Evaluation of Line Integrals | 线积分的计算

📚 Evaluation of Line Integrals | 线积分的计算

Line integrals extend the familiar concept of a single-variable integral to integration along a curve in two- or three-dimensional space. In the IB HL Mathematics course (calculus option or Further Maths), you are expected to compute line integrals of both scalar functions and vector fields, interpret them in terms of arc length or work, and understand when the value depends only on endpoints. Mastering these ideas opens the door to advanced physics and engineering applications.

线积分将熟悉的单变量积分概念推广到沿二维或三维空间中一条曲线的积分。在 IB 高等数学课程（微积分选项或进阶数学）中，你需要掌握标量函数和向量场沿曲线的积分计算，理解它们在弧长或做功意义上的解释，并判断何时积分值仅依赖于端点。掌握这些思想将为高等物理和工程应用打开大门。

1. Introduction to Line Integrals | 线积分简介

Unlike a standard definite integral ∫_a^b f(x) dx, where integration is performed along the x‑axis, a line integral follows a curve C in the xy‑ne (or in space). The integrand can be a scalar function f(x,y) or a vector field F(x,y) = ⟨P(x,y), Q(x,y)⟩. The curve C must be given in parametric form r(t) = ⟨x(t), y(t)⟩ with t ranging from a to b.

与标准定积分 ∫_a^b f(x) dx 沿 x 轴进行不同，线积分沿着 xy 平面（或空间中）的一条曲线 C 进行。被积函数可以是标量函数 f(x,y) 或向量场 F(x,y) = ⟨P(x,y), Q(x,y)⟩。曲线 C 必须以参数形式 r(t) = ⟨x(t), y(t)⟩ 给出，其中参数 t 从 a 变化到 b。

Two principal types appear in the syllabus: the line integral with respect to arc length (scalar) and the line integral of a vector field (often called work or circulation). You will need to choose the appropriate formula, parametrize the curve, and evaluate the resulting ordinary integral.

教学大纲中出现两种主要类型：关于弧长的线积分（标量）和向量场的线积分（常称为功或环流量）。你需要选择合适的公式，对曲线进行参数化，并计算最终得到的常义积分。

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2. Scalar Line Integrals – Definition | 标量线积分的定义

For a scalar function f(x,y) defined on a curve C, the line integral with respect to arc length s is given by

对于定义在曲线 C 上的标量函数 f(x,y)，关于弧长 s 的线积分由下式给出

∫_C f(x,y) ds = ∫_a^b f(x(t), y(t)) √[(dx/dt)² + (dy/dt)²] dt

Here ds = √[(dx/dt)² + (dy/dt)²] dt is the arc‑ Geometrically, it accumulates the function values multiplied by tiny arc lengths along C.

这里 ds = √[(dx/dt)² + (dy/dt)²] dt 是弧长微元。这种积分可给出密度为 f(x,y) 的金属丝的总质量，或一曲边栅栏的面积。在几何上，它将函数值乘以沿 C 的微小弧长并累加。

The extension to three dimensions is straightforward: if C is parametrized by r(t)=⟨x(t),y(t),z(t)⟩, then ds = √[(x')²+(y')²+(z')²] dt and the integral becomes ∫_C f(x,y,z) ds.

推广到三维是直接的：若 C 由 r(t)=⟨x(t),y(t),z(t)⟩ 参数化，则 ds = √[(x')²+(y')²+(z')²] dt，积分变为 ∫_C f(x,y,z) ds。

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3. Evaluating Scalar Line Integrals | 标量线积分的计算

To evaluate ∫_C e curve exactly once and determine the parameter limits a and b. Second, compute the speed |r'(t)| = √[(dx/dt)² + (dy/dt)²] and form the integrand f(x(t),y(t)) |r'(t)|. Third, integrate the resulting expression with respect to t from a to b.

要计算 ∫_C f(x,y) ds，按三步进行。首先，为 C 选取一个恰好描出曲线一次的参数化 r(t)，并确定参数范围 a 和 b。其次，计算速率 |r'(t)| = √[(dx/dt)² + (dy/dt)²] 并构造被积函数 f(x(t),y(t)) |r'(t)|。最后，将所得表达式对 t 从 a 到 b 积分。

the upper half of the unit circle from (1,0) to (−1,0). A natural parametrization is r(t)=⟨cos t, sin t⟩, 0 ≤ t ≤ π. Then |r'(t)| = √[(-sin t)²+(cos t)²] = 1. If f(x,y)=x+y, then ∫_C f ds = ∫₀^π (cos t + sin t)·1 dt = 2.

