From cb562f607b18540df333d6d9a911cf51c91884d0 Mon Sep 17 00:00:00 2001 From: Roy Seitz Date: Thu, 15 Apr 2021 23:49:49 +0200 Subject: Mehr Folien und einige Typos. --- vorlesungen/slides/10/vektorfelder.mp | 266 ++++++++++++++++++++++++++++++++++ 1 file changed, 266 insertions(+) create mode 100644 vorlesungen/slides/10/vektorfelder.mp (limited to 'vorlesungen/slides/10/vektorfelder.mp') diff --git a/vorlesungen/slides/10/vektorfelder.mp b/vorlesungen/slides/10/vektorfelder.mp new file mode 100644 index 0000000..f488327 --- /dev/null +++ b/vorlesungen/slides/10/vektorfelder.mp @@ -0,0 +1,266 @@ +% +% Stroemungsfelder linearer Differentialgleichungen +% +% (c) 2015 Prof Dr Andreas Mueller, Hochschule Rapperswil +% 2021-04-14, Roy Seitz, Copied for SeminarMatrizen +% +verbatimtex +\documentclass{book} +\usepackage{times} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsfonts} +\usepackage{txfonts} +\begin{document} +etex; + +input TEX; + +TEXPRE("%&latex" & char(10) & +"\documentclass{book}" & +"\usepackage{times}" & +"\usepackage{amsmath}" & +"\usepackage{amssymb}" & +"\usepackage{amsfonts}" & +"\usepackage{txfonts}" & +"\begin{document}"); +TEXPOST("\end{document}"); + +% +% Vektorfeld in der Ebene mit Lösungskurve +% so(2) +% +beginfig(1) + +% Scaling parameter +numeric unit; +unit := 150; + +% Some points +z1 = (-1.5, 0) * unit; +z2 = ( 1.5, 0) * unit; +z3 = ( 0, -1.5) * unit; +z4 = ( 0, 1.5) * unit; + +pickup pencircle scaled 1pt; +drawarrow (z1 shifted (-10,0))--(z2 shifted (10,0)); +drawarrow (z3 shifted (0,-10))--(z4 shifted (0,10)); +label.top(btex $x_1$ etex, z2 shifted (10,0)); +label.rt(btex $x_2$ etex, z4 shifted (0,10)); + +% Draw circles +for x = 0.2 step 0.2 until 1.4: + path p; + p = (x,0); + for a = 5 step 5 until 355: + p := p--(x*cosd(a), x*sind(a)); + endfor; + p := p--cycle; + pickup pencircle scaled 1pt; + draw p scaled unit withcolor red; +endfor; + +% Define DGL +def dglField(expr x, y) = + %(-0.5 * (x + y), -0.5 * (y - x)) + (-y, x) +enddef; + +% Draw arrows for each grid point +pickup pencircle scaled 0.5pt; +for x = -1.5 step 0.1 until 1.55: + for y = -1.5 step 0.1 until 1.55: + drawarrow ((x, y) * unit) + --(((x,y) * unit) shifted (8 * dglField(x,y))) + withcolor blue; + endfor; +endfor; + +endfig; + + + +% +% Vektorfeld in der Ebene mit Lösungskurve +% X \in sl(2, R) +% +beginfig(2) + +numeric unit; +unit := 150; + +z0 = ( 0, 0); +z1 = (-1.5, 0) * unit; +z2 = ( 1.5, 0) * unit; +z3 = ( 0, -1.5) * unit; +z4 = ( 0, 1.5) * unit; + +pickup pencircle scaled 1pt; +drawarrow (z1 shifted (-10,0))--(z2 shifted (10,0)); +drawarrow (z3 shifted (0,-10))--(z4 shifted (0,10)); +label.top(btex $x_1$ etex, z2 shifted (10,0)); +label.rt(btex $x_2$ etex, z4 shifted (0,10)); + +% Draw flow lines +for y = -1.4 step 0.2 until 1.4: + path p; + p = (-1.5,y) -- (1.5, y); + pickup pencircle scaled 1pt; + draw p scaled unit withcolor red; +endfor; + +def dglField(expr x, y) = + (y, 0) +enddef; + +% Draw arrows for each grid point +pickup pencircle scaled 0.5pt; +for x = -1.5 step 0.1 until 1.55: + for y = -1.5 step 0.1 until 1.55: + drawarrow ((x, y) * unit) + --(((x,y) * unit) shifted (8 * dglField(x,y))) + withcolor blue; + endfor; +endfor; + +endfig; + + + + +% +% Vektorfeld in der Ebene mit Lösungskurve +% Y \in sl(2, R) +% +beginfig(3) + +numeric unit; +unit := 150; + +z0 = ( 0, 0); +z1 = (-1.5, 0) * unit; +z2 = ( 1.5, 0) * unit; +z3 = ( 0, -1.5) * unit; +z4 = ( 0, 1.5) * unit; + +pickup pencircle scaled 1pt; +drawarrow (z1 shifted (-10,0))--(z2 shifted (10,0)); +drawarrow (z3 shifted (0,-10))--(z4 shifted (0,10)); +label.top(btex $x_1$ etex, z2 shifted (10,0)); +label.rt(btex $x_2$ etex, z4 shifted (0,10)); + +% Draw flow lines +for x = -1.4 step 0.2 until 1.4: + path p; + p = (x, -1.5) -- (x, 1.5); + pickup pencircle scaled 1pt; + draw p scaled unit withcolor red; +endfor; + +def dglField(expr x, y) = + (0, x) +enddef; + +% def dglFieldp(expr z) = +% dglField(xpart z, ypart z) +% enddef; +% +% def curve(expr z, l) = +% path p; +% p := z; +% for t = 0 step 1 until l: +% p := p--((point (length p) of p) shifted (0.01 * dglFieldp(point (length p) of p))); +% endfor; +% draw p scaled unit withcolor red; +% enddef; +% +% numeric outerlength; +% outerlength = 200; +% curve(( 0.1, 0), outerlength); +% curve(( 0.2, 0), outerlength); +% +% numeric innerlength; +% innerlength = 500; +% +% for a = 0 step 30 until 330: +% curve(0.05 * (cosd(a), sind(a)), innerlength); +% endfor; + +% Draw arrows for each grid point +pickup pencircle scaled 0.5pt; +for x = -1.5 step 0.1 until 1.55: + for y = -1.5 step 0.1 until 1.55: + drawarrow ((x, y) * unit) + --(((x,y) * unit) shifted (8 * dglField(x,y))) + withcolor blue; + endfor; +endfor; + +endfig; + + +% +% Vektorfeld in der Ebene mit Lösungskurve +% H \in sl(2, R) +% +beginfig(4) + +numeric unit; +unit := 150; + +z0 = ( 0, 0); +z1 = (-1.5, 0) * unit; +z2 = ( 1.5, 0) * unit; +z3 = ( 0, -1.5) * unit; +z4 = ( 0, 1.5) * unit; + +pickup pencircle scaled 1pt; +drawarrow (z1 shifted (-25,0))--(z2 shifted (25,0)); +drawarrow (z3 shifted (0,-10))--(z4 shifted (0,10)); +label.top(btex $x_1$ etex, z2 shifted (25,0)); +label.rt(btex $x_2$ etex, z4 shifted (0,10)); + +def dglField(expr x, y) = + (x, -y) +enddef; + +def dglFieldp(expr z) = + dglField(xpart z, ypart z) +enddef; + +def curve(expr z, l) = + path p; + p := z; + for t = 0 step 1 until l: + p := p--((point (length p) of p) shifted (0.01 * dglFieldp(point (length p) of p))); + endfor; + draw p scaled unit withcolor red; +enddef; + +for i = -1 step 2 until 1: + for k = -1 step 2 until 1: + curve((1.3 * i, 1.5 * k), 18); + curve((1.1 * i, 1.5 * k), 35); + curve((0.9 * i, 1.5 * k), 55); + curve((0.7 * i, 1.5 * k), 80); + curve((0.5 * i, 1.5 * k), 114); + curve((0.3 * i, 1.5 * k), 165); + curve((0.1 * i, 1.5 * k), 275); + endfor; +endfor; + +% Draw arrows for each grid point +pickup pencircle scaled 0.5pt; +for x = -1.5 step 0.1 until 1.55: + for y = -1.5 step 0.1 until 1.55: + drawarrow ((x, y) * unit) + --(((x,y) * unit) shifted (8 * dglField(x,y))) + withcolor blue; + endfor; +endfor; + +endfig; + + + +end; -- cgit v1.2.1