F(x,y)=[F1(x,y),F2(x,y)]我有定义为的矢量场
F1(x,y)= -(2*y/x^2)*(2*ln(cosh(x)) + (y-2*x)*tanh(x) - x*y*(1-(tanh(x))^2))
F2(x,y)= -(4/x)*(x*y-ln(cosh(x)) - y*tanh(x))
在 Mathematica 中，我可以通过指令轻松绘制相应的流图
StreamPlot[{F1[a, c], F2[a, c]}, {a, -10, 10}, {c, -10, 10}, AspectRatio -> 1/GoldenRatio]
产生
我一直在尝试使用 Tikz 生成类似的东西。事实上，我尝试使用和\addplot3规范化quiver，但奇点破坏了代码（例如，在 [-10,10]x[-10,10] 域中）。此外，我尝试将所有内容拆分为非奇异子域，但尽管可以编译，但由于“直箭头”，外观很糟糕。下面是一段没有中断的相应代码及其输出：
\documentclass{standalone}
\usepackage{amsmath, tikz, pgfplots}
\pgfplotsset{compat = newest}
\usepgfplotslibrary{colormaps}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
zmin = 0, zmax = 1,
axis equal image,
xtick distance = 1,
ytick distance = 1,
view = {0}{90},
scale = 1.25,
title = {$F$},
height=7cm,
xlabel = {$x$},
ylabel = {$y$},
colormap/viridis,
colorbar,
colorbar style = {
ylabel = {Vector Length}
}
]
\addplot3 [
point meta = {sqrt(( (2*y / x^2)*( 2*ln(cosh(x)) + (y-2*x)*tanh(x) - x*y*(1-(tanh(x))^2) ) )^2 + ( (4 / x)*( x*y - ln(cosh(x)) - y*tanh(x) )^2)},
quiver = {
u = {-(2*y / x^2)*( 2*ln(cosh(x)) + (y-2*x)*tanh(x) - x*y*(1-(tanh(x))^2) ) / sqrt(( (2*y / x^2)*( 2*ln(cosh(x)) + (y-2*x)*tanh(x) - x*y*(1-(tanh(x))^2) ) )^2 + ( (4 / x)*( x*y - ln(cosh(x)) - y*tanh(x) )^2)},
v = {-(4 / x)*( x*y - ln(cosh(x)) - y*tanh(x) ) / sqrt(( (2*y / x^2)*( 2*ln(cosh(x)) + (y-2*x)*tanh(x) - x*y*(1-(tanh(x))^2) ) )^2 + ( (4 / x)*( x*y - ln(cosh(x)) - y*tanh(x) )^2)},
scale arrows = 0.15,
},
quiver/colored = {mapped color},
-stealth,
domain = 0:5,
domain y = 0:5,
] {0};
\end{axis}
\end{tikzpicture}
\end{document}
请注意，上述代码通过F1=u, F2=v规范化箭头的长度（随后通过 颜色刻度图例指示point meta）来规范化中的 que 向量场。
我想在 Tikz 中获得类似于第一种情况的东西（带箭头的曲线，而不是许多直箭头），能够处理“奇点”。
先感谢您 ;）。
答案1
我尝试使用基本 TikZ 获得“流图”。每条轨迹（或轨道）都是从一个点开始构建的，并分别在时间上向前和向后移动。考虑到您的具体问题，我的代码不包括封闭轨道的情况。必须在命令中添加不同的分支\streamPlot。
stream plot by name上图使用了由和构成的轨道集stream plot TB（顶部 底部）。请注意，这些是pic对象，因此如果全局使用缩放，则应小心处理它们。它们基于\streamPlot。
