📚 1 Vector fields and field lines | 矢量场与场线
Vector fields offer a powerful way to visualise how a vector quantity changes over a plane or in space. From the flow of a fluid to the force of gravity, field lines (also called streamlines) translate abstract equations into intuitive curves. In IB Mathematics, especially in the HL Calculus option, you learn to construct, analyse and sketch vector fields, and to interpret field lines through differential equations and the notions of divergence and curl.
矢量场提供了一种直观方式来观察矢量在平面或空间中的变化。从流体流动到万有引力，场线（也称流线）将抽象的方程转化为清晰的曲线。在IB数学中，特别是在高水平微积分选修部分，你将学习如何构造、分析并绘制矢量场，以及通过微分方程和散度、旋度的概念来解读场线。
1. What is a Vector Field? | 什么是矢量场？
A vector field in two dimensions is a function that assigns a vector to every point (x, y) in some region. We write it as F(x, y) = P(x, y) i + Q(x, y) j, where P and Q are scalar functions. For example, F(x, y) = x i + y j defines a radial field that points directly away from the origin, with magnitude increasing as you move outward.
二维矢量场是一个为区域内每一点 (x, y) 赋予一个矢量的函数。我们写作 F(x, y) = P(x, y) i + Q(x, y) j，其中 P 和 Q 是标量函数。例如，F(x, y) = x i + y j 定义了一个径向场，其方向直接从原点向外，大小随着远离原点而增加。
Vector fields can model velocity of a moving fluid, gravitational force per unit mass, or electric field strength. In each case the vector at a point tells you the direction and magnitude of the influence at that location. Understanding the geometry of these fields is a key skill in multivariable calculus.
矢量场可以模拟运动流体的速度、单位质量的引力或电场强度。在每种情况下，某一点处的矢量告诉你在该位置影响的方向和大小。理解这些场的几何结构是多变量微积分中的一项关键技能。
2. Graphical Representation of Vector Fields | 矢量场的图形表示
To sketch a vector field by hand or with technology, we choose a grid of sample points and draw a small arrow at each point representing F(x, y). The arrow’s direction must match that of the vector, and its length is usually scaled to prevent overlapping. A common convention is to multiply by a scaling factor so that the longest arrow fits the grid nicely.
为了手绘或借助技术绘制矢量场，我们选取一组采样网格点，并在每一点画出代表 F(x, y) 的小箭头。箭头的方向必须与矢量一致，其长度通常会按比例缩放以避免重叠。常见的做法是乘以一个缩放因子，使最长的箭头能很好地适配网格。
For instance, the field F(x, y) = -y i + x j produces arrows that circulate around the origin. The length grows with distance from the origin, so a plot will show shorter arrows near the centre and longer ones further out, all tangent to circles centred at the origin.
例如，场 F(x, y) = -y i + x j 产生的箭头围绕原点环流。箭头的长度随到原点的距离增大，因此在图中靠近中心处箭头较短，远离中心处箭头较长，并且所有箭头都与以原点为圆心的圆相切。
When sketching by hand, it is often enough to draw arrows at a few key points and note symmetries. Software like GeoGebra or Desmos can generate dynamic plots that reveal the field’s structure instantly, which is especially helpful when analysing field lines.
手绘时，通常在几个关键点画出箭头并注意对称性就足够了。像 GeoGebra 或 Desmos 这样的软件可以生成动态绘图，立即揭示场的结构，这在分析场线时尤为有用。
3. Field Lines (Streamlines) – Definition | 场线（流线）的定义
A field line, or streamline, is a curve that is everywhere tangent to the vector field. If you imagine placing a tiny particle in the field and letting it be carried along, its trajectory would trace out a field line. Mathematically, at every point on the curve the tangent vector dr = dx i + dy j is parallel to the field vector F(x, y).
场线，或称流线，是一条处处与矢量场相切的曲线。想象将一个微小粒子放入场中并任其随场运动，它的轨迹就是一条场线。数学上，曲线上每一点处的切向量 dr = dx i + dy j 都与场向量 F(x, y) 平行。
Field lines cannot cross each other unless the field is zero at that point or the direction is undefined. They provide a dense representation of the field: the closer the lines, the stronger the field. This is completely analogous to the way electric field lines or magnetic field lines are drawn in physics.
场线不能互相交叉，除非在该点处场为零或者方向无定义。它们提供了场的密集表示：线条越密集，场越强。这与物理学中绘制电场线或磁感线的方式完全类似。
4. Deriving the Differential Equation for Field Lines | 场线微分方程的推导
Because the tangent vector dr is parallel to F, their components are proportional: dx / P(x, y) = dy / Q(x, y), assuming P and Q are non-zero. Rearranging gives the first-order ordinary differential equation for the field lines:
由于切向量 dr 与 F 平行，它们的分量成比例：dx / P(x, y) = dy / Q(x, y)，假设 P 与 Q 非零。整理后得到场线的一阶常微分方程：
dy / dx = Q(x, y) / P(x, y)
If P = 0 at some points, the field is vertical there, and we can instead write dx / dy = P / Q to obtain vertical segments. In any case, solving this ODE yields a family of curves parameterised by an integration constant; each curve is a field line.
