Sketching Slope Fields · 绘制方向场
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| slope field/sləʊp fiːld/ | 斜率场 | xié lǜ chǎng |
A map of slopes across the plane
- A differential equation $\dfrac{dy}{dx}=F(x,y)$ gives a slope at every point.
- Draw a short segment with that slope at each point of a grid, and you get a slope field 斜率场.
- It's a "map" showing which way solution curves must travel everywhere.
- You don't need to solve the equation to draw it — just evaluate the slope.
平面上的斜率地图
- 微分方程 $\dfrac{dy}{dx}=F(x,y)$ 在每一点给出一个斜率。
- 在网格的每一点画一条具有该斜率的短线段,就得到一个斜率场。
- 它是一张"地图",显示解曲线在各处必须朝哪个方向走。
- 你无需解方程就能画它——只要求出斜率。
A slope field for dy/dx = ay · dy/dx = ay 的方向场
Each dash has slope $\tfrac{dy}{dx}$ at its point — flat near $y=0$ and steeper far away for $\tfrac{dy}{dx}=ay$. · 每段短线在点处斜率为$\tfrac{dy}{dx}$——靠近$y=0$处平缓，远离$\tfrac{dy}{dx}=ay$处陡峭。
A slope field shows, at each point, the... · 方向场显示每一点处的...
Each dash is the tangent direction there. · 每段短线代表该点的切线方向。
To draw a slope field you compute slopes only — you do ____ need to solve the equation. · 绘制方向场只需计算斜率——你____需要解方程。
The equation gives the slope directly. · 方程直接给出斜率。
Compute the slope at each point
- At a grid point $(x,y)$, plug into $\dfrac{dy}{dx}=F(x,y)$ to get the number — that's the segment's slope.
- Slope $0$ → a horizontal dash; large positive → a steep upward dash; negative → downward.
- Repeat over a lattice of points to fill the field.
- The equation tells you the slope directly; no antiderivative needed.
在每点计算斜率
- 在网格点 $(x,y)$,代入 $\dfrac{dy}{dx}=F(x,y)$ 得到那个数——就是线段的斜率。
- 斜率 $0$ → 一条水平短线;大的正数 → 一条陡峭上斜线;负数 → 下斜。
- 在一个点阵上重复,填满整个场。
- 方程直接告诉你斜率;不需要原函数。
For · 支持 $\dfrac{dy}{dx}=x+y$, find the slope of the dash at the point $(2,3)$. · 对于$\dfrac{dy}{dx}=x+y$，找出点$(2,3)$处短线的斜率。
$2+3=5$.
For · 支持 $\dfrac{dy}{dx}=xy$, the slope at $(3,2)$ is... · 对于$\dfrac{dy}{dx}=xy$，在$(3,2)$处的斜率是...
$xy=3\cdot2=6$ (use both coordinates). · $xy=3\cdot2=6$（使用两个坐标）。
Spotting patterns quickly
- If $F$ depends only on $x$: the slope is the same all along a vertical line (columns look identical).
- If $F$ depends only on $y$: the slope is the same all along a horizontal line (rows look identical).
- For $\dfrac{dy}{dx}=y$: slopes grow with height — flat near $y=0$, steep far away.
- Recognizing these patterns speeds up sketching enormously.
快速发现规律
- 若 $F$ 只依赖 $x$:沿一条竖直线斜率相同(各列看起来一样)。
- 若 $F$ 只依赖 $y$:沿一条水平线斜率相同(各行看起来一样)。
- 对 $\dfrac{dy}{dx}=y$:斜率随高度增大——在 $y=0$ 附近平、离得远则陡。
- 认出这些规律能极大加快作图。
If $\dfrac{dy}{dx}$ depends only on $x$, the dashes in each vertical column are parallel. · 若$\dfrac{dy}{dx}$仅取决于$x$，则同一垂直列中的短线互相平行。
Same $x$ → same slope down the column. · 相同的$x$ → 同列斜率相同。
Equilibrium lines
- Where $F(x,y)=0$, the slope is $0$ — a row (or curve) of horizontal dashes.
- For $\dfrac{dy}{dx}=y$, that happens at $y=0$: a whole horizontal line of flat segments.
- These lines often mark equilibrium behavior — solutions can level off there.
- They're the easiest points to plot, so start with them.
平衡线
- 在 $F(x,y)=0$ 之处,斜率为 $0$——一行(或一条曲线)的水平短线。
- 对 $\dfrac{dy}{dx}=y$,这发生在 $y=0$:整条水平线都是平线段。
- 这些线常标记平衡行为——解可以在此处趋平。
- 它们是最容易画的点,所以从它们开始。
For · 支持 $\dfrac{dy}{dx}=y$, the dashes are horizontal along the line... · 对于$\dfrac{dy}{dx}=y$，沿线...的短线是水平的。
Slope · 斜率 $=y=0$ along $y=0$. · 沿 $=y=0$ 方向的斜率 $y=0$。
A slope field shows the differential equation's slopes, not the solution curves themselves. The little segments are tangent directions; a solution is a curve that stays tangent to them. And compute the slope with the point's $(x,y)$, using $y$ too when $F$ depends on it — not just $x$.
斜率场显示微分方程的斜率,而非解曲线本身。那些小线段是切线方向;解是一条始终与它们相切的曲线。而且用点的 $(x,y)$ 计算斜率,当 $F$ 依赖 $y$ 时也要用 $y$——不只是 $x$。
Sketch a few slopes for $\dfrac{dy}{dx}=x$.
- At $(1,0)$: slope $=1$. At $(2,5)$: slope $=2$. At $(-1,3)$: slope $=-1$.
- Since $F=x$ (no $y$), every point in a vertical column has the same slope.
- Columns of parallel dashes: flat at $x=0$, tilting more as $|x|$ grows.
为 $\dfrac{dy}{dx}=x$ 画几个斜率。
- 在 $(1,0)$:斜率 $=1$。在 $(2,5)$:斜率 $=2$。在 $(-1,3)$:斜率 $=-1$。
- 因为 $F=x$(无 $y$),竖直列上的每一点斜率相同。
- 一列列平行短线:$x=0$ 处平,$|x|$ 增大时越倾斜。
A slope field draws a short segment of slope $\frac{dy}{dx}=F(x,y)$ at each grid point — a map of tangent directions, no solving needed. Compute the slope at each $(x,y)$; look for patterns (columns match if $F$ depends only on $x$, rows if only on $y$) and horizontal dashes where $F=0$.
斜率场在每个网格点画一条斜率为 $\frac{dy}{dx}=F(x,y)$ 的短线段——一张切线方向地图,无需求解。在每个 $(x,y)$ 计算斜率;寻找规律(若 $F$ 只依赖 $x$ 则各列相同,只依赖 $y$ 则各行相同),$F=0$ 之处是水平短线。