我需要两者，因为我认为需要更接近矢量场的奇点才能获得良好的图像。例如，使用 Mathematica 构建的图像并没有真正显示原点附近发生的事情。
关于所考虑的矢量场的一些注释：一些重写是为了得到一种可能用于 TikZ 的形式（计算精度较低，但足以用于绘图）。下面是查看这两个组件的一种方式。我在X需要理解小值的行为X。
评论
查看第一幅图中的流图，可以注意到在轨迹“相遇”的地方发生了一些事情。为了理解这个问题，我们可以将矢量场与曲线一起绘制U（x，y）= 0（蓝色）和V（x，y）= 0（红色的）。
另请参阅下面用 Python 构建的最后一张图片（这就是它位于我的答案末尾的原因）。向量场的奇点并不完全遵循蓝色曲线（假设）。但我们在流图（图 1）中看不到这一点。这就是为什么我为某个初始条件添加了红色相应微分方程的解。它使用函数
ode-trajectory。所有计算背后的算法都是按照 风格开发的经典龙格-库塔算法
runge-kutta。
代码 代码相当长，主要是因为矢量场分量各定义了三次（不同上下文所需），并且图形元素有三种类型：矢量场、流图、柯西问题的轨迹。
\documentclass[11pt, margin=17pt]{standalone}
\usepackage{amsmath, amssymb}
\usepackage{ifthen}
\usepackage{tikz}
\usetikzlibrary{math, calc, arrows.meta}
\usetikzlibrary{decorations.markings}
\pgfkeys{/tikz/.cd,
x bound start/.store in=\xBoundStart,
x bound end/.store in=\xBoundEnd,
y bound start/.store in=\yBoundStart,
y bound end/.store in=\yBoundEnd
}
\makeatletter
\tikzset{%
bounds/.style n args={4}{%
evaluate={%
\xBoundStart = #1;
\xBoundEnd = #2;
\yBoundStart = #3;
\yBoundEnd = #4;
}
},
pics/vector field background/.style={% axes + grid for a vector field
code={%
\begin{scope}[every node/.style={scale=.8},
evaluate={%
\iIni = {int(\xBoundStart)};
\iEnd = {int(\xBoundEnd)};
\jIni = {int(\yBoundStart)};
\jEnd = {int(\yBoundEnd)};
}]
\draw[gray!50] (\xBoundStart, \yBoundStart)
grid (\xBoundEnd, \yBoundEnd);
\fill[black] (0, 0) circle (1pt);
\foreach \i in {\iIni, ..., \iEnd}{%
\path (\i, \yBoundStart) node[below=1ex] {$\i$};
}
\foreach \j in {\jIni, ..., \jEnd}{%
\path (\xBoundStart, \j) node[left=1ex] {$\j$};
}
\end{scope}
}
},
flow/.style={% create a directional arrow along a trajectory
decoration={
markings,% switch on markings
mark=at position .5 with {\arrow[#1]{Latex}}
}, postaction=decorate
},
-flow/.style={%
decoration={
markings,% switch on markings
mark=at position .5 with {\arrowreversed[#1]{Latex}}
}, postaction=decorate
},
runge-kutta/.style={% to be invoked before using ode-trajectory, ...