例如，设 C 为从 (1,0) 到 (−1,0) 的上半单位圆。一个自然的参数化是 r(t)=⟨cos t, sin t⟩, 0 ≤ t ≤ π。那么 |r'(t)| = √[(-sin t)²+(cos t)²] = 1。若 f(x,y)=x+y，则 ∫_C f ds = ∫₀^π (cos t + sin t)·1 dt = 2。

Always check whether the curve is oriented; scalar line integrals do not depend on orientation (ds is always positive). Reversing the path leaves the value unchanged.

始终检查曲线的方向；标量线积分不依赖于方向（ds 恒正）。反向改变路径，积分值不变。

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4. Vector Line Integrals – Work | 向量线积分与功

The line integral of a vector field F = ⟨P, Q⟩ along a curve C is defined as

向量场 F = ⟨P, Q⟩ 沿曲线 C 的线积分定义为

∫_C F · dr = ∫_C P dx + Q dy = ∫_a^b F(x(t),y(t)) · r'(t) dt

g along C. The symbolic form P dx + Q dy directly corresponds to the dot product with dr = ⟨dx, dy⟩. When the curve is parametrized, dx = x'(t) dt and dy = y'(t) dt.

该积分表示力 F 对沿 C 运动的粒子所做的功。符号形式 P dx + Q dy 直接对应于与 dr = ⟨dx, dy⟩ 的点乘。当曲线被参数化时，dx = x'(t) dt 且 dy = y'(t) dt。

If the curve is closed (start and end points coincide), the integral is called the circulation of F around C and is often denoted ∮_C F · dr. IB HL candidates should be able to compute such integrals and interpret their sign.

若曲线闭合（起点与终点重合），该积分称为 F 绕 C 的环流量，常记为 ∮_C F · dr。IB HL 考生应能计算这类积分并解释其符号。

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5. Computing Vector Line Integrals | 计算向量线积分

To compute ∫_C d express the curve parametrically. Then substitute x=x(t), y=y(t) into P and Q, compute r' a to t=b. Alternatively, if the integrand is given as P dx + Q dy, directly replace dx = x'(t) dt and dy = y'(t) dt.

要计算 ∫_C F · dr，首先将 F 写成分量形式并用参数表示曲线。然后将 x=x(t), y=y(t) 代入 P 和 Q，计算 r'(t)，作点乘，并从 t=a 到 t=b 积分。另一种方式是，若被积式以 P dx + Q dy 给出，则直接代换 dx = x'(t) dt 和 dy = y'(t) dt。

For example, let F = ⟨y, x²⟩ and C be the parabola y = x² from (0,0) to (1,1). A convenient parametrization is r(t)=⟨t, t²⟩, 0≤t≤1. Then r'(t)=⟨1, 2t⟩, F(r(t))=⟨t², t²⟩. The dot product is t²·1 + t²·2t = t² + 2t³. Integrating yields ∫₀¹ (t²+2t³) dt = [t³/3 + t⁴/2]₀¹ = 1/3 + 1/2 = 5/6.

例如，设 F = ⟨y, x²⟩，C 为从 (0,0) 到 (1,1) 的抛物线 y = x²。一个方便的参数化是 r(t)=⟨t, t²⟩, 0≤t≤1。那么 r'(t)=⟨1, 2t⟩, F(r(t))=⟨t², t²⟩。点乘为 t²·1 + t²·2t = t² + 2t³。积分得 ∫₀¹ (t²+2t³) dt = [t³/3 + t⁴/2]₀¹ = 1/3 + 1/2 = 5/6。

Watch out for orientation: if the curve is traversed in the opposite direction, r'(t) changes sign and the integral changes sign. Always verify the orientation implied by the parametrization.

注意方向：若曲线沿相反方向行进，r'(t) 变号，积分值也随之变号。始终要确认参数化所隐含的方向。

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6. Parametrizing Curves | 曲线参数化

written as r(t)=A + t(B−A), 0≤t≤1. A circle of radius R centred at the origin admits r(t)=⟨R cos t, R sin t⟩, with t covering the required arc. For y = g(x), a simple choice is r(t)=⟨t, g(t)⟩.

