A-Level Mathematics: Differentiation of Vector Functions | A-Level 数学：向量函数的微分

📚 A-Level Mathematics: Differentiation of Vector Functions | A-Level 数学：向量函数的微分

This article explains how to differentiate vector functions, a key topic in A-Level Further Mathematics. We will cover the definition, component-wise rules, product rules, and important applications in kinematics and geometry.

本文讲解 A-Level 进阶数学中向量函数的微分。我们将介绍定义、分量求导法则、数量积与向量积的求导规则，以及在运动学与几何中的重要应用。

1. What is a Vector Function? | 什么是向量函数？

A vector function assigns a vector to each value of a real variable, often time \(t\) or arc length \(s\). In three dimensions, we write:

向量函数将每一个实数变量（通常是时间 \(t\) 或弧长 \(s\)）对应一个向量。在三维空间中，我们写成：

r(t) = x(t)i + y(t)j + z(t)k

Here \(x(t), y(t), z(t)\) are scalar functions, and i, j, k are fixed unit vectors along the coordinate axes.

其中 \(x(t), y(t), z(t)\) 是数量函数，i、j、k 是沿坐标轴方向的固定单位向量。

For example, the function r(t) = cos t i + sin t j + t k describes a helix that spirals upwards as t increases.

例如，r(t) = cos t i + sin t j + t k 描述了一条随着 t 增大而螺旋上升的空间曲线。

2. The Derivative of a Vector Function | 向量函数的导数

The derivative of a vector function is defined in the same way as the derivative of a scalar function:

向量函数的导数与数量函数的导数以相同方式定义：

r'(t) = lim_Δt→0 [r(t + Δt) − r(t)] / Δt

This limit exists if each component x(t), y(t), z(t) is differentiable.

当每个分量 x(t)、y(t)、z(t) 均可导时，该极限存在。

Since i, j, k are constant vectors, we can differentiate component by component:

因为 i、j、k 是常向量，我们可以逐分量求导：

r'(t) = x'(t)i + y'(t)j + z'(t)k

This simple idea is the heart of vector differentiation.

这个简单的思想是向量微分运算的核心。

3. Component-Wise Differentiation | 分量求导法则

In practice, we differentiate each scalar component using ordinary rules of calculus, then keep the unit vectors i, j, k unchanged.

实际操作中，我们用普通微积分法则对每个数量分量求导，同时保持单位向量 i、j、k 不变。

For example, if r(t) = t² i + 3t j + e²ᵗ k, then:

例如，若 r(t) = t² i + 3t j + e²ᵗ k，则：

r'(t) = 2t i + 3j + 2e²ᵗ k

We can also write the second derivative by differentiating again:

我们还可以再次求导得到二阶导数：

r”(t) = 2i + 4e²ᵗ k

Altogether, the derivative of a vector function is simply the vector of derivatives of its components.

总之，向量函数的导数就是其各分量导数组成的向量。

4. Differentiation Rules for Scalar and Vector Products | 数量积与向量积的求导法则

For differentiable vector functions u(t) and v(t), and a differentiable scalar function φ(t), the following rules hold:

对于可导的向量函数 u(t)、v(t) 以及可导的数量函数 φ(t)，下列求导法则成立：

d/dt (u + v) = u’ + v’

d/dt (φu) = φ’u + φu’

d/dt (u · v) = u’ · v + u · v’

The dot product rule is also called the scalar product rule. The order of multiplication does not matter for dot products because u · v = v · u.

点积法则也称为数量积法则。由于 u · v = v · u，点积乘法顺序不影响结果。

For the cross product, order matters:

对于叉积，顺序很重要：

d/dt (u × v) = u’ × v + u × v’

Do not reverse the cross product factors, since u’ × v ≠ v × u’ in general.

切勿颠倒叉积中的因子，因为通常 u’ × v ≠ v × u’。

There is also a chain rule: if s is a function of t, then d/dt r(s(t)) = r'(s(t)) s'(t).

链式法则同样适用：若 s 是 t 的函数，则 d/dt r(s(t)) = r'(s(t)) s'(t)。

5. Velocity and Acceleration | 速度与加速度

The most common application in mechanics is kinematics. If a particle has position vector r(t), then its velocity is:

力学中最常见的应用是运动学。若质点位置向量为 r(t)，则其速度为：

v(t) = dr/dt

and its acceleration is:

其加速度为：

a(t) = d²r/dt²

The speed of the particle is the magnitude of the velocity, |v(t)|.

质点的速率是速度的大小，即 |v(t)|。

Consider uniform circular motion: r(t) = R cos(ωt)i + R sin(ωt)j, where R and ω are constants.

考虑匀速圆周运动：r(t) = R cos(ωt)i + R sin(ωt)j，其中 R 和 ω 为常数。

v(t) = −Rω sin(ωt)i + Rω cos(ωt)j

a(t) = −ω² [R cos(ωt)i + R sin(ωt)j] = −ω² r

Notice that a is directed towards the centre of the circle and that v · r = 0, so the velocity is perpendicular to the position vector.

请注意，加速度指向圆心，而且 v · r = 0，因此速度与位置向量垂直。

6. Tangents and Normals to Space Curves | 空间曲线的切线与法线

If a curve is given by a vector function r(t), then the tangent vector at any point is r'(t), provided r'(t) ≠ 0.

