Integration of Vector Functions | 向量函数的积分

📚 Integration of Vector Functions | 向量函数的积分

In A-Level Further Mathematics, vector functions are often used to describe the position, velocity, or acceleration of a particle in space. Integrating a vector function means integrating each of its components separately with respect to a single variable, usually time (t) or a parameter. This process is essential for recovering position from velocity, or velocity from acceleration.

在 A-Level 进阶数学中，向量函数通常用于描述空间中质点的位置、速度或加速度。对向量函数求积分，意味着对其每一个分量分别关于同一个变量（通常是 t 或某个参数）进行积分。这一过程对于从速度恢复位置、或从加速度恢复速度至关重要。

---

1. Definition of a Vector Function | 向量函数的定义

A vector function in three dimensions can be written as (mathbf{r}(t) = x(t)mathbf{i} + y(t)mathbf{j} + z(t)mathbf{k}), where (x(t)), (y(t)), and (z(t)) are scalar functions of the parameter (t). In two dimensions, the (mathbf{k}) component is omitted.

三维空间中的向量函数可以写作 (mathbf{r}(t) = x(t)mathbf{i} + y(t)mathbf{j} + z(t)mathbf{k})，其中 (x(t))、(y(t)) 和 (z(t)) 是关于参数 t 的标量函数。在二维情形下，(mathbf{k}) 分量省略。

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

---

2. Integrating a Vector Function | 向量函数的积分

To integrate a vector function, integrate each component separately. If (mathbf{r}(t) = x(t)mathbf{i} + y(t)mathbf{j} + z(t)mathbf{k}), then the indefinite integral is:

要对向量函数求积分，只需对每个分量分别积分。若 (mathbf{r}(t) = x(t)mathbf{i} + y(t)mathbf{j} + z(t)mathbf{k})，其不定积分为：

(int mathbf{r}(t),dt = left(int x(t),dtright)mathbf{i} + left(int y(t),dtright)mathbf{j} + left(int z(t),dtright)mathbf{k} + mathbf{c})

where (mathbf{c} = c_1mathbf{i} + c_2mathbf{j} + c_3mathbf{k}) is a constant vector of integration.

其中 (mathbf{c} = c_1mathbf{i} + c_2mathbf{j} + c_3mathbf{k}) 是积分常数向量。

---

3. Worked Example: Indefinite Integral | 实例：不定积分

Given (mathbf{v}(t) = 3t^2mathbf{i} + 2cos t,mathbf{j} - e^tmathbf{k}), find (int mathbf{v}(t),dt).

已知 (mathbf{v}(t) = 3t^2mathbf{i} + 2cos t,mathbf{j} - e^tmathbf{k})，求 (int mathbf{v}(t),dt)。

Integrating component by component:

逐分量积分：

(int 3t²,dt = t³ + C₁,quad int 2cos t,dt = 2sin t + C₂,quad int -e^t,dt = -e^t + C₃)

Therefore:

因此：

(int mathbf{v}(t),dt = (t³ + C₁)mathbf{i} + (2sin t + C₂)mathbf{j} + (-e^t + C₃)mathbf{k})

---

4. Definite Integral of a Vector Function | 向量函数的定积分

For a definite integral from (t=a) to (t=b), integrate each component and evaluate at the limits. The result is a constant vector.

对于从 t=a 到 t=b 的定积分，对每个分量积分并在上下限处求值。结果是一个常数向量。

(int_a^b mathbf{r}(t),dt = left(int_a^b x(t),dtright)mathbf{i} + left(int_a^b y(t),dtright)mathbf{j} + left(int_a^b z(t),dtright)mathbf{k})

---

5. Worked Example: Definite Integral | 实例：定积分

If (mathbf{a}(t) = t,mathbf{i} + t²,mathbf{j}), find (int_0¹ mathbf{a}(t),dt).