任何线积分的成功计算都依赖于适当的参数化。常见曲线包括：从 A 到 B 的直线段可写作 r(t)=A + t(B−A), 0≤t≤1。中心在原点的半径为 R 的圆可采用 r(t)=⟨R cos t, R sin t⟩，用 t 覆盖所需弧段。对 y = g(x)，一个简单的选择是 r(t)=⟨t, g(t)⟩。

In three dimensions, a helix, for instance, might be r(t)=⟨cos t, sin t, t⟩ s the curve exactly once and that the orientation matches the problem statement. The speed |r'(t)| should be non‑zero on the interval.

在三维中，例如螺旋线，可为 r(t)=⟨cos t, sin t, t⟩。始终要检查参数化是否恰好覆盖曲线一次，且方向与题目一致。速率 |r'(t)| 在区间上应非零。

A useful trick: when the curve is expressed explicitly as y = f(x), the arc‑length differential simplifies to ds = √[1 + (f'(x))²] dx, and the scalar line integral becomes a single‑variable integral ∫_a^b g(x) √[1+(f'(x))²] dx.

一个有用的技巧：当曲线显式表示为 y = f(x) 时，弧长微元简化为 ds = √[1 + (f'(x))²] dx，标量线积分变为单变量积分 ∫_a^b g(x) √[1+(f'(x))²] dx。

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7. The Fundamental Theorem for Line Integrals | 线积分基本定理

the vector field F is conservative, i.e. there exists a potential function f such that F = ∇f, then the line integral depends only on the endpoints A and B:

若向量场 F 是保守场，即存在势函数 f 使得 F = ∇f，则线积分仅依赖于端点 A 和 B：

∫_C F · dr = f(B) − f(A)

ever F is a gradient field. The proof follows by applying the chain rule: d/dt [f(r(t))] = ∇f · r'(t) = F · r'(t).

这是线积分的基本定理。只要 F 是梯度场，它对任意从 A 到 B 的曲线 C 都成立。证明来自链式法则：d/dt [f(r(t))] = ∇f · r'(t) = F · r'(t)。

Such fields are called path‑independent. If C is closed, the integral around any closed loop is zero. Identifying conservative fields dramatically simplifies computations because you only need to know f at the endpoints.

这样的场称为路径无关的。若 C 闭合，绕任何闭回路的积分为零。识别保守场可极大简化计算，因为你只需知道 f 在端点的值。

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8. Path Independence and Conservative Fields | 路径无关性与保守场

A vector field F = ⟨P, Q⟩ if and only if ∂Q/∂x = ∂P/∂y (in two dimensions). This condition is derived from Clairaut's theorem on mixed partials: if F = ∇f, then ∂P/∂y = f_xy = f_yx = ∂Q/∂x. The converse holds on simply connected domains.

定义在开连通区域上的二维向量场 F = ⟨P, Q⟩ 是保守场当且仅当 ∂Q/∂x = ∂P/∂y。这一条件源自混合偏导数的克莱罗定理：若 F = ∇f，则 ∂P/∂y = f_xy = f_yx = ∂Q/∂x。在单连通区域上其逆亦真。

To check path independence, compute the partial derivatives. If they are equal everywhere, any line integral between two points gives the same value regardless of the curve chosen. In IB HL questions, you may be asked to verify conservativity and then use the fundamental theorem or find a potential.

要检验路径无关性，计算偏导数。若它们处处相等，则任意两点间的线积分值与所选曲线无关。在 IB HL 考题中，你可能需要验证保守性，然后利用基本定理或寻找势函数。

the domain is not simply connected (e.g. has a hole). The IB syllabus usually restricts to simply connected regions, so the test is sufficient.

注意：∂Q/∂x = ∂P/∂y 是必要条件，但如果区域非单连通（如有洞），则并不充分。 IB 大纲通常限制在单连通区域，因此该检验是充分的。

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9. Finding the Potential Function | 求势函数

When the field F = ⟨P, Q⟩ with f = ∫ P dx + g(y), differentiate with respect to y and set equal to Q to determine g'(y); integrate to get g(y).

