IB Mathematics: Tangent Planes and Normals to Surfaces | IB数学：曲面的切平面与法线

📚 IB Mathematics: Tangent Planes and Normals to Surfaces | IB数学：曲面的切平面与法线

Welcome to TutorHao’s IB Mathematics Revision Series. In this article we explore tangent planes and normals to surfaces, a core topic in multivariable calculus for IB Mathematics Analysis and Approaches HL. You will learn how to compute these objects for implicit, explicit and parametric surfaces, and how to interpret them geometrically.

欢迎来到 TutorHao 的 IB 数学复习系列。本文将探讨曲面的切平面与法线，这是 IB 数学分析与方法（AA）HL 中多元微积分的核心内容。你将学习如何对隐式、显式和参数化曲面计算这些量，并理解其几何意义。

1. Partial Derivatives and Differential Notation | 偏导数与微分记号

Before writing plane equations, recall that the partial derivative ∂f/∂x measures the rate of change of f with respect to x while holding y and z fixed. For a function f(x,y), the differential is df = (∂f/∂x)dx + (∂f/∂y)dy.

在写出平面方程之前，请回顾：偏导数 ∂f/∂x 表示在 y 和 z 保持不变时 f 关于 x 的变化率。对于函数 f(x,y)，其微分为 df = (∂f/∂x)dx + (∂f/∂y)dy。

df = (∂f/∂x)dx + (∂f/∂y)dy

This notation is essential when deriving tangent planes. The total differential tells us how small changes in the input variables produce changes in the output.

这一记号在推导切平面时至关重要。全微分告诉我们输入变量的微小变化如何引起输出的变化。

2. Implicit and Explicit Forms of a Surface | 曲面的隐式与显式表示

A surface in 3D can be described explicitly as z = f(x,y) or implicitly by F(x,y,z) = 0. For example, a sphere of radius a can be written explicitly as z = √(a² – x² – y²) (upper half) or implicitly as x² + y² + z² – a² = 0.

三维空间中的曲面可以用显式形式 z = f(x,y) 描述，也可以用隐式形式 F(x,y,z) = 0 描述。例如，半径为 a 的球面可显式写为 z = √(a² – x² – y²)（上半球），也可隐式写为 x² + y² + z² – a² = 0。

In IB problems, you must recognise which form is given and choose the corresponding formula. The implicit form is more general because it includes vertical tangent planes and closed surfaces.

在 IB 题目中，你必须识别出给出的形式并选择相应公式。隐式形式更一般，因为它可以包括竖直切平面和封闭曲面。

3. The Gradient Vector and Its Geometric Meaning | 梯度向量及其几何意义

For a differentiable function F(x,y,z), the gradient is the vector of first partial derivatives:

对于可微函数 F(x,y,z)，梯度是一阶偏导数组成的向量：

∇F = (∂F/∂x, ∂F/∂y, ∂F/∂z)

At a point P₀ on the surface F(x,y,z) = 0, the gradient ∇F(P₀) is perpendicular to the tangent plane of the surface at P₀. Thus it serves as a normal vector for the tangent plane.

在曲面 F(x,y,z) = 0 上的点 P₀ 处，梯度 ∇F(P₀) 垂直于曲面在该点的切平面。因此，它可作为切平面的一个法向量。

The gradient points in the direction of the greatest rate of increase of F, which is why it is normal to a level surface F = constant.

梯度指向 F 增长最快的方向，因此它垂直于等值面 F = 常数。

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

Suppose P₀ = (x₀, y₀, z₀) lies on the implicit surface F(x,y,z) = 0 and ∇F(P₀) ≠ 0. Then the tangent plane at P₀ is given by:

设 P₀ = (x₀, y₀, z₀) 位于隐式曲面 F(x,y,z) = 0 上，且 ∇F(P₀) ≠ 0。则曲面在 P₀ 处的切平面方程为：

∂F/∂x(P₀)(x – x₀) + ∂F/∂y(P₀)(y – y₀) + ∂F/∂z(P₀)(z – z₀) = 0

This equation comes from the fact that (x – x₀, y – y₀, z – z₀) lies in the tangent plane exactly when it is perpendicular to ∇F(P₀).

该方程来自以下事实：向量 (x – x₀, y – y₀, z – z₀) 位于切平面中，当且仅当它与 ∇F(P₀) 垂直。

Example: For the sphere x² + y² + z² = 14 at P₀ = (1, 2, 3), we have F = x² + y² + z² – 14. The gradient is (2x, 2y, 2z), so ∇F(P₀) = (2, 4, 6). The tangent plane is 2(x – 1) + 4(y – 2) + 6(z – 3) = 0, or x + 2y + 3z = 14.

示例：对于球面 x² + y² + z² = 14，在

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