若 (mathbf{a}(t) = t,mathbf{i} + t²,mathbf{j})，求 (int_0¹ mathbf{a}(t),dt)。

Compute each component:

计算各分量：

(int_0¹ t,dt = frac{1}{2},quad int_0¹ t²,dt = frac{1}{3})

Hence:

因此：

(int_0¹ mathbf{a}(t),dt = frac{1}{2}mathbf{i} + frac{1}{3}mathbf{j})

---

6. Physical Interpretation: Velocity and Displacement | 物理意义：速度与位移

In kinematics, if (mathbf{v}(t)) is the velocity vector of a particle, then its displacement from (t=t₁) to (t=t₂) is (int_t1^t₂ mathbf{v}(t),dt). If (mathbf{a}(t)) is acceleration, then (int mathbf{a}(t),dt = mathbf{v}(t) + mathbf{c}).

在运动学中，若 (mathbf{v}(t)) 是质点的速度向量，则从 t=t₁ 到 t=t₂ 的位移为 (int_t1^t₂ mathbf{v}(t),dt)。若 (mathbf{a}(t)) 是加速度，则 (int mathbf{a}(t),dt = mathbf{v}(t) + mathbf{c})。

---

广告 / Sponsored

7. Using Initial Conditions to Find the Constant Vector | 利用初始条件求常数向量

When finding the position vector (mathbf{r}(t)) from a known velocity (mathbf{v}(t)), the constant vector (mathbf{c}) is determined by substituting an initial condition, such as (mathbf{r}(0) = mathbf{r}_0).

当从已知速度 (mathbf{v}(t)) 求位置向量 (mathbf{r}(t)) 时，常数向量 (mathbf{c}) 通过代入初始条件（例如 (mathbf{r}(0) = mathbf{r}_0)）来确定。

Example: Given (mathbf{v}(t) = 2t,mathbf{i} + 3,mathbf{j}) and (mathbf{r}(0) = mathbf{i} - mathbf{j}), find (mathbf{r}(t)).

例：已知 (mathbf{v}(t) = 2t,mathbf{i} + 3,mathbf{j}) 且 (mathbf{r}(0) = mathbf{i} - mathbf{j})，求 (mathbf{r}(t))。

(mathbf{r}(t) = int mathbf{v}(t),dt = t^2mathbf{i} + 3t,mathbf{j} + mathbf{c})

Substitute (t=0): (mathbf{c} = mathbf{i} - mathbf{j}). Therefore:

代入 t=0：(mathbf{c} = mathbf{i} - mathbf{j})。因此：

(mathbf{r}(t) = (t² + 1)mathbf{i} + (3t - 1)mathbf{j})

---

8. Integrating Vector Functions with Unit Vectors | 含单位向量的向量函数积分

Sometimes vector functions are expressed in terms of unit vectors (mathbf{i}), (mathbf{j}), (mathbf{k}) that are constant in direction; integration is straightforward. However, in polar or spherical coordinates, basis vectors may change with position, and integration becomes more complex. In A-Level, only fixed Cartesian unit vectors are used.

有时向量函数以方向恒定的单位向量 (mathbf{i})、(mathbf{j})、(mathbf{k}) 表示，积分很直接。然而在极坐标或球坐标中，基向量会随位置变化，积分更为复杂。在 A-Level 中，仅使用固定的直角坐标单位向量。

---

9. Common Mistakes and Tips | 常见错误与提示

• Do not forget the constant vector of integration for indefinite integrals.
• For definite integrals, integrate each component separately and then evaluate.
• Be careful with signs when integrating negative components.
• Always check whether initial conditions are given to determine the constant vector.

• 不要忘记不定积分中的常数向量。
• 定积分要分别对每个分量积分后再代入上下限。
• 积分负分量时注意符号。
• 务必检查是否给出初始条件以确定常数向量。

---

10. Summary | 小结

Integration of vector functions is a direct extension of scalar integration. By integrating each component independently, you can recover position from velocity, velocity from acceleration, and solve many practical problems in mechanics. Mastering this skill requires careful bookkeeping of components and constants.

向量函数的积分是标量积分的直接推广。通过对每个分量独立积分，可以从速度恢复位置、从加速度恢复速度，并解决力学中的许多实际问题。掌握这一技能需要仔细处理各分量和常数。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）