Doppelintegral_ Volumenberechnung
Author: Werner Knobloch 2011

R_x(α) = \left(\begin{array}&  1& 0& 0\  0& cos(α)& -sin(α)\  0& sin(α)& cos(α)\\end{array}\right)

R_y(\alpha) = \left(\begin{array} & cos(β)& 0& -sin(β)\  0& 1& 0\ sin(β)& 0& cos(β) \ \end{array}\right)

R_z(γ) = \left(\begin{array}&  cos(γ) & -sin(γ) &0 \  sin(γ) & cos(γ) & 0 \0& 0&1\ \end{array}\right)

R_{xyz} = R_x R_y R_z

R_y(b) \cdot R_z(γ) = \left(\begin{array}  & cos(β)cos(γ) & - cos(β)sin(γ) &- sin(β) \  sin(γ) & cos(γ) & 0 \ sin(β)cos(γ) & - sin(β)sin(γ)& cos(β)\ \end{array}\right)

R_{xyz}(α,β,γ) =R_x(α) \cdot R_y(β) \cdot R_z(γ) = \left(\begin{array}  &  cos(β)cos(γ) & - cos(β)sin(γ) &- sin(β) \ cos(α) sin(γ) - sin(α) sin(β) cos(γ)    & cos(α) cos(γ) + sin(α) sin(β) sin(γ) & - sin(α) cos(β)  \ sin(α) sin(γ) + cos(α) sin(β) cos(γ) & sin(α) cos(γ) - cos(α) sin(β) sin(γ)& cos(α) cos(β)   \ \end{array}\right)

PR_{xy}(α,β,γ) =P_{xy} \cdot R_x(α) \cdot R_y(β) \cdot R_z(γ)=\left(\begin{array}  & cos(β)cos(γ) & - cos(β)sin(γ) &- sin(β) \ cos(α) sin(γ) - sin(α) sin(β) cos(γ)    & cos(α) cos(γ) + sin(α) sin(β) sin(γ) & - sin(α) cos(β)  \ \end{array}\right)

f(x,y_0)

f(x_0,y)

