📚 Tangent Plane and Normal to a Surface | 曲面的切平面与法线
In multivariable calculus, understanding the local behaviour of a surface requires the tangent plane and normal line. These concepts generalise the tangent line of a curve to three dimensions and are vital for linear approximation, optimisation, and geometric analysis. In the IB syllabus, students learn to determine tangent planes and normal lines for both implicit surfaces F(x, y, z) = 0 and explicit surfaces z = f(x, y). This article provides a complete, step‑by‑step guide with rigorous definitions, practical examples, and common pitfalls.
在多元微积分中，要理解曲面的局部性质，就需要掌握切平面和法线。这两个概念将曲线的切线推广到三维空间，对于线性近似、优化以及几何分析至关重要。在 IB 课程中，学生需要学会求隐式曲面 F(x, y, z) = 0 和显式曲面 z = f(x, y) 的切平面与法线方程。本文通过严谨的定义、实用的例题和常见误区，提供一个完整的逐步学习指南。
1. Surfaces in Three Dimensions | 三维空间中的曲面
A surface in ℝ³ can be represented explicitly as z = f(x, y) or implicitly by an equation of the form F(x, y, z) = 0. An explicit example is the paraboloid z = x² + y², while an implicit example is the ellipsoid x²/4 + y²/9 + z²/16 = 1. Both representations are fundamental. When we investigate a surface near a point, we look for its best linear approximation — the tangent plane. A surface is said to be smooth at a point if it has a unique, well‑defined tangent plane there, which requires the function involved to be differentiable.
ℝ³ 中的曲面可以用显式方程 z = f(x, y) 或隐式方程 F(x, y, z) = 0 来表示。抛物面 z = x² + y² 是显式例子，而椭球面 x²/4 + y²/9 + z²/16 = 1 是隐式例子。这两种表示法都是基础。当我们考察曲面上某点附近的形态时，就要寻求其最佳线性近似——切平面。如果曲面在某点具有唯一、明确确定的切平面，则该曲面在该点光滑，这要求所涉及的函数是可微的。
2. Gradient Vector and Level Surfaces | 梯度向量与等值面
Consider a differentiable function of three variables, F(x, y, z). Its gradient is the vector field ∇F = (∂F/∂x, ∂F/∂y, ∂F/∂z). A level surface is a set of points satisfying F(x, y, z) = k, where k is a constant. The gradient evaluated at a point P on the level surface is orthogonal to the surface at P. This crucial property gives us the normal vector to the surface, which is the building block for the tangent plane and normal line. Geometrically, the gradient points in the direction of maximum rate of increase of F.
考虑一个可微的三元函数 F(x, y, z)。其梯度为向量场 ∇F = (∂F/∂x, ∂F/∂y, ∂F/∂z)。等值面是满足 F(x, y, z) = k（k 为常数）的点集。在等值面上点 P 处求得的梯度正交于该点处的曲面。这一关键性质给出了曲面的法向量，它是建立切平面和法线的基础。在几何上，梯度指向 F 增加速率最快的方向。
3. Tangent Plane as a Linear Approximation | 切平面作为线性近似
For a smooth surface, the tangent plane at P is the collection of all tangent lines to curves lying on the surface that pass through P. If the surface is given explicitly by z = f(x, y), the tangent plane provides the linear approximation: for small increments Δx, Δy, the change Δz ≈ ∂f/∂x(P) · Δx + ∂f/∂y(P) · Δy. This generalises the differential of a single‑variable function. The tangent plane can thus be used to estimate values of f near a known point, which is extremely useful in applied problems.
对于光滑曲面，点 P 处的切平面是曲面上经过 P 的所有曲线的切线的全体。如果曲面由显式 z = f(x, y) 给出，则切平面给出了线性近似：对于微小增量 Δx、Δy，变动量 Δz ≈ ∂f/∂x(P) · Δx + ∂f/∂y(P) · Δy。这推广了单变量函数的微分。因此，切平面可用于在已知点附近估计 f 的值，这在应用问题中非常有用。
4. Equation of Tangent Plane for Implicit Surface F(x,y,z)=0 | 隐式曲面 F(x,y,z)=0 的切平面方程
Given an implicit surface defined by F(x, y, z) = 0 and a point P(x₀, y₀, z₀) satisfying the equation, the gradient vector ∇F(P) is normal to the surface. Any vector lying in the tangent plane is orthogonal to ∇F(P). Hence, the tangent plane equation is derived from the dot product of the normal vector with any displacement vector from P being zero:
给定由 F(x, y, z) = 0 定义的隐式曲面和满足该方程的点 P(x₀, y₀, z₀)，梯度向量 ∇F(P) 是曲面的法向量。切平面内的任何向量都与 ∇F(P) 正交。因此，切平面方程可由法向量与从 P 出发的任意位移向量的点积为零得到：
∂F/∂x(P) · (x – x₀) + ∂F/∂y(P) · (y – y₀) + ∂F/∂z(P) · (z – z₀) = 0
It is essential that all partial derivatives are evaluated at the point P. The equation can be expanded and simplified to a linear equation of the form ax + by + cz = d. This standard form describes a plane through P with normal vector (a, b, c) = ∇F(P).