evaluate={%
int \N@ode, \i, \j;
\N@ode = 4;
for \i in {1, ..., \N@ode}{%
for \j in {1, ..., \N@ode}{%
\a@ode{\i,\j} = 0;
};
};
\a@ode{2,1} = 1/2; \a@ode{3,2} = 1/2; \a@ode{4,3} = 1;
\c@ode{1} = 0; \c@ode{2} = 1/2; \c@ode{3} = 1/2; \c@ode{4} = 1;
\b@ode{1} = 1/6; \b@ode{2} = 1/3; \b@ode{3} = 1/3; \b@ode{4} = 1/6;
}
},
% for ode in two dimensions
ode-trajectory/.style args={RHTx=#1, RHTy=#2, from=#3, to=#4, steps=#5}{%
insert path={coordinate (tmp) let \p1 = (tmp) in},
evaluate={%
real \showpointr, \testTmp;
\showpointr = 1;
int \steps@ode, \i, \j, \k, \s;
\t@ode{0} = #3;
\u@ode{0} = \x1*0.0352778; % converts pt to cm
\v@ode{0} = \y1*0.0352778;
\steps@ode = int(#5);
\t@ode{\steps@ode} = #4;
\h = (\t@ode{\steps@ode} -\t@ode{0})/\steps@ode;
for \s in {1, 2, ..., \steps@ode}{
\k = int(\s -1);
\t@ode{0,1} = \t@ode{\k};
\x@ode{0,1} = \u@ode{\k};
\y@ode{0,1} = \v@ode{\k};
\qx@ode{0,1} = #1(\t@ode{0,1}, \x@ode{0,1}, \y@ode{0,1});
\qy@ode{0,1} = #2(\t@ode{0,1}, \x@ode{0,1}, \y@ode{0,1});
for \i in {2, ..., \N@ode}{%
\t@ode{0,\i} = \t@ode{\k} +\h*\c@ode{\i};
\x@ode{0,\i} = \u@ode{\k};
\y@ode{0,\i} = \v@ode{\k};
for \j in {1, ..., {int(\i -1)}}{%
\x@ode{0,\i} = \x@ode{0,\i} +\h*\a@ode{\i,\j}*\qx@ode{0,\j};
\y@ode{0,\i} = \y@ode{0,\i} +\h*\a@ode{\i,\j}*\qy@ode{0,\j};
};
\qx@ode{0,\i} = #1(\t@ode{0,\i}, \x@ode{0,\i}, \y@ode{0,\i});
\qy@ode{0,\i} = #2(\t@ode{0,\i}, \x@ode{0,\i}, \y@ode{0,\i});
};
\t@ode{\s} = \t@ode{\k} +\h;
\u@ode{\s} = \u@ode{\k};
\v@ode{\s} = \v@ode{\k};
for \i in {1, 2, ..., \N@ode}{%
\u@ode{\s} = \u@ode{\s} +\h*\b@ode{\i}*\qx@ode{0,\i};
\v@ode{\s} = \v@ode{\s} +\h*\b@ode{\i}*\qy@ode{0,\i};
};
};
\k = \steps@ode-1;
\testTmp = pow(pow(\u@ode{\steps@ode} -\u@ode{\k}, 2)
+pow(\v@ode{\steps@ode} -\v@ode{\k}, 2), .5);
},
insert path={%
\foreach \s in {1, ..., \steps@ode}{
-- (\u@ode{\s}, \v@ode{\s})
}
}
},
m vector field/.style n args={5}{% u / v / Nx / Ny / arrow scale
insert path = {%
\pgfextra{%
\let\tikz@mode@save=\tikz@mode
\let\tikz@options@save=\tikz@options%
}
},
evaluate={%
coordinate \tmp@P, \tmp@V;
integer \Nx, \Ny;
\Nx = #3;
\Ny = #4;
real \dx, \dy, \tmp, \uVF, \vVF, \nVF, \s@cst;
\s@cst = .49;
\dx = (\xBoundEnd -\xBoundStart)/\Nx;
\dy = (\yBoundEnd -\yBoundStart)/\Ny;
for \i in {1, ..., \Nx}{%
for \j in {1, ..., \Ny}{%
\uVF = #1(\xBoundStart -\dx/2 +\i*\dx, \yBoundStart -\dy/2 +\j*\dy);
\vVF = #2(\xBoundStart -\dx/2 +\i*\dx, \yBoundStart -\dy/2 +\j*\dy);
\nVF = pow(\uVF*\uVF +\vVF*\vVF, .5);
if \nVF > \s@cst then {%
\uVF = \uVF/\nVF*.5;
\vVF = \vVF/\nVF*.5;
};
\tmp@P = (\xBoundStart +\i*\dx, \yBoundStart +\j*\dy);
\tmp@V = (#5*\uVF, #5*\vVF);
{
\draw[arrows={-Latex[width=2.5pt, length=2pt]}]
\pgfextra{\let\tikz@mode=\tikz@mode@save
\let\tikz@options=\tikz@options@save}
($(\tmp@P) -.5*(\tmp@V)$) -- ++(\tmp@V);
};
};
};
}
}
}
\newcommand{\streamPlot}[7]{% RHTx, RHTy, h, px, py, min, color
\tikzmath{%
integer \steps@ode, \i, \j, \k, \s\, flag;
real \h, \nor;
\flag = 0;
\s = 0;
\k = 0;
\h = #3;
\u@ode{0} = #4;
\v@ode{0} = #5;
\t@ode{0} = 0;
},
\whiledo{\equal{\flag}{0}}{%
\tikzmath{
\s = \s +1;
\t@ode{0,1} = 0;
\x@ode{0,1} = \u@ode{\k};
\y@ode{0,1} = \v@ode{\k};
\qx@ode{0,1} = #1(\t@ode{0,1}, \x@ode{0,1}, \y@ode{0,1});
\qy@ode{0,1} = #2(\t@ode{0,1}, \x@ode{0,1}, \y@ode{0,1});
for \i in {2, ..., \N@ode}{%
\t@ode{0,\i} = \t@ode{\k} +\h*\c@ode{\i};
\x@ode{0,\i} = \u@ode{\k};
\y@ode{0,\i} = \v@ode{\k};
for \j in {1, ..., {int(\i -1)}}{%
\x@ode{0,\i} = \x@ode{0,\i} +\h*\a@ode{\i,\j}*\qx@ode{0,\j};
\y@ode{0,\i} = \y@ode{0,\i} +\h*\a@ode{\i,\j}*\qy@ode{0,\j};
};
\qx@ode{0,\i} = #1(\t@ode{0,\i}, \x@ode{0,\i}, \y@ode{0,\i});
\qy@ode{0,\i} = #2(\t@ode{0,\i}, \x@ode{0,\i}, \y@ode{0,\i});
};
\t@ode{\s} = \t@ode{\k} +\h;
\u@ode{\s} = \u@ode{\k};
\v@ode{\s} = \v@ode{\k};
for \i in {1, 2, ..., \N@ode}{%
\u@ode{\s} = \u@ode{\s} +\h*\b@ode{\i}*\qx@ode{0,\i};
\v@ode{\s} = \v@ode{\s} +\h*\b@ode{\i}*\qy@ode{0,\i};
};
\nor = pow(%
pow(\u@ode{\s} -\u@ode{\k}, 2) +pow(\v@ode{\s} -\v@ode{\k}, 2),
.5);
if \nor<#6 then {\flag = 1;};
if \u@ode{\s}<\xBoundStart then {\flag = 1;};
if \u@ode{\s}>\xBoundEnd then {\flag = 1;};
if \v@ode{\s}<\yBoundStart then {\flag = 1;};
if \v@ode{\s}>\yBoundEnd then {\flag = 1;};
\k = \k +1;
}
},
\tikzmath{%
if \s>1 then {%
if #3>0 then {%
{%
\draw[#7, flow={#7}] (\u@ode{0}, \v@ode{0})
\foreach \i [parse=true] in {1, ..., \s-1}{
-- (\u@ode{\i}, \v@ode{\i})
};
};
} else {%
{%
\draw[#7, -flow={#7}] (\u@ode{0}, \v@ode{0})
\foreach \i [parse=true] in {1, ..., \s-1}{
-- (\u@ode{\i}, \v@ode{\i})
};
};
};
};
}
}
\makeatother
\tikzset{
pics/stream plot by name/.style args={RHTx=#1, RHTy=#2, dT=#3, pName=#4,
pNumber=#5, min=#6, color=#7}{%
code={
\foreach \i in {1, ..., #5}{%
\tikzmath{%
coordinate \P;
\P = (#4-\i);
\px = \Px*1pt/1cm; % converts pt to cm
\py = \Py*1pt/1cm;
}
% \draw (\px, \py) circle (2pt);
\streamPlot{#1}{#2}{#3}{\px}{\py}{#6}{#7};
\streamPlot{#1}{#2}{-#3}{\px}{\py}{#6}{#7};
}
}
},
pics/stream plot TB/.style args={%
RHTx=#1, RHTy=#2, dT=#3, uSteps=#4, min=#5, color=#6}{%
code={
\tikzmath{%
real \px, \py, \dx;
\dx = (\xBoundEnd -\xBoundStart)/#4;
}
\foreach \i in {0, ..., #4}{%
\tikzmath{%
\px = \xBoundStart +\i*\dx;
\py = \yBoundStart;
}
\streamPlot{#1}{#2}{#3}{\px}{\py}{#5}{#6};
\streamPlot{#1}{#2}{-#3}{\px}{\py}{#5}{#6};
}
\foreach \i in {0, ..., #4}{%
\tikzmath{%
\px = \xBoundStart +\i*\dx;
\py = \yBoundEnd;
}
\streamPlot{#1}{#2}{#3}{\px}{\py}{#5}{#6};
\streamPlot{#1}{#2}{-#3}{\px}{\py}{#5}{#6};
}
}
}
}
\begin{document}
%\iffalse
\begin{tikzpicture}[every node/.style={%
right, align=left, inner sep=3ex
}]
\path (-1.3, 0) node {$U(x, y)$}
(0, 0) node[] {$\displaystyle
= y\left[
-2\,\frac{\ln(\cosh x)}{x^2}
+\frac{x -\tanh x}{x^2}\,y
+\frac{\tanh x}{x}\,(2 -y\,\tanh x)
\right]$}
++(0, -1) node[] {$\displaystyle
= y\left[
1 +\frac{2}{9}\,x^4
-\bigg(\frac{2}{3}\,x -\frac{8}{15}\,x^3\bigg)y
\right] +[5]$};
\path (-1.3, -3) node {$V(x, y)$}
(0, -3) node[] {$\displaystyle
= 2\,\frac{\ln(\cosh x)}{x} +2\,\frac{\tanh x -x}{x}\,y$}
++(0, -1) node[] {$\displaystyle
= x\left[
1 -\frac{1}{3}\,x^2
-\bigg( \frac{2}{3}\,x -\frac{4}{15}\,x^3 \bigg)y
\right] +[5]$};
\end{tikzpicture}
%\fi
\tikzmath{%
function Ux(\t, \u, \v) {%
real \a, \b, \c;
if abs(\u)>.05 then {%
\a = -2*ln(cosh(\u))/(\u*\u);
\b = (\u -tanh(\u))/(\u*\u)*\v;
\c = (tanh(\u)/\u)*(2 - tanh(\u)*\v);
} else {%
\a = 1 -2/3*\u*\v;
\b = 8/15*pow(\u, 3)*\v;
\c = 2/9*pow(\u, 4);
};
return {(\a +\b +\c)*\v};
};
function Uy(\t, \u, \v) {%
real \a, \b;
if abs(\u)>.05 then {%
\a = ln(cosh(\u))/\u;
\b = (tanh(\u) -\u)/\u;
return {2*\a +2*\b*\v};
} else {
\a = 1 -pow(\u, 2)/3;
\b = -(2/3)*\u +(4/15)*pow(\u, 3);
return {\u*(\a +\b*\v)};
};
};
function fUy(\u) {%
real \a, \b;
\a = ln(cosh(\u))/\u;
\b = (\u -tanh(\u))/\u;
return \a/\b;
};
function UxVF(\u, \v) {%
real \a, \b, \c;
if abs(\u)>.05 then {%
\a = -2*ln(cosh(\u))/(\u*\u);
\b = (\u -tanh(\u))/(\u*\u)*\v;
\c = (tanh(\u)/\u)*(2 - tanh(\u)*\v);
} else {%
\a = 1 -2/3*\u*\v;
\b = 8/15*pow(\u, 3)*\v;
\c = 2/9*pow(\u, 4);
};
return {(\a +\b +\c)*\v};
};
function fUx(\u) {%
real \a, \b;
\a = 2*(tanh(\u)/\u) -2*ln(cosh(\u))/(\u*\u);
\b = -(\u -tanh(\u))/(\u*\u) +(tanh(\u)/\u)*tanh(\u);
return \a/\b;
};
function UyVF(\u, \v) {%
real \a, \b;
if abs(\u)>.05 then {%
\a = ln(cosh(\u))/\u;
\b = (tanh(\u) -\u)/\u;
return {2*\a +2*\b*\v};
} else {
\a = 1 -pow(\u, 2)/3;
\b = -(2/3)*\u +(4/15)*pow(\u, 3);
return {\u*(\a +\b*\v)};
};
};
}
\iffalse
\begin{tikzpicture}
\draw[gray!50, very thin] (-3.5, -3.25) grid (3.5, 3.25);
\draw[->] (-3.5, 0) -- (3.5, 0) node[above right] {$x$};
\draw[->] (0,-3.5) -- (0, 3.5) node[left] {$y$};
\draw[blue!70!black, thick, variable=\t, domain=.5:3.25, samples=200]
plot (\t, {fUx(\t)});
\draw[blue!70!black, thick] (-3.25, 0) -- (3.25, 0);
\draw[blue!70!black, thick, variable=\t, domain=-3.25:-.5, samples=200]
plot (\t, {fUx(\t)});
\draw[red, thin, variable=\t, domain=.5:3.25, samples=200]
plot (\t, {fUy(\t)});
\draw[red] (-0, -3.25) -- (0, 3.25);
\draw[red, thin, variable=\t, domain=-3.25:-.5, samples=200]
plot (\t, {fUy(\t)});
\end{tikzpicture}
\fi
\iffalse
\begin{tikzpicture}[runge-kutta, bounds={-4}{4}{-4}{4}, scale=1.2]
\path pic[scale=1.2] {vector field background};
\draw[blue!70!black, thick,
variable=\t, domain=.35:\xBoundEnd, samples=200]
plot (\t, {fUx(\t)});
\draw[blue!70!black, thick] (\xBoundStart, 0) -- (\xBoundEnd, 0);
\draw[blue!70!black, thick,
variable=\t, domain=\xBoundStart:-.35, samples=200]
plot (\t, {fUx(\t)});
\draw[red, thin, variable=\t, domain=.35:\xBoundEnd, samples=200]
plot (\t, {fUy(\t)});
\draw[red] (-0, \xBoundStart) -- (0, \xBoundEnd);
\draw[red, thin, variable=\t, domain=\xBoundStart:-.35, samples=200]
plot (\t, {fUy(\t)});
\draw[green!50!black, thick, m vector field={UxVF}{UyVF}{35}{35}{.4}];
\end{tikzpicture}
\fi
\iffalse
\begin{tikzpicture}[runge-kutta, bounds={-4}{4}{-3.5}{3.5}]
\path pic {vector field background};
% A points
\path (0, 1.5) coordinate (A-1);
\path (0, -1.5) coordinate (A-2);
\path (2.2, \yBoundEnd) coordinate (A-3);
\path (-2.2, \yBoundStart) coordinate (A-4);
% B points
\tikzmath{
integer \N;
\N = 4;
}
\foreach \i in {1, ..., \N}{%
\path (\i*360/\N: .5) coordinate (B-\i);
}
\path pic {stream plot by name={RHTx=Ux, RHTy=Uy, dT=.05,
pName=A, pNumber=4, min=.005, color=blue!60!black}};
\path pic {stream plot by name={RHTx=Ux, RHTy=Uy, dT=.05,
pName=B, pNumber=\N, min=.005, color=blue!60!black}};
\path pic {stream plot TB={RHTx=Ux, RHTy=Uy, dT=.05,
uSteps=11, min=.005, color=blue!60!black}};
\draw[red, very thick] (2.7, -3)
[ode-trajectory={RHTx=Ux, RHTy=Uy, from=0, to=15, steps=1500}];
\end{tikzpicture}
\fi
\end{document}
的水平曲线乌矢量场的分量。图 1 中的红色曲线是此处的黑色轨迹！