Werner Knobloch 2011

http://www.mp.haw-hamburg.de/pers/Vassilevskaya/download/m2/int-mv/volumen-1.pdf

Werner Knobloch 2011

Doppelintegral: Volumenberechnung

Δ = 0.3 (slider from 0.1 to 1, step 0.05)
E_3: {{0}, {0}, {1}}
E_2: {{0}, {1}, {0}}
β = 6.283185307179586 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
α = 5.532693728822025 rad (slider from 0*° to 6.283185307179586, step 0.017453292519943295)
γ = 3.961897402027128 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
Direktabbildung: {{(cos(β) * cos(γ)), ((-cos(β)) * sin(γ)), (-sin(β))}, {(sin(α) * sin(γ)) + ((cos(α) * sin(β)) * cos(γ)), (sin(α) * cos(γ)) - ((cos(α) * sin(β)) * sin(γ)), (cos(α) * cos(β))}}
e3: (Direktabbildung * E_3)
e_3: (Element[Element[e3, 1], 1], Element[Element[e3, 2], 1])
e2: (Direktabbildung * E_2)
e_2: (Element[Element[e2, 1], 1], Element[Element[e2, 2], 1])
E_1: {{1}, {0}, {0}}
e1: (Direktabbildung * E_1)
e_1: (Element[Element[e1, 1], 1], Element[Element[e1, 2], 1])
ymax = 3.141592653589793 (slider from 0.1 to 4, step 0.01)
ymin: (-ymax)
n_y: round((ymax - ymin) / Δ)
xmax = 3.141592653589793 (slider from 0.1 to 4, step 0.01)
xmin: (-xmax)
n_x: round((xmax - xmin) / Δ)
f(x, y) = 3 + (cos(x) * cos(y))
F_x = Integral(f, x)
c_7 = 0
c_6 = 0
c_5 = 0
c_4 = 0
c_3 = 0
T_y = 6
T_x = -4.8 (slider from -10 to 10, step 0.1)
t_x = 2.1
t_y = 1.5 (slider from 0.1 to 2, step 0.1)
R_x: {{1, 0, 0}, {0, cos(α), (-sin(α))}, {0, sin(α), cos(α)}}
R_y: {{cos(β), 0, (-sin(β))}, {0, 1, 0}, {sin(β), 0, cos(β)}}
R_z: {{cos(γ), (-sin(γ)), 0}, {sin(γ), cos(γ), 0}, {0, 0, 1}}
R_{xyz}: ((R_x * R_y) * R_z)
P_{xy}: {{1, 0, 0}, {0, 0, 1}}
PR_{xy}: (((P_{xy} * R_x) * R_y) * R_z)
V_1: e_1
V_2: e_2
V_3: e_3
Text14: "R_y = \left(\begin{array} " + (cos(β)) + "  & 0& " + ((-sin(β))) + " \\  0& 1& 0\\ " + (sin(β)) + "  & 0& " + (cos(β)) + " \\ \end{array}\right)"
DirektRot: {{(cos(β) * cos(γ)), ((-cos(β)) * sin(γ)), (-sin(β))}, {(cos(α) * sin(γ)) - ((sin(α) * sin(β)) * cos(γ)), (cos(α) * cos(γ)) + ((sin(α) * sin(β)) * sin(γ)), ((-sin(α)) * cos(β))}, {(sin(α) * sin(γ)) + ((cos(α) * sin(β)) * cos(γ)), (sin(α) * cos(γ)) - ((cos(α) * sin(β)) * sin(γ)), (cos(α) * cos(β))}}
a = 0 (slider from -5 to 5, step 1)
b = 1 (slider from -5 to 5, step 1)
c = 0 (slider from -5 to 5, step 1)
Demopunkt = Raumpunkt(a, b, c, Direktabbildung)
Trennebenenhöhe = 3.2 (slider from 0 to 5, step 0.1)
O = Raumpunkt(0, 0, 0, Direktabbildung)
A_{xy} = Raumpunkt(Trennebenenhöhe, Trennebenenhöhe, 0, Direktabbildung)
B_{xy} = Raumpunkt(Trennebenenhöhe, (-Trennebenenhöhe), 0, Direktabbildung)
C_{xy} = Raumpunkt((-Trennebenenhöhe), (-Trennebenenhöhe), 0, Direktabbildung)
D_{xy} = Raumpunkt((-Trennebenenhöhe), Trennebenenhöhe, 0, Direktabbildung)
A_{yz} = Raumpunkt(0, Trennebenenhöhe, Trennebenenhöhe, Direktabbildung)
B_{yz} = Raumpunkt(0, Trennebenenhöhe, (-Trennebenenhöhe), Direktabbildung)
C_{yz} = Raumpunkt(0, (-Trennebenenhöhe), (-Trennebenenhöhe), Direktabbildung)
D_{yz} = Raumpunkt(0, (-Trennebenenhöhe), Trennebenenhöhe, Direktabbildung)
Vieleck2, a_1, b_1, c_1, d_1 = Polygon(A_{yz}, B_{yz}, C_{yz}, D_{yz})
A_{xz} = Raumpunkt(Trennebenenhöhe, 0, Trennebenenhöhe, Direktabbildung)
B_{xz} = Raumpunkt(Trennebenenhöhe, 0, (-Trennebenenhöhe), Direktabbildung)
C_{xz} = Raumpunkt((-Trennebenenhöhe), 0, (-Trennebenenhöhe), Direktabbildung)