若空间曲线由向量函数 r(t) 给出，则曲线上任意一点的切向量为 r'(t)，需要 r'(t) ≠ 0。

The equation of the tangent line at t = t₀ is:

在 t = t₀ 处的切线方程为：

r = r(t₀) + λ r'(t₀)

where λ is a real parameter.

其中 λ 是实数参数。

The unit tangent vector is defined as T = r'(t) / |r'(t)|. A normal vector is any vector perpendicular to T. The principal normal vector is found by differentiating T:

单位切向量定义为 T = r'(t) / |r'(t)|。法向量是与 T 垂直的任意向量；主法向量可通过微分 T 得到：

N = dT/dt / |dT/dt|

This geometry is often tested with vector-valued position functions, especially in Further Mathematics papers.

这类几何问题经常在进阶数学考试中出现，尤其伴随向量形式的位置函数一起考查。

7. Second Derivatives and Constant Length Vectors | 二阶导数与模长恒定的向量

If the magnitude of r(t) is constant for all t, then the particle or point lies on a sphere centred at the origin. Differentiating |r|² = constant gives a useful result.

如果 r(t) 的模长对所有 t 都恒定，那么该点位于以原点为球心的球面上。对 |r|² = 常数求导会得到一个有用的结论。

Since |r|² = r · r, differentiating both sides gives:

因为 |r|² = r · r，两边求导得到：

2 r · r’ = 0, so r · r’ = 0

Therefore r and r’ are perpendicular. This is a common exam trick: whenever a problem says the length of a position vector is fixed, immediately use r · r’ = 0.

因此 r 与 r’ 互相垂直。这是一个常见考试技巧：只要题目说位置向量长度恒定，就立即使用 r · r’ = 0。

In general, the derivative of |r| with respect to t is:

一般地，|r| 对 t 的导数为：

d/dt |r| = (r · r’) / |r|

This formula is often confused with |r’|, so read questions carefully.

这个公式常与 |r’| 混淆，因此审题时务必仔细。

8. Worked Example | 综合例题

Let us work through a typical A-Level style example.

下面我们完成一个典型的 A-Level 例题。

Given r(t) = (t³ − 2t)i + e²ᵗ j + t² k, find r'(t), r”(t), and the tangent line at t = 0.

已知 r(t) = (t³ − 2t)i + e²ᵗ j + t² k，求 r'(t)、r”(t) 以及 t = 0 处的切线方程。

First differentiate each component:

首先逐分量求导：

r'(t) = (3t² − 2)i + 2e²ᵗ j + 2t k

Then differentiate again:

再求一次导数：

r”(t) = 6t i + 4e²ᵗ j + 2k

At t = 0, we have r(0) = 0i + 1j + 0k = j and r'(0) = −2i + 2j.

在 t = 0 时，r(0) = 0i + 1j + 0k = j，r'(0) = −2i + 2j。

Therefore the tangent line is:

因此切线方程为：

r = j + λ(−2i + 2j)

or equivalently r = (0, 1, 0) + λ(−2, 2, 0), where λ is a parameter.

也可写为 r = (0, 1, 0) + λ(−2, 2, 0)，其中 λ 为参数。

This gives a straight line passing through (0, 1, 0) in the direction (−2, 2, 0).

这表示一条经过点 (0, 1, 0) 且方向为 (−2, 2, 0) 的直线。

9. Common Mistakes and Exam Tips | 常见错误与考试技巧

One common mistake is forgetting that i, j, k are constant in Cartesian coordinates. This is true in standard A-Level work, but not in moving coordinate systems such as polar coordinates.

一个常见错误是忘记在笛卡尔坐标系中 i、j、k 是常向量。这在标准 A-Level 题目中成立，但在极坐标等运动坐标系中并不成立。

Another mistake is writing d/dt |r| = |r’|. The correct expression is d/dt |r| = (r · r’) / |r|.

另一个错误是写出 d/dt |r| = |r’|。正确表达式应为 d/dt |r| = (r · r’) / |r|。

When differentiating a cross product, keep the order exactly as u’ × v + u × v’, never reverse it.

对叉积求导时，必须保持顺序为 u’ × v + u × v’，绝不能颠倒。

Remember to use the chain rule whenever a component contains a composite function, such as e²ᵗ or sin(ωt).

只要分量含有复合函数，如 e²ᵗ 或 sin(ωt)，就要记得使用链式法则。

Finally, for constant-length vectors, r · r’ = 0 is one of the quickest ways to gain marks in a vector calculus question.

最后，遇到模长恒定的向量时，r · r’ = 0 是向量微积分题目中一个非常高效的得分点。

10. Conclusion | 总结

Differentiating vector functions is mostly a matter of differentiating each component separately. The derivative rules for sums, scalar multiples, dot products, cross products, and the chain rule all extend naturally from ordinary calculus.

向量函数的微分本质上就是分别对每个分量求导。对加法、数量倍数、点积、叉积以及链式法则的求导规则都可以自然地从普通微积分推广而来。

These ideas are essential for understanding velocity, acceleration, tangent lines, and the geometry of space curves in A-Level Further Mathematics.

这些思想对于理解 A-Level 进阶数学中的速度、加速度、切线以及空间曲线几何性质至关重要。

With regular practice, vector differentiation becomes a reliable and high-scoring topic.

通过经常练习，向量微分会成为一个可靠且容易得分的专题。

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A-Level Mathematics: Differentiation of Vector Functions | A-Level 数学：向量函数的微分