如果在某些点 P = 0，场在该处是竖直的，我们可以改写为 dx / dy = P / Q 以获得竖直段。无论如何，求解这个常微分方程会得到一个由积分常数参数化的曲线族；每条曲线就是一条场线。
Arrows on the field lines indicate the direction of the vector field. The direction can be determined by evaluating F at a point on the curve: the arrow points in the direction of the vector. This completes the geometric picture of the vector field.
场线上的箭头标示了矢量场的方向。可通过在曲线上取一点计算 F 来确定方向：箭头指向矢量的方向。这便完成了矢量场的几何图像。
5. Example 1 – A Rotational Field | 例1 – 旋转场
Consider the vector field F(x, y) = -y i + x j. Here P = -y and Q = x. The differential equation for field lines is dy/dx = x / (-y) = -x / y. Separating variables gives y dy = -x dx.
考虑矢量场 F(x, y) = -y i + x j。这里 P = -y，Q = x。场线的微分方程为 dy/dx = x / (-y) = -x / y。分离变量得 y dy = -x dx。
Integrating both sides yields (1/2) y2 = -(1/2) x2 + C, which simplifies to x2 + y2 = 2C. Therefore, the field lines are concentric circles centred at the origin. The direction is counterclockwise, as seen by evaluating F(1,0) = 0 i + 1 j, which points upward at the rightmost point, indicating anticlockwise circulation.
两边积分得 (1/2) y2 = -(1/2) x2 + C，化简为 x2 + y2 = 2C。因此，场线是以原点为圆心的同心圆。方向为逆时针，可通过计算 F(1,0) = 0 i + 1 j 看出，在最右侧点矢量向上，表示逆时针环流。
This field is a classic example of a rotational field with zero divergence but non-zero curl. The circular field lines clearly visualise a ‘swirling’ motion, much like the velocity field of water draining from a basin.
这个场是具有零散度但非零旋度的旋转场的经典示例。圆形的场线清晰地呈现出“涡旋”运动，就像从水槽排出的水的速度场一样。
6. Example 2 – A Radial Field | 例2 – 径向场
Take the simple radial field F(x, y) = x i + y j. Then P = x, Q = y, and the field line equation is dy/dx = y / x. Separating variables: dy / y = dx / x for x ≠ 0, y ≠ 0. Integration gives ln|y| = ln|x| + C, so |y| = eC |x|, which represents straight lines through the origin y = k x.
考虑简单的径向场 F(x, y) = x i + y j。此时 P = x，Q = y，场线方程为 dy/dx = y / x。分离变量（当 x ≠ 0, y ≠ 0 时）：dy / y = dx / x。积分得 ln|y| = ln|x| + C，故 |y| = eC |x|，这表示通过原点的直线 y = k x。
Thus the field lines are rays emanating from the origin. The direction is outward for positive x and y, as seen from F(1,1) = i + j. This field has zero curl but positive divergence, indicating a source at the origin. If the sign were reversed, the lines would point inward, representing a sink.
因此场线是从原点发出的射线。从 F(1,1) = i + j 可知，对于正的 x 和 y，方向向外。这个场旋度为零但散度为正，表明原点处存在源。如果符号相反，场线将指向内部，代表汇。
7. Gradient Fields and Potential Functions | 梯度场与势函数
A vector field F is called a gradient field (or conservative field) if there exists a scalar function φ(x, y) such that F = ∇φ = (∂φ/∂x) i + (∂φ/∂y) j. The function φ is called a potential function. For example, if φ(x, y) = (1/2)(x2 + y2), then F = x i + y j, the radial field we just examined.
若存在标量函数 φ(x, y) 使得 F = ∇φ = (∂φ/∂x) i + (∂φ/∂y) j，则称矢量场 F 为梯度场（或保守场）。函数 φ 称为势函数。例如，若 φ(x, y) = (1/2)(x2 + y2)，则 F = x i + y j，即我们刚检视过的径向场。
Field lines of a gradient field are everywhere perpendicular to the level curves of φ (curves where φ is constant). In the radial example, the level curves are circles x2 + y2 = const, and the field lines are rays; indeed circles and rays intersect at right angles. This orthogonality is a powerful geometric property of gradient fields.