务必注意，所有偏导数都要在点 P 处取值。该方程展开并化简后，可写成 ax + by + cz = d 的线性形式。这个标准形式描述了一个经过 P 且以 (a, b, c) = ∇F(P) 为法向量的平面。
5. Normal Line to a Surface | 曲面的法线
The normal line at point P passes through P and is parallel to the gradient vector ∇F(P). Its parametric equations are:
点 P 处的法线经过 P 且平行于梯度向量 ∇F(P)。其参数方程为：
x = x₀ + t · ∂F/∂x(P), y = y₀ + t · ∂F/∂y(P), z = z₀ + t · ∂F/∂z(P)
where t is a real parameter. If none of the partial derivatives is zero, symmetric equations can be written by eliminating t: (x – x₀)/Fₓ(P) = (y – y₀)/Fᵧ(P) = (z – z₀)/F₂(P). The normal line is crucial for understanding the direction in which the surface is steepest and is used in optimisation and physics.
其中 t 为实数参数。若所有偏导数均不为零，消去 t 后可写出对称式方程：(x – x₀)/Fₓ(P) = (y – y₀)/Fᵧ(P) = (z – z₀)/F₂(P)。法线对于理解曲面最陡的方向至关重要，并用于优化和物理学中。
6. Tangent Plane to an Explicit Surface z = f(x,y) | 显式曲面 z = f(x,y) 的切平面
An explicit surface z = f(x, y) can be treated as a special implicit surface by letting F(x, y, z) = f(x, y) – z = 0. The gradient is then (fₓ, fᵧ, –1). Substituting into the general tangent plane formula gives the famous linearisation formula:
显式曲面 z = f(x, y) 可通过设 F(x, y, z) = f(x, y) – z = 0 而视为一种特殊的隐式曲面。其梯度为 (fₓ, fᵧ, –1)。代入一般切平面公式，就得到著名的线性化公式：
z – z₀ = fₓ(x₀, y₀)(x – x₀) + fᵧ(x₀, y₀)(y – y₀)
This form clearly shows how the surface behaves locally like a linear function of x and y. The notation fₓ and fᵧ represent the partial derivatives ∂f/∂x and ∂f/∂y, respectively. The normal vector (fₓ, fᵧ, –1) is directed downward in the z‑component, reflecting the “z – f(x,y) = 0” structure.
此形式清楚地表明，曲面在局部像是一个关于 x 和 y 的线性函数。记号 fₓ 和 fᵧ 分别表示偏导数 ∂f/∂x 和 ∂f/∂y。法向量 (fₓ, fᵧ, –1) 的 z 分量向下，这反映了“z – f(x,y) = 0”的结构。
7. Step‑by‑Step Worked Example | 逐步求解示例
Consider the implicit surface x² + y² + z² = 9 at the point (2, 2, 1). We will find the tangent plane and the normal line. First, verify the point lies on the surface: 2² + 2² + 1² = 4 + 4 + 1 = 9, which is correct.
考虑隐式曲面 x² + y² + z² = 9 在点 (2, 2, 1) 处的切平面与法线。首先验证点是否在曲面上：2² + 2² + 1² = 4 + 4 + 1 = 9，正确。
Define F(x, y, z) = x² + y² + z² – 9 = 0. Compute the gradient: ∇F = (2x, 2y, 2z).
定义 F(x, y, z) = x² + y² + z² – 9 = 0。计算梯度：∇F = (2x, 2y, 2z)。
Evaluate the gradient at P(2, 2, 1): ∇F(2, 2, 1) = (4, 4, 2). This vector is normal to the surface.
在点 P(2, 2, 1) 处求梯度：∇F(2, 2, 1) = (4, 4, 2)。该向量即为曲面的法向量。
Tangent plane equation: 4(x – 2) + 4(y – 2) + 2(z – 1) = 0. Simplify: 4x + 4y + 2z = 8 + 8 + 2 = 18, or dividing by 2: 2x + 2y + z = 9.
切平面方程：4(x – 2) + 4(y – 2) + 2(z – 1) = 0。化简得：4x + 4y + 2z = 18，或除以 2：2x + 2y + z = 9。
Normal line: using the gradient as direction vector, parametric equations are x = 2 + 4t, y = 2 + 4t, z = 1 + 2t. A simplified direction vector (2, 2, 1) can also be used: x = 2 + 2s, y = 2 + 2s, z = 1 + s.
法线：以梯度为方向向量，参数方程为 x = 2 + 4t, y = 2 + 4t, z = 1 + 2t。亦可使用简化方向向量 (2, 2, 1)：x = 2 + 2s, y = 2 + 2s, z = 1 + s。
8. Finding Tangent Plane Using Partial Derivatives | 使用偏导数求切平面
For an explicit surface, we can directly apply the linearisation formula. Take z = 3x² + 2y² at the point (1, –1, 5). First confirm 3(1)² + 2(–1)² = 3 + 2 = 5. The partial derivatives are fₓ = 6x and fᵧ = 4y. At (1, –1), fₓ = 6 and fᵧ = –4.
对于显式曲面，可直接套用线性化公式。考虑 z = 3x² + 2y² 在点 (1, –1, 5) 处。首先确认 3(1)² + 2(–1)² = 3 + 2 = 5。偏导数 fₓ = 6x，fᵧ = 4y。在 (1, –1) 处，fₓ = 6，fᵧ = –4。
Tangent plane equation: z – 5 = 6(x – 1) + (–4)(y – (–1)) = 6(x – 1) – 4(y + 1). Expanding gives z – 5 = 6x – 6 – 4y – 4, or z = 6x – 4y – 5. You can verify the point satisfies this plane equation.
切平面方程：z – 5 = 6(x – 1) + (–4)(y – (–1)) = 6(x – 1) – 4(y + 1)。展开得 z – 5 = 6x – 6 – 4y – 4，即 z = 6x – 4y – 5。可以验证原点的确满足此平面方程。
Had we used the implicit approach, F = 3x² + 2y² – z = 0, ∇F = (6x, 4y, –1) → at (1, –1, 5) = (6, –4, –1). The dot product gives 6(x–1) – 4(y+1) – (z–5) = 0, which simplifies to the same equation.
若采用隐式方法，设 F = 3x² + 2y² – z = 0，∇F = (6x, 4y, –1) → 在 (1, –1, 5) 处为 (6, –4, –1)。点积给出 6(x–1) – 4(y+1) – (z–5) = 0，化简后得到相同方程。
9. Geometric Interpretation of Gradient | 梯度的几何解释
The gradient vector ∇F at a point is not just a normal vector — it also indicates the direction in which the function F increases most rapidly. On a level surface F = constant, this direction is perpendicular to the surface. Consequently, the tangent plane collects all directions in which the rate of change of F is zero to first order. This dual interpretation helps students link partial derivatives, directional derivatives, and the geometry of surfaces.
点处的梯度向量 ∇F 不仅是法向量，它还指明了函数 F 增长最快的方向。在等值面 F = 常数 上，该方向垂直于曲面。因此，切平面汇集了所有使 F 的一阶变化率为零的方向。这种双重解读有助于学生将偏导数、方向导数与曲面的几何性质联系起来。
10. Common Pitfalls and How to Avoid Them | 常见陷阱与避免方法
Pitfall 1 – Forgetting to check the point lies on the surface. Always substitute the coordinates into the defining equation before finding the tangent plane.
陷阱 1 – 忘记检查点是否在曲面上。在求切平面之前，务必将坐标代入定义方程进行验证。
Pitfall 2 – Wrong gradient for explicit surfaces. When using F(x, y, z) = f(x, y) – z, the partial derivative with respect to z is –1, not 0. Losing this –1 results in an incorrect normal vector and a vertical shift error.
陷阱 2 – 显式曲面的梯度错误。当使用 F(x, y, z) = f(x, y) – z 时，关于 z 的偏导数是 –1 而非 0。忽略这个 –1 将导致错误的法向量以及铅垂方向的位移误差。
Pitfall 3 – Sign errors in the tangent plane equation. The dot product must equal zero: ∇F · (r – r₀) = 0. A common mistake is writing the equation as ∇F · r = 0, which is only valid if the surface passes through the origin and ∇F(P) is taken there.
陷阱 3 – 切平面方程中的符号错误。点积必须等于零：∇F · (r – r₀) = 0。常见的错误是写成 ∇F · r = 0，这只在曲面过原点且 ∇F(P) 在该点取值时才成立。
Pitfall 4 – Confusing the tangent plane with the normal line. The tangent plane uses the gradient as a perpendicular direction; the normal line uses it as a direction vector. Always check the geometric meaning.
陷阱 4 – 混淆切平面与法线。切平面以梯度为垂直方向，而法线则以梯度为方向向量。要始终检查几何含义。
11. Practice Problems and Summary | 练习与总结
Try these problems to solidify your understanding:
尝试以下问题来巩固理解：
- Problem 1: Find the tangent
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导