \le f(a, b)\)，则称 \((a, b)\) 为局部极大值点；若不等式反向，则为局部极小值点。鞍点是既非极大值也非极小值、但两个偏导数均为零的点。

For critical points, we use the second derivative test. Let \(D = f_{xx} f_{yy} – (f_{xy})^2\). If \(D > 0\) and \(f_{xx} > 0\), we have a local minimum; if \(D > 0\) and \(f_{xx} < 0\), a local maximum; if \(D < 0\), a saddle point; if \(D = 0\), the test is inconclusive.

对于临界点，我们使用二阶导数判别法。令 \(D = f_{xx} f_{yy} – (f_{xy})^2\)。若 \(D > 0\) 且 \(f_{xx} > 0\)，则为局部极小值；若 \(D > 0\) 且 \(f_{xx} < 0\)，则为局部极大值；若 \(D < 0\)，则为鞍点；若 \(D = 0\)，判别法失效。

7. Common Surfaces in IB Mathematics | IB数学中的常见曲面

Several standard surfaces recur throughout IB problems. Recognising their equations and graphs is crucial for quick interpretation.

在IB问题中，有几类标准曲面反复出现。快速识别它们的方程和图像至关重要。

Plane: \(z = ax + by + c\) → 平面：\(z = ax + by + c\)
Elliptic paraboloid: \(z = \frac{x^2}{a^2} + \frac{y^2}{b^2}\) → 椭圆抛物面：\(z = \frac{x^2}{a^2} + \frac{y^2}{b^2}\)
Hyperbolic paraboloid: \(z = \frac{x^2}{a^2} – \frac{y^2}{b^2}\) → 双曲抛物面：\(z = \frac{x^2}{a^2} – \frac{y^2}{b^2}\)
Ellipsoid: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\) → 椭球面：\(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\)
Cone: \(z^2 = \frac{x^2}{a^2} + \frac{y^2}{b^2}\) → 锥面：\(z^2 = \frac{x^2}{a^2} + \frac{y^2}{b^2}\)

8. Graphing with Technology | 利用技术绘图

Graphing calculators and software such as GeoGebra, Desmos 3D, or Python’s Matplotlib allow us to plot surfaces, level curves, and tangent planes. In IB exams, you may be asked to interpret such graphs rather than produce them by hand.

图形计算器以及 GeoGebra、Desmos 3D 或 Python 的 Matplotlib 等软件可以用来绘制曲面、等高线和切平面。在IB考试中，你更可能被要求解释这些图像，而不是手工绘制。

When using technology, always check the viewing window and scaling, because a surface like \(z = x^2 + y^2\) looks different if the \(z\)-axis is compressed or stretched.

使用技术工具时，务必注意观察窗口和缩放比例，因为像 \(z = x^2 + y^2\) 这样的曲面，若 \(z\) 轴被压缩或拉伸，其视觉效果会完全不同。

9. Interpreting Graphs in Exam Questions | 考试题中的图像解读

Many IB questions present a surface or a contour map and ask for properties such as approximate critical points, the sign of partial derivatives, or the shape of cross-sections. You should be able to read from the graph whether \(f_x\) is positive or negative at a given point by observing the slope in the \(x\)-direction.

许多IB题目会给出曲面或等高线图，并要求判断临界点的大致位置、偏导数的符号或截面的形状。你应该能够通过观察曲面在 \(x\) 方向的斜率，判断出给定点处 \(f_x\) 是正还是负。

For example, on a contour map, if moving in the positive \(x\)-direction crosses level curves of increasing \(c\), then \(f_x > 0\). Similarly, the spacing of contours reveals the magnitude of the gradient.

例如，在等高线图上，如果沿 \(x\) 正方向移动时穿过 \(c\) 值增大的等高线，则 \(f_x > 0\)。同样，等高线的间距揭示了梯度的大小。

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

The gradient vector \(\nabla f = (f_x, f_y)\) points in the direction of steepest ascent of the surface, and its magnitude is the rate of increase in that direction. The directional derivative \(D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u}\) gives the slope in the direction of a unit vector \(\mathbf{u}\).

梯度向量 \(\nabla f = (f_x, f_y)\) 指向曲面上升最陡的方向，其大小就是该方向上的增长率。方向导数 \(D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u}\) 给出沿单位向量 \(\mathbf{u}\) 方向的斜率。

On a contour map, the gradient at a point is perpendicular to the level curve passing through that point. This geometric interpretation is often tested in IB questions.

在等高线图上，某点的梯度垂直于经过该点的等高线。这一几何解释经常在IB题目中被考查。

11. Applications: Optimisation with Two Variables | 应用：二元函数的最优化

Finding the maximum or minimum of a function of two variables on a closed and bounded region often requires combining interior critical points with boundary analysis. This is a standard application of partial derivatives.

在闭且有界区域上求二元函数的最大值或最小值，通常需要结合内部临界点分析和边界分析。这是偏导数的一个标准应用。

If no constraint is given, we set \(f_x = 0\) and \(f_y = 0\) simultaneously. For constrained problems, the method of Lagrange multipliers may be used, which will be covered in a separate article.

若没有约束条件，我们同时令 \(f_x = 0\) 和 \(f_y = 0\)。对于有约束的问题，可以使用拉格朗日乘数法，这将在另一篇文章中介绍。

12. Summary and Key Points | 总结与要点

Graphing functions of two variables is about visualising a surface. Key skills include determining the domain, sketching or interpreting level curves, computing partial derivatives, and using the second derivative test for classification.

二元函数图像的核心是将曲面可视化。关键技能包括：确定定义域、绘制或解释等高线、计算偏导数，以及使用二阶导数判别法对临界点进行分类。

Always remember that the graph is a geometric object, and every algebraic property of \(f\) has a geometric meaning. Mastering both views will help you succeed in IB Mathematics.

始终记住：图像是一个几何对象，\(f\) 的每个代数性质都有几何意义。掌握这两种视角将帮助你在IB数学中取得成功。

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Graphing Functions of Two Variables | 二元函数的图像表示