梯度场的场线处处与 φ 的等高线（φ 为常数的曲线）垂直。在径向例子中，等高线是圆 x2 + y2 = 常数，场线是射线；圆与射线确实垂直相交。这种正交性是梯度场一个非常强大的几何性质。
Physically, if F represents a force, the potential φ relates to potential energy, and the force points in the direction of steepest descent of φ. Thus field lines are the paths of steepest descent, which are orthogonal to equipotential surfaces.
物理上，若 F 表示力，势 φ 便关联势能，力指向 φ 下降最快的方向。因此场线就是最速下降路径，与等势面正交。
8. Conservative Fields and Path Independence | 保守场与路径无关
A key test for a vector field to be conservative in a simply connected region is that its curl (in 2D, the scalar curl) vanishes: ∂Q/∂x – ∂P/∂y = 0. If this holds, the line integral ∫C F·dr between two points depends only on the endpoints, not the path taken. The integral can be computed as φ(end) – φ(start).
在单连通区域内，矢量场为保守场的关键检验是其旋度（二维中的标量旋度）为零：∂Q/∂x – ∂P/∂y = 0。如果成立，则两点之间的线积分 ∫C F·dr 只取决于端点，而与路径无关。积分可计算为 φ(终点) – φ(起点)。
For example, F(x, y) = 2xy i + x2 j has P = 2xy, Q = x2. Then ∂Q/∂x = 2x and ∂P/∂y = 2x, so the curl is zero. A potential function is φ = x2y. In contrast, the rotational field F = -y i + x j has ∂Q/∂x – ∂P/∂y = 1 – (-1) = 2 ≠ 0, so it is not conservative; work done around a closed loop is non-zero.
例如，F(x, y) = 2xy i + x2 j 的 P = 2xy，Q = x2。则 ∂Q/∂x = 2x 且 ∂P/∂y = 2x，旋度为零。它的一个势函数是 φ = x2y。相比之下，旋转场 F = -y i + x j 的 ∂Q/∂x – ∂P/∂y = 1 – (-1) = 2 ≠ 0，因此非保守；绕闭合回路做功非零。
The condition ∂Q/∂x = ∂P/∂y is exactly the Clairaut compatibility condition that ensures the existence of a potential function. In IB problems, this is the most common tool for proving conservativity without finding φ explicitly.
条件 ∂Q/∂x = ∂P/∂y 恰好是保证势函数存在的克莱罗相容性条件。在IB题目中，这是证明保守性而不必显式求出 φ 的最常用工具。
9. Divergence and Curl – A Brief Look | 散度与旋度概览
In two dimensions, divergence measures the net outflow per unit area: div F = ∂P/∂x + ∂Q/∂y. Positive divergence at a point indicates a source; negative indicates a sink. Curl (scalar curl) measures the circulation density: curl F = ∂Q/∂x – ∂P/∂y. A non-zero curl suggests a rotational tendency.
在二维中，散度衡量单位面积的净流出量：div F = ∂P/∂x + ∂Q/∂y。某点处的散度为正表示源，为负表示汇。旋度（标量旋度）衡量环流密度：curl F = ∂Q/∂x – ∂P/∂y。非零旋度意味着旋转趋势。
Let’s compute for the two main examples: For F = x i + y j, div = 1+1 = 2, curl = 0 – 0 = 0. For F = -y i + x j, div = 0+0 = 0, curl = 1 – (-1) = 2. A field that is both divergence-free and curl-free everywhere (except possibly isolated points) is called harmonic and plays a vital role in potential theory.
我们来计算两个主要例子：对于 F = x i + y j，div = 1+1 = 2，curl = 0 – 0 = 0。对于 F = -y i + x j，div = 0+0 = 0，curl = 1 – (-1) = 2。一个处处既无散又无旋的场（除可能孤立的点外）称为调和场，它在势论中扮演着核心角色。
The table below summarises these two representative fields:
下表总结了这两个代表性场：
| Vector Field | div F | curl F | Field line shape |
|---|---|---|---|
| F = x i + y j | 2 (source everywhere) | 0 (irrotational) | Rays from origin |
| F = -y i + x j | 0 (incompressible) | 2 (rotational) | Circles around origin |
Combined with the concept of field lines, divergence and curl offer a complete local description of the vector field’s behaviour: expansion/contraction and local circulation.
结合场线的概念，散度和旋度为矢量场的局部行为提供了完整描述：膨胀/收缩和局部环流。
10. Real-World Contexts and Applications | 实际背景与应用
Vector fields and their field lines are not abstract – they are the language of classical field theories. In gravitation, the gravitational field per unit mass is g(x, y) = – (GM / r2) r̂, which points radially inward; field lines converge to the mass. This is an irrotational, divergent field with a sink at the origin.
矢量场及其场线并非抽象概念——它们是经典场论的语言。在引力中，单位质量的引力场为 g(x, y) = – (GM / r2) Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply