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-rw-r--r--vorlesungen/slides/10/vektorfelder.mp241
1 files changed, 168 insertions, 73 deletions
diff --git a/vorlesungen/slides/10/vektorfelder.mp b/vorlesungen/slides/10/vektorfelder.mp
index f488327..e63b2d5 100644
--- a/vorlesungen/slides/10/vektorfelder.mp
+++ b/vorlesungen/slides/10/vektorfelder.mp
@@ -48,17 +48,17 @@ 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;
+% % 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) =
@@ -66,6 +66,10 @@ def dglField(expr x, y) =
(-y, x)
enddef;
+pair A;
+A := (1, 0);
+draw A scaled unit withpen pencircle scaled 8bp withcolor red;
+
% Draw arrows for each grid point
pickup pencircle scaled 0.5pt;
for x = -1.5 step 0.1 until 1.55:
@@ -78,11 +82,9 @@ endfor;
endfig;
-
-
%
% Vektorfeld in der Ebene mit Lösungskurve
-% X \in sl(2, R)
+% Euler(1)
%
beginfig(2)
@@ -101,18 +103,28 @@ 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:
+def dglField(expr x, y) =
+ (-y, x)
+enddef;
+
+def dglFieldp(expr z) =
+ dglField(xpart z, ypart z)
+enddef;
+
+def curve(expr z, l, s) =
path p;
- p = (-1.5,y) -- (1.5, y);
- pickup pencircle scaled 1pt;
+ p := z;
+ for t = 0 step 1 until l:
+ p := p--((point (length p) of p) shifted (s * dglFieldp(point (length p) of p)));
+ endfor;
draw p scaled unit withcolor red;
-endfor;
-
-def dglField(expr x, y) =
- (y, 0)
enddef;
+pair A;
+A := (1, 0);
+draw A scaled unit withpen pencircle scaled 8bp withcolor red;
+curve(A, 0, 1);
+
% Draw arrows for each grid point
pickup pencircle scaled 0.5pt;
for x = -1.5 step 0.1 until 1.55:
@@ -125,12 +137,9 @@ endfor;
endfig;
-
-
-
%
% Vektorfeld in der Ebene mit Lösungskurve
-% Y \in sl(2, R)
+% Euler(2)
%
beginfig(3)
@@ -149,42 +158,82 @@ 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:
+def dglField(expr x, y) =
+ (-y, x)
+enddef;
+
+def dglFieldp(expr z) =
+ dglField(xpart z, ypart z)
+enddef;
+
+def curve(expr z, l, s) =
path p;
- p = (x, -1.5) -- (x, 1.5);
- pickup pencircle scaled 1pt;
+ p := z;
+ for t = 0 step 1 until l:
+ p := p--((point (length p) of p) shifted (s * dglFieldp(point (length p) of p)));
+ endfor;
draw p scaled unit withcolor red;
+enddef;
+
+pair A;
+A := (1, 0);
+draw A scaled unit withpen pencircle scaled 8bp withcolor red;
+curve(A, 1, 0.5);
+
+% 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;
-def dglField(expr x, y) =
- (0, x)
-enddef;
+endfig;
-% 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);
+% Vektorfeld in der Ebene mit Lösungskurve
+% Euler(3)
%
-% numeric innerlength;
-% innerlength = 500;
-%
-% for a = 0 step 30 until 330:
-% curve(0.05 * (cosd(a), sind(a)), innerlength);
-% endfor;
+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 (-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));
+
+def dglField(expr x, y) =
+ (-y, x)
+enddef;
+
+def dglFieldp(expr z) =
+ dglField(xpart z, ypart z)
+enddef;
+
+def curve(expr z, l, s) =
+ path p;
+ p := z;
+ for t = 0 step 1 until l:
+ p := p--((point (length p) of p) shifted (s * dglFieldp(point (length p) of p)));
+ endfor;
+ draw p scaled unit withcolor red;
+enddef;
+
+pair A;
+A := (1, 0);
+draw A scaled unit withpen pencircle scaled 8bp withcolor red;
+curve(A, 3, 0.25);
% Draw arrows for each grid point
pickup pencircle scaled 0.5pt;
@@ -198,12 +247,11 @@ endfor;
endfig;
-
%
% Vektorfeld in der Ebene mit Lösungskurve
-% H \in sl(2, R)
+% Euler(4)
%
-beginfig(4)
+beginfig(5)
numeric unit;
unit := 150;
@@ -215,40 +263,88 @@ z3 = ( 0, -1.5) * unit;
z4 = ( 0, 1.5) * unit;
pickup pencircle scaled 1pt;
-drawarrow (z1 shifted (-25,0))--(z2 shifted (25,0));
+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 (25,0));
+label.top(btex $x_1$ etex, z2 shifted (10,0));
label.rt(btex $x_2$ etex, z4 shifted (0,10));
def dglField(expr x, y) =
- (x, -y)
+ (-y, x)
enddef;
def dglFieldp(expr z) =
dglField(xpart z, ypart z)
enddef;
-def curve(expr z, l) =
+def curve(expr z, l, s) =
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)));
+ p := p--((point (length p) of p) shifted (s * 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);
+pair A;
+A := (1, 0);
+draw A scaled unit withpen pencircle scaled 8bp withcolor red;
+curve(A, 7, 0.125);
+
+% 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
+% Euler(5)
+%
+beginfig(6)
+
+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));
+
+def dglField(expr x, y) =
+ (-y, x)
+enddef;
+
+def dglFieldp(expr z) =
+ dglField(xpart z, ypart z)
+enddef;
+
+def curve(expr z, l, s) =
+ path p;
+ p := z;
+ for t = 0 step 1 until l:
+ p := p--((point (length p) of p) shifted (s * dglFieldp(point (length p) of p)));
+ endfor;
+ draw p scaled unit withcolor red;
+enddef;
+
+pair A;
+A := (1, 0);
+draw A scaled unit withpen pencircle scaled 8bp withcolor red;
+curve(A, 99, 0.01);
+
% Draw arrows for each grid point
pickup pencircle scaled 0.5pt;
for x = -1.5 step 0.1 until 1.55:
@@ -262,5 +358,4 @@ endfor;
endfig;
-
end;
generated by cgit v1.2.1 (git 2.18.0) at 2024-06-01 10:46:07 +0200