D_{xz} = Raumpunkt((-Trennebenenhöhe), 0, Trennebenenhöhe, Direktabbildung)
Vieleck3, a_2, b_2, c_2, d_2 = Polygon(A_{xz}, B_{xz}, C_{xz}, D_{xz})
x_{Achse} = Line(V_1, O)
y_{Achse} = Line(O, V_2)
z_{Achse} = Line(O, V_3)
Achsen = true
Koordinatenebenen = false
x_0 = -0.14159265358979312 (slider from ymin to ymax, step 0.1)
y_0 = -1.441592653589793 (slider from xmin to xmax, step 0.1)
Raster_{x0} = Segment((x_0 * V_1) + (ymin * V_2), (x_0 * V_1) + (ymax * V_2))
Raster_{y0} = Segment((y_0 * V_2) + (xmin * V_1), (y_0 * V_2) + (xmax * V_1))
P_{x0y0}: (x_0 * e_1) + (y_0 * e_2)
f_0: f(x_0, y_0)
P_0: (x_0 * e_1) + (y_0 * e_2) + (f(x_0, y_0) * e_3)
f_x = Derivative(f, x)
f_y = Derivative(f, y)
f_{xy} = Derivative(f_x, y)
s_0 = Segment(P_{x0y0}, P_0)
Punkte_{y0} = Sequence(((xmin + (k * Δ)) * e_1) + (y_0 * e_2) + (f(xmin + (k * Δ), y_0) * e_3), k, 0, n_x)
Punkte = Sequence(Sequence[((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2) + (f(xmin + (k * Δ), ymin + (l * Δ)) * e_3), k, 0, n_x], l, 0, n_y)
Punkte_{x0} = Sequence((x_0 * e_1) + ((ymin + (k * Δ)) * e_2) + (f(x_0, ymin + (k * Δ)) * e_3), k, 0, n_y)
Vieleck1, a_{xy}, b_{xy}, c_{xy}, d_{xy} = Polygon(A_{xy}, B_{xy}, C_{xy}, D_{xy})
Text6: "P_{x,y}=" + (LaTeX[P_{xy}]) + ""
Text13: "R_x(α) = \left(\begin{array} & 1& 0& 0\\  0& " + (LaTeX[cos(α)]) + "  & " + (LaTeX[(-sin(α))]) + " \\  0& " + (LaTeX[sin(α)]) + "  & " + (LaTeX[cos(α)]) + " \\\end{array}\right)"
Text15: "R_z(γ) = \left(\begin{array}  " + (LaTeX[cos(γ)]) + "  & " + (LaTeX[(-sin(γ))]) + "   &0 \\  " + (LaTeX[sin(γ)]) + "  & " + (LaTeX[cos(γ)]) + " & 0 \\0& 0&1\\ \end{array}\right)"
Text12: "PR_{xy}(α,β,γ) =P_{xy} \cdot R_x(α) \cdot R_y(β) \cdot R_z(γ= \left(\begin{array} & " + (LaTeX[(cos(β) * cos(γ))]) + " & " + (LaTeX[((-cos(β)) * sin(γ))]) + " & " + (LaTeX[(-sin(β))]) + " \\" + (LaTeX[(sin(α) * sin(γ)) + ((cos(α) * sin(β)) * cos(γ))]) + " & " + (LaTeX[(sin(α) * cos(γ)) - ((cos(α) * sin(β)) * sin(γ))]) + " & " + (LaTeX[(cos(α) * cos(β))]) + "  \\
 \end{array}\right)"
Text11: "R_{xyz}(α,β,γ) =R_x(α) \cdot R_y(β) \cdot R_z(γ) = \left(\begin{array} & " + (LaTeX[(cos(β) * cos(γ))]) + " & " + (LaTeX[((-cos(β)) * sin(γ))]) + " & " + (LaTeX[(-sin(β))]) + " \\ " + (LaTeX[(cos(α) * sin(γ)) - ((sin(α) * sin(β)) * cos(γ))]) + "    &  " + (LaTeX[(cos(α) * cos(γ)) + ((sin(α) * sin(β)) * sin(γ))]) + "  & " + (LaTeX[((-sin(α)) * cos(β))]) + "  \\ " + (LaTeX[(sin(α) * sin(γ)) + ((cos(α) * sin(β)) * cos(γ))]) + " & " + (LaTeX[(sin(α) * cos(γ)) - ((cos(α) * sin(β)) * sin(γ))]) + " & " + (LaTeX[(cos(α) * cos(β))]) + "  \\ \end{array}\right)"
Textraster = Sequence(Sequence[(T_x + ((j * t_x) * 2), T_y - (i * t_y)), j, 0, 3], i, 0, 14)
Text9: "R_{xyz}(α,β,γ) =R_x(α) \cdot R_y(β) \cdot R_z(γ) = \left(\begin{array} & cos(" + (LaTeX[β]) + ") cos(" + (LaTeX[γ]) + ") & - cos(" + (LaTeX[β]) + ") sin(" + (LaTeX[γ]) + ") &- sin(" + (LaTeX[β]) + ") \\                                     cos(" + (LaTeX[α]) + ")  sin(" + (LaTeX[γ]) + ") - sin(" + (LaTeX[α]) + ")  sin(" + (LaTeX[β]) + ") cos(" + (LaTeX[γ]) + ")    & cos(" + (LaTeX[α]) + ")  cos(" + (LaTeX[γ]) + ") + sin(" + (LaTeX[α]) + ") sin(" + (LaTeX[β]) + ") sin(" + (LaTeX[γ]) + ") & - sin(" + (LaTeX[α]) + ") cos(" + (LaTeX[β]) + ")  \\ sin(" + (LaTeX[α]) + ") sin(" + (LaTeX[γ]) + ") + cos(" + (LaTeX[α]) + ") sin(" + (LaTeX[β]) + ") cos(" + (LaTeX[γ]) + ") & sin(" + (LaTeX[α]) + ") cos(" + (LaTeX[γ]) + ") - cos(" + (LaTeX[α]) + ") sin(" + (LaTeX[β]) + ") sin(" + (LaTeX[γ]) + ") & cos(" + (LaTeX[α]) + ") cos(" + (LaTeX[β]) + ")   \\ \end{array}\right)"
Matrizen = false
m_x: f_x(x_0, y_0)
m_y: f_y(x_0, y_0)
d = 0.7
A_{Tangentialebene}: ((x_0 - d) * e_1) + (y_0 * e_2) + ((f(x_0, y_0) - (m_x * d)) * e_3)
B_{Tangentialebene}: ((x_0 + d) * e_1) + (y_0 * e_2) + ((f(x_0, y_0) + (m_x * d)) * e_3)
C_{Tangentialebene}: (x_0 * e_1) + ((y_0 - d) * e_2) + ((f(x_0, y_0) - (m_y * d)) * e_3)
D_{Tangentialebene}: (x_0 * e_1) + ((y_0 + d) * e_2) + ((f(x_0, y_0) + (m_y * d)) * e_3)
yTangente = Segment(A_{Tangentialebene}, B_{Tangentialebene})
xTangente = Segment(C_{Tangentialebene}, D_{Tangentialebene})
E_{Tangentialebene}: C_{Tangentialebene} + A_{Tangentialebene} - P_0
F_{Tangentialebene}: D_{Tangentialebene} + A_{Tangentialebene} - P_0
G_{Tangentialebene}: D_{Tangentialebene} - A_{Tangentialebene} + P_0
H_{Tangentialebene}: C_{Tangentialebene} - A_{Tangentialebene} + P_0
Tangentialebene, e_{Tangentialebene}, f_{Tangentialebene}, g_{Tangentialebene}, h_{Tangentialebene} = Polygon(E_{Tangentialebene}, F_{Tangentialebene}, G_{Tangentialebene}, H_{Tangentialebene})
xAbleitung = true
yAbleitung = true
P1: ((x_0 + m_x) * e_1) + ((y_0 + m_y) * e_2) + (f(x_0, y_0) * e_3)
P'_1: ((x_0 + m_x) * e_1) + ((y_0 + m_y) * e_2)
Gradient = Vector(P_{x0y0}, P'_1)
Gradient_1 = true
Raster = true
Gitterpunkte = false
Text19: "P_0= (" + x_0 + "," + y_0 + ", " + (f(x_0, y_0)) + " )"
Normalenrichtung: ((x_0 - f_x(x_0, y_0)) * e_1) + ((y_0, (-f_y(x_0, y_0))) * e_2) + e_3
Norm: sqrt(((-f_x(x_0, y_0)))^(2) + ((-f_y(x_0, y_0)))^(2) + 1)
N: Normalenrichtung / Norm + (f(x_0, y_0) * e_3)
Normale = Vector(P_0, N)
Normalenvektor = true
f_{ymin}(x) = 1 - (1 / 4 * x^(2)) - (1 / 9 * ymin^(2))
F_{ymin} = Integral(f_{ymin})
F_{xmin}(y) = F_x((-xmax), y)
F_{xmax}(y) = F_x(xmax, y)
Fläche(y) = F_{xmax}(y) - F_{xmin}(y)
Volumen = Integral(Fläche, ymin, ymax)
Grenzwände = true
Integrationssäulen = true
Integral_xRaster = Sequence(Sequence[PolyLine[(0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2), (0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + ((l + 1) * Δ)) * e_2), (0, 0) + ((xmin + ((k + 1) * Δ)) * e_1) + ((ymin + ((l + 1) * Δ)) * e_2), (0, 0) + ((xmin + ((k + 1) * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2), (0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2)], k, 0, n_x - 1], l, 0, 0)
Integral_yOrdinaten = Sequence(Sequence[Segment[(0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2), (0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2) + (f(xmin + (k * Δ), ymin + (l * Δ)) * e_3)], k, 0, 1], l, 0, n_y)
Integral_xOrdinaten = Sequence(Sequence[Segment[(0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2), (0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2) + (f(xmin + (k * Δ), ymin + (l * Δ)) * e_3)], k, 0, n_x], l, 0, 1)
Integral_yRaster = Sequence(Sequence[PolyLine[(0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2), (0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + ((l + 1) * Δ)) * e_2), (0, 0) + ((xmin + ((k + 1) * Δ)) * e_1) + ((ymin + ((l + 1) * Δ)) * e_2), (0, 0) + ((xmin + ((k + 1) * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2), (0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2)], k, 0, 0], l, 0, n_y - 1)
