Line Integrals along Paths Parallel to Coordinate Axes | IB数学：平行于坐标轴的线积分路径

📚 Line Integrals along Paths Parallel to Coordinate Axes | IB数学：平行于坐标轴的线积分路径

A line integral along an axis-parallel path is one of the simplest ways to learn about integration over curves. When the path is horizontal or vertical, the general expression \(\int_C P\,dx + Q\,dy\) reduces immediately to an ordinary one-variable integral.

沿着平行于坐标轴的路径计算线积分，是学习曲线积分最直观的入门方式。当路径为水平或竖直时，一般表达式 \(\int_C P\,dx + Q\,dy\) 会立刻化简为普通的一元积分。

1. What is a Line Integral? | 什么是线积分？

For a vector field \(F = P(x,y)\,\mathbf{i} + Q(x,y)\,\mathbf{j}\), the line integral of \(F\) along an oriented curve \(C\) is defined as

对于向量场 \(F = P(x,y)\,\mathbf{i} + Q(x,y)\,\mathbf{j}\)，沿有向曲线 \(C\) 的线积分定义为

∫_C F · dr = ∫_C P dx + Q dy

Here \(dr = (dx, dy)\), and the integral is often called the circulation or work integral, depending on the physical context.

其中 \(dr = (dx, dy)\)。这个积分在物理中常被称为环量或做功积分。

If the curve is made of segments parallel to the coordinate axes, then on every segment either \(dx = 0\) or \(dy = 0\). This removes one term and makes the integration straightforward.

如果曲线由平行于坐标轴的线段组成，那么在每一段上要么 \(dx = 0\)，要么 \(dy = 0\)。这会使积分中的一项消失，从而使计算变得非常直接。

2. Paths Parallel to the Axes | 平行于坐标轴的路径

A horizontal path is described by \(y = c\), where \(c\) is a constant. Along such a path, \(dy = 0\).

水平路径可以表示为 \(y = c\)，其中 \(c\) 是常数。在这条路径上，\(dy = 0\)。

A vertical path is described by \(x = c\), where \(c\) is a constant. Along such a path, \(dx = 0\).

竖直路径可以表示为 \(x = c\)，其中 \(c\) 是常数。在这条路径上，\(dx = 0\)。

These two cases are the building blocks for more complicated piecewise paths.

这两种情况是构造更复杂分段路径的基本单元。

Notice that on a horizontal segment only the \(P\) component contributes; on a vertical segment only the \(Q\) component contributes.

注意：在水平线段上只有 \(P\) 分量有贡献；在竖直线段上只有 \(Q\) 分量有贡献。

3. Horizontal Path: y = Constant | 水平路径：y 为常数

Suppose \(C\) is a horizontal segment from \((a,c)\) to \((b,c)\). Since \(y = c\), we have \(dy = 0\).

设 \(C\) 是从 \((a,c)\) 到 \((b,c)\) 的水平线段。因为 \(y = c\)，所以 \(dy = 0\)。

∫_C F · dr = ∫_a^b P(x,c) dx

For example, let \(F = (xy,\; x^2)\) and take the path from \((1,2)\) to \((4,2)\). Here \(y = 2\), so

例如，设 \(F = (xy,\; x^2)\)，路径是从 \((1,2)\) 到 \((4,2)\)。此时 \(y = 2\)，于是

∫₁⁴ x(2) dx = ∫₁⁴ 2x dx = x²

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