Equation of the Tangent Plane | 切平面方程

📚 Equation of the Tangent Plane | 切平面方程

In single-variable calculus, the derivative gives the slope of the tangent line, which provides the best linear approximation of a function at a point. In multivariable calculus, we extend this idea to surfaces: the tangent plane replaces the tangent line, serving as the best linear approximation to a surface z = f(x, y) at a given point (x₀, y₀). Understanding how to find the equation of a tangent plane is essential for local linearity, optimisation, and error estimation in IB Mathematics analysis and applications.

在单变量微积分中，导数给出切线的斜率，提供了函数在一点的最佳线性近似。在多变量微积分中，我们将这一概念推广到曲面：切平面取代了切线，成为曲面 z = f(x, y) 在给定点 (x₀, y₀) 的最佳线性近似。掌握如何求切平面方程对于理解局部线性、优化以及误差估计等内容在 IB 数学分析与应用中都至关重要。

1. Intuitive Idea of a Tangent Plane | 切平面的直观概念

Imagine a smooth surface like a hill. At any point on the hill, there is a flat plane that just ‘touches’ the surface without cutting through it — this is the tangent plane. In a small neighbourhood around the point, the surface and the plane are almost indistinguishable. This is the geometric basis for linear approximation: the tangent plane approximates the surface locally in the same way a tangent line approximates a curve.

想象一个光滑的曲面，比如一座小山。在山上的任意一点，都有一个正好“接触”曲面而不穿过它的平坦平面——这就是切平面。在点附近的小邻域内，曲面与切平面几乎无法区分。这是线性近似的几何基础：切平面以局部方式近似曲面，就像切线近似曲线一样。

2. Review of Partial Derivatives | 偏导数复习

For a function of two variables z = f(x, y), the partial derivative f_x = ∂f/∂x measures the rate of change of f with respect to x while y is held constant. Similarly, f_y = ∂f/∂y measures the rate of change with respect to y, with x constant. At a point (x₀, y₀), f_x(x₀, y₀) gives the slope of the tangent line in the x-direction, and f_y(x₀, y₀) gives the slope in the y-direction. These two slopes determine the orientation of the tangent plane.

对于二元函数 z = f(x, y)，偏导数 f_x = ∂f/∂x 衡量在 y 固定时 f 关于 x 的变化率。类似地，f_y = ∂f/∂y 衡量在 x 固定时关于 y 的变化率。在点 (x₀, y₀) 处，f_x(x₀, y₀) 给出沿 x 方向的切线斜率，f_y(x₀, y₀) 给出沿 y 方向的切线斜率。这两个斜率决定了切平面的方向。

3. Tangent Plane Equation for z = f(x, y) | 显函数 z = f(x, y) 的切平面方程

If f is differentiable at (x₀, y₀), then the tangent plane to the surface at the point P(x₀, y₀, z₀) with z₀ = f(x₀, y₀) has the equation:

如果 f 在 (x₀, y₀) 处可微，那么在点 P(x₀, y₀, z₀)（其中 z₀ = f(x₀, y₀)）的切平面方程为：

z − z₀ = f_x(x₀, y₀)(x − x₀) + f_y(x₀, y₀)(y − y₀)

This formula comes directly from the idea that small changes in z are approximated by the sum of the partial changes: Δz ≈ f_x Δx + f_y Δy. The tangent plane is the graph of this linear function in x and y.

这个公式直接源于微小变化近似：Δz ≈ f_x Δx + f_y Δy。切平面正是这个关于 x 和 y 的线性函数的图像。

4. Linear Approximation Interpretation | 线性近似视角

Rearranging the tangent plane equation gives the linearisation L(x, y) of f at (x₀, y₀):

重新整理切平面方程可得到 f 在 (x₀, y₀) 的线性化 L(x, y)：

L(x, y) = f(x₀, y₀) + f_x(x₀, y₀)(x − x₀) + f_y(x₀, y₀)(y − y₀)

This is the first-order Taylor polynomial in two variables. For points (x, y) near (x₀, y₀), L(x, y) provides a quick estimate of f(x, y). The quality of the approximation depends on the smoothness of f and the size of the step away from (x₀, y₀).

这是二元函数的一阶泰勒多项式。对于 (x₀, y₀) 附近的点 (x, y)，L(x, y) 可快速估算 f(x, y)。近似的好坏取决于 f 的光滑程度和离开 (x₀, y₀) 的步长大小。

5. Worked Example: Paraboloid | 示例：抛物面

Find the tangent plane to the surface z = x² + y² at the point (1, 2, 5).

求曲面 z = x² + y² 在点 (1, 2, 5) 的切平面方程。

Step 1: Compute partial derivatives. f_x = 2x, f_y = 2y. At (1, 2): f_x(1,2) = 2, f_y(1,2) = 4. Also z₀ = 5.

步骤1：计算偏导数。f_x = 2x，f_y = 2y。在 (1, 2) 处：f_x(1,2) = 2，f_y(1,2) = 4。同时 z₀ = 5。

Step 2: Substitute into the tangent plane equation:

步骤2：代入切平面方程：

z − 5 = 2(x − 1) + 4(y − 2)

Simplify to get the plane in standard form:

化简为标准形式：

z = 2x + 4y − 5

This linear function approximates the paraboloid near (1, 2). For instance, at (1.1, 2.1) the actual value is 1.21 + 4.41 = 5.62, while the tangent plane gives z = 2(1.1) + 4(2.1) − 5 = 2.2 + 8.4 − 5 = 5.6, a very close match.

这个线性函数在 (1, 2) 附近近似该抛物面。例如在 (1.1, 2.1) 处，真实值为 1.21+4.41=5.62，而切平面给出 z=2(1.1)+4(2.1)−5=5.6，非常接近。

6. Normal Vector and the Tangent Plane | 法向量与切平面

The tangent plane can also be described using its normal vector. For a surface given by z = f(x, y), rewrite it as F(x, y, z) = f(x, y) − z = 0. The gradient of F gives a normal vector:

切平面也可以通过其法向量来描述。对于由 z = f(x, y) 给出的曲面，重写为 F(x, y, z) = f(x, y) − z = 0。F 的梯度给出了一个法向量：

n = ∇F = ⟨f_x, f_y, −1⟩

At the point (x₀, y₀, z₀), the components f_x(x₀, y₀) and f_y(x₀, y₀) determine the normal direction. The tangent plane equation is then n · ⟨x − x₀, y − y₀, z − z₀⟩ = 0, which expands to exactly the same formula as before. Recognising this connection helps when dealing with implicitly defined surfaces.

在点 (x₀, y₀, z₀) 处，分量 f_x(x₀, y₀) 和 f_y(x₀, y₀) 决定了法线方向。切平面方程即为 n · ⟨x − x₀, y − y₀, z − z₀⟩ = 0，展开后与前面的公式完全一致。认识到这一联系有助于处理隐式定义的曲面。

7. Tangent Plane to an Implicit Surface | 隐式曲面的切平面

When a surface is defined implicitly by F(x, y, z) = 0 (for example, a sphere x² + y² + z² − R² = 0), the gradient ∇F = ⟨F_x, F_y, F_z

当曲面由隐函数 F(x, y, z) = 0 定义时（例如球面 x² + y² + z² − R² = 0），在曲面上一点 P₀ 处计算的梯度 ∇F = ⟨F_x, F_y, F_z

F_x(P₀)(x − x₀) + F_y(P₀)(y − y₀) + F_z(P₀)(z − z₀) = 0

This method does not require solving for z explicitly. For the sphere x² + y² + z² = 25 at the point (3, 4, 0), we have F_x = 2x → 6, F_y = 8, F_z = 0. The tangent plane equation is 6(x − 3) + 8(y − 4) + 0(z − 0) = 0, or 3x + 4y = 25.

这种方法不需要显式解出 z。对于球面 x² + y² + z² = 25 在点 (3, 4, 0) 处，有 F_x=2x→6，F_y=8，F_z=0。切平面方程为 6(x−3)+8(y−4)=0，即 3x+4y=25。

8. Using the Gradient for Quick Derivation | 利用梯度快速推导

Whether the surface is given explicitly or implicitly, remember the key: the gradient ∇F (or the vector from the partial derivatives of f) is orthogonal to the tangent plane. Therefore, the tangent plane consists of all points (x, y, z) such that the dot product of the normal vector with the displacement vector is zero. This viewpoint unifies both cases and is particularly useful when a question mixes forms.

无论曲面是显式还是隐式给出，记住关键：梯度 ∇F（或由 f 的偏导构成的向量）垂直于切平面。因此，切平面由所有满足法向量与位移向量点积为零的点 (x, y, z) 组成。这一观点将两种情形统一起来，当题目混合形式时尤其有用。

For instance, if asked for the tangent plane to a level surface of a function of three variables, such as T(x, y, z) = constant, the gradient ∇T is the normal vector, and the tangent plane equation follows directly.

例如，如果需要求三元函数等值面 T(x, y, z) = 常数的切平面，梯度 ∇T 就是法向量，切平面方程可直接写出。

9. Differentiability and Existence Conditions | 可微性与存在条件

Not every surface has a tangent plane at every point. For the tangent plane to exist at (x₀, y₀), f must be differentiable there. This requires that the partial derivatives f_x and f_y exist near (x₀, y₀) and are continuous at that point. If there is a sharp corner, cusp, or discontinuity, the tangent plane is not defined. In IB examinations, you usually work with polynomials and elementary functions that are smooth everywhere except possibly at isolated points.

并非每个曲面在每一点都有切平面。要使切平面在 (x₀, y₀) 存在，f 必须在该点可微。这就要求偏导数 f_x 和 f_y 在 (x₀, y₀) 附近存在且在该点连续。如果有尖角、尖点或不连续，则切平面不存在。在 IB 考试中，你通常处理的是处处光滑（个别孤立点除外）的多项式和初等函数。

Checking differentiability: For a function given by a single analytical expression, continuity of the first partial derivatives on an open region containing the point is sufficient.

可微性检查：对于由单一解析表达式给出的函数，其一阶偏导数在包含该点的开区域上连续是充分的。

10. Applications: Approximation and Error Estimation | 应用：近似与误差估计

The tangent plane is a practical tool for approximating function values without a calculator. For example, if you need √(4.1² + 2.9²) quickly, you can use the function f(x, y) = √(x² + y²) at (4, 3). Compute f_x=x/√(x²+y²) → 4/5=0.8, f_y=3/5=0.6, f(4,3)=5. The tangent plane gives L(4.1,2.9)=5+0.8(0.1)+0.6(−0.1)=5.02. The true value is about 5.01996, so the approximation is excellent.

切平面是一个实用工具，可在无计算器时估算函数值。例如，需要快速计算 √(4.1² + 2.9²)，可用函数 f(x, y)=√(x²+y²) 在 (4,3) 处。计算 f_x=x/√(x²+y²)→4/5=0.8，f_y=3/5=0.6，f(4,3)=5。切平面给出 L(4.1,2.9)=5+0.8(0.1)+0.6(−0.1)=5.02。真实值约 5.01996，近似极佳。

This linear approximation method also underpins error propagation formulas: when measurements x and y have small uncertainties Δx and Δy, the uncertainty in a computed quantity z = f(x, y) is approximately |f_xΔx + f_yΔy|, directly derived from the tangent plane.

这种线性近似方法也是误差传递公式的基础：当测量量 x 和 y 有微小不确定度 Δx 和 Δy 时，计算量 z = f(x, y) 的不确定度近似为 |f_xΔx + f_yΔy|，这直接源自切平面。

11. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Forgetting to evaluate partial derivatives at the given point: Always plug in (x₀, y₀) to f_x and f_y before writing the equation.
Mixing up x and y increments: The coefficient in front of (x − x₀) must be f_x(x₀, y₀), not f_y.
Using the wrong sign for the normal component: For z = f(x, y), the normal vector is ⟨f_x, f_y, −1⟩, not ⟨f_x, f_y, 1⟩.
Assuming a tangent plane exists without checking differentiability: While most IB problems use smooth functions, be aware that functions like f(x, y) = √(|xy|) fail at the origin.

忘记在给定点处计算偏导数：在写出方程之前，务必将 (x₀, y₀) 代入 f_x 和 f_y。
混淆 x 和 y 的增量：(x − x₀) 前面的系数必须是 f_x(x₀, y₀)，而不是 f_y。
法向量分量符号错误：对于 z = f(x, y)，法向量为 ⟨f_x, f_y, −1⟩，而非 ⟨f_x, f_y, 1⟩。
未检查可微性即假定切平面存在：虽然大多数 IB 题目使用光滑函数，但要注意如 f(x, y)=√(|xy|) 这样的函数在原点不存在切平面。

12. Summary and Key Formulae | 总结与关键公式

A concise table helps memorise the two main formulas:

一个简明的表格有助于记忆这两个主要公式：

Surface type 曲面类型	Equation of tangent plane 切平面方程
Explicit: z = f(x, y)	z − z₀ = f_x(x₀, y₀)(x − x₀) + f_y(x₀, y₀)(y − y₀)
Implicit: F(x, y, z) = 0	F_x(P₀)(x − x₀) + F_y(P₀)(y − y₀) + F_z(P₀)(z − z₀) = 0

Mastering these two templates will allow you to tackle any tangent plane problem in IB Mathematics, whether it appears in the context of calculus, geometry, or linear approximation.

掌握这两个模板将使你能够解决 IB 数学中任何切平面问题，无论它出现在微积分、几何还是线性近似的语境中。

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Equation of the Tangent Plane | 切平面方程