Tangent Plane Equations: Construction and Applications | 切平面方程的建立与应用

📚 Tangent Plane Equations: Construction and Applications | 切平面方程的建立与应用

For a smooth surface in three-dimensional space, the tangent plane is the plane that best approximates the surface near a given point. It generalises the tangent line from single-variable calculus and provides the foundation for local linearisation, error analysis, and many geometric computations.

对于三维空间中的光滑曲面，切平面是在某点附近最能逼近该曲面的平面。它是一元微积分中切线概念的推广，为局部线性化、误差分析和许多几何计算奠定了基础。

1. The Intuition: Local Linearization | 直观理解：局部线性化

In one dimension, a differentiable curve y = f(x) can be approximated near x = a by its tangent line: y ≈ f(a) + f'(a)(x-a). This is the best first-degree approximation, and its error shrinks faster than |x-a| as x → a.

在一元微积分中，可微曲线 y = f(x) 在 x = a 附近可以用切线来近似：y ≈ f(a) + f'(a)(x-a)。这是一次近似中的最优选择，其误差在 x → a 时比 |x-a| 更快地趋于零。

For a surface z = f(x,y), there are two independent directions x and y. Instead of one tangent line, we need a plane whose slopes in the x- and y-directions match the partial derivatives of f at the point.

对于曲面 z = f(x,y)，存在两个独立的方向 x 和 y。因此我们需要的不是一条切线，而是一个平面；该平面在 x 方向与 y 方向上的斜率必须与函数在该点的偏导数一致。

2. Tangent Plane for Explicit Surfaces z = f(x,y) | 显式曲面的切平面

If f has continuous first partial derivatives at (a,b), then the tangent plane to the surface z = f(x,y) at the point P(a,b,f(a,b)) is given by

z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)

where f_x and f_y denote the partial derivatives ∂f/∂x and ∂f/∂y evaluated at (a,b).

其中 f_x 和 f_y 表示偏导数 ∂f/∂x 与 ∂f/∂y 在点 (a,b) 处的取值。

To see why, consider the curve obtained by fixing y = b. Its tangent vector at P is v₁ = (1,0,f_x(a,b)). Similarly, fixing x = a gives v₂ = (0,1,f_y(a,b)). Both vectors lie in the tangent plane, so the plane must contain their span.

我们可以这样理解：先在曲面上固定 y = b，得到一条曲线，它在 P 点的切向量为 v₁ = (1,0,f_x(a,b))；再固定 x = a，得到另一条曲线，其切向量为 v₂ = (0,1,f_y(a,b))。这两个向量都位于切平面内，因此切平面必须包含它们张成的平面。

For example, for z = x² + y² at (1,1), we have f(1,1)=2, f_x=2, f_y=2, so the tangent plane is z = 2 + 2(x-1) + 2(y-1) = 2x + 2y – 2.

例如，对于 z = x² + y² 在点 (1,1) 处，f(1,1)=2，f_x=2，f_y=2，所以切平面为 z = 2 + 2(x-1) + 2(y-1) = 2x + 2y – 2。

3. Tangent Plane for Implicit Surfaces F(x,y,z) = 0 | 隐式曲面的切平面

Many surfaces are given implicitly as F(x,y,z) = 0. If P(a,b,c) lies on the surface and ∇F(P) ≠ 0, then the tangent plane at P is

F_x(P)(x-a) + F_y(P)(y-b) + F_z(P)(z-c) = 0

This equation can be written compactly as ∇F(P) · (x-a, y-b, z-c) = 0, which says that the gradient vector is perpendicular to every displacement vector in the tangent plane.

该方程可简洁地写为 ∇F(P) · (x-a, y-b, z-c) = 0，即梯度向量垂直于切平面中的所有位移向量。

The same formula applies to level surfaces F(x,y,z) = k with constant k, because the equation F = k can be rewritten as F – k = 0.

同样的公式也适用于等值面 F(x,y,z) = k（其中 k 为常数），因为方程 F = k 可以改写为 F – k = 0。

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