当场 F = ⟨P, Q⟩ 满足 ∂Q/∂x = ∂P/∂y 时，可通过将 P 对 x 积分、将 Q 对 y 积分并匹配结果来找到势函数 f。从 f = ∫ P dx + g(y) 开始，对 y 求导并令其等于 Q 以确定 g'(y)，然后积分得到 g(y)。

For example, let F = ⟨2xy, x² + 3y²⟩ F is conservative. Integrating P: f = ∫ 2xy dx = x²y + g(y). Differentiate: ∂f/∂y = x² + g'(y) must equal Q = x² + 3y², giving g'(y) = 3y², so g(y) = y³ + C. Hence f(x,y) = x²y + y³ + C.

例如，设 F = ⟨2xy, x² + 3y²⟩。则 ∂P/∂y = 2x = ∂Q/∂x，故 F 是保守的。对 P 积分：f = ∫ 2xy dx = x²y + g(y)。求导后 ∂f/∂y = x² + g'(y) 必须等于 Q = x² + 3y²，得出 g'(y) = 3y²，所以 g(y) = y³ + C。因此 f(x,y) = x²y + y³ + C。

The constant C can be ignored when evaluating ∫_C g this algorithm ensures you can quickly exploit the fundamental theorem in exam settings.

计算 ∫_C F · dr = f(B) − f(A) 时，常数 C 可忽略。练习这一算法可确保你在考试中快速利用基本定理。

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10. Worked Examples | 例题解析

Example 1 (scalar): Evaluate ∫_C (x² + y²) ds where C is the straight line from (0,0) to (2,2). Parametrize by r(t)=⟨t, t⟩, 0≤t≤2. Then |r'(t)| = √1²+1² = √2. The integral becomes ∫₀² (t²+t²) √2 dt = √2 ∫₀² 2t² dt = 2√2 [t³/3]₀² = (16√2)/3.

例1（标量）：计算 ∫_C (x² + y²) ds，其中 C 为从 (0,0) 到 (2,2) 的直线。参数化 r(t)=⟨t, t⟩, 0≤t≤2。则 |r'(t)| = √1²+1² = √2。积分变为 ∫₀² (t²+t²) √2 dt = √2 ∫₀² 2t² dt = 2√2 [t³/3]₀² = (16√2)/3。

Example 2 (vector, conservative): Given F = ⟨y, x⟩, find ∫_C any curve. Note ∂P/∂y = 1, ∂Q/∂x = 1, so F is conservative. A potential is f = xy. Then ∫_C F · dr = f(0,1) − f(1,0) = 0 − 0 = 0. (If done by parametrization, any path yields 0.)

例2（向量，保守）：给定 F = ⟨y, x⟩，求 ∫_C F · dr 从 (1,0) 到 (0,1) 沿任意曲线。注意到 ∂P/∂y = 1, ∂Q/∂x = 1，故 F 保守。势函数为 f = xy。则 ∫_C F · dr = f(0,1) − f(1,0) = 0 − 0 = 0。（若用参数化计算，任何路径均得 0。）

Example 3 (non‑conservative): Evaluate ∫_C F · dr where F = ⟨−y, x⟩ and C is the unit circle once counter‑clockwise. Parametrize r(t)=⟨cos t, sin t⟩, 0≤t≤2π. Then r'(t)=⟨−sin t, cos t⟩, F(r(t))=⟨−sin t, cos t⟩. Dot product = sin²t + cos²t = 1. Integrate ∫₀^2π 1 dt = 2π. This non‑zero circulation indicates a non‑conservative field (∂Q/∂x = 1 ≠ ∂P/∂y = −1).

例3（非保守）：计算 ∫_C F · dr，其中 F = ⟨−y, x⟩，C 为单位圆逆时针一周。参数化 r(t)=⟨cos t, sin t⟩, 0≤t≤2π。则 r'(t)=⟨−sin t, cos t⟩, F(r(t))=⟨−sin t, cos t⟩。点乘 = sin²t + cos²t = 1。积分 ∫₀^2π 1 dt = 2π。这个非零的环流量表明场非保守（∂Q/∂x = 1 ≠ ∂P/∂y = −1）。

Regular practice with such examples builds the confidence to handle line integrals under timed conditions, correctly choosing between direct parametrization and the fundamental theorem.

经常练习这类例题能让你建立起在限时条件下处理线积分的信心，并正确选择直接参数化或使用基本定理。

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