Integral_yKacheln = Sequence(Sequence[PolyLine[Element[Element[Punkte, i], j], Element[Element[Punkte, i], j + 1], Element[Element[Punkte, i + 1], j + 1], Element[Element[Punkte, i + 1], j]], i, 1, n_y], j, 1, 1)
Integral_xKacheln = Sequence(Sequence[PolyLine[Element[Element[Punkte, i], j], Element[Element[Punkte, i], j + 1], Element[Element[Punkte, i + 1], j + 1], Element[Element[Punkte, i + 1], j]], i, 1, 1], j, 1, n_x)
Gitter_y = Sequence(Sequence[Segment[Element[Element[Punkte, l], k], Element[Element[Punkte, l + 1], k]], k, 1, n_x + 1], l, 1, n_y)
Gitter_{x0} = Sequence(Segment[Element[Punkte_{x0}, k], Element[Punkte_{x0}, k + 1]], k, 1, n_x)
Gitter_{y0} = Sequence(Segment[Element[Punkte_{y0}, k], Element[Punkte_{y0}, k + 1]], k, 1, n_y)
Ordinaten_1 = Sequence(Sequence[Segment[(0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2), (0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2) + (f(xmin + (k * Δ), ymin + (l * Δ)) * e_3)], k, 0, n_x], l, 0, 0)
Ordinaten_2 = Sequence(Sequence[Segment[(0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2), (0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2) + (f(xmin + (k * Δ), ymin + (l * Δ)) * e_3)], k, 0, n_x], l, n_y, n_y)
Ordinaten_{mehr} = Sequence(Sequence[Segment[(0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2), (0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2) + (f(xmin + (k * Δ), ymin + (l * Δ)) * e_3)], k, 0, n_x], l, 0, 3)
xmax_{Raster}: round((xmax * 1.5))
ymax_{Raster}: round((ymax * 1.5))
xmin_{Raster}: (-xmax_{Raster})
ymin_{Raster}: (-ymax_{Raster})
n_xRaster: round((xmax_{Raster} - xmin_{Raster}) / Δ)
n_yRaster: round((ymax_{Raster} - ymin_{Raster}) / Δ)
Ordinaten_4 = Sequence(Sequence[Segment[(0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2), (0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2) + (f(xmin + (k * Δ), ymin + (l * Δ)) * e_3)], k, n_x, n_x], l, 0, n_y)
Ordinaten_3 = Sequence(Sequence[Segment[(0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2), (0, 0) + ((xmin + (k * Δ)) * e_1) + ((ymin + (l * Δ)) * e_2) + (f(xmin + (k * Δ), ymin + (l * Δ)) * e_3)], k, 0, 0], l, 0, n_y)
Animation_α = false
Animation_β = false
Animation_γ = false
speed_α = If(Animation_α, 1, 0)
speed_β = If(Animation_β, 1, 0)
speed_γ = If(Animation_γ, 1, 0)
Gitter_x = Sequence(Sequence[Segment[Element[Element[Punkte, l], k], Element[Element[Punkte, l], k + 1]], k, 1, n_x], l, 1, n_y + 1)
Normale_x: (-f_x(x_0, y_0))
Normale_y: (-f_y(x_0, y_0))
Normale_z = 1
Tangentialfläche = true
Raster_x = Sequence(Segment[(xmin_{Raster} * V_1) + ((ymin_{Raster} + (k * Δ)) * V_2), ((xmin_{Raster} + (n_xRaster * Δ)) * V_1) + ((ymin_{Raster} + (k * Δ)) * V_2)], k, 0, n_yRaster)
Raster_y = Sequence(Segment[(ymin_{Raster} * V_2) + ((xmin_{Raster} + (k * Δ)) * V_1), ((ymin_{Raster} + (n_yRaster * Δ)) * V_2) + ((xmin_{Raster} + (k * Δ)) * V_1)], k, 0, n_xRaster)
Text17: "\scalebox{1.1} {f_y=" + (LaTeX[f_y]) + "}"
Text16: "\scalebox{1.1} {f_x=" + (LaTeX[f_x]) + "}"
Text2: "\scalebox{1.1} {f(x,y) = " + (LaTeX[f]) + "}"
Text23: "\scalebox{1.0}{V= \int_{" + (LaTeX[ymin]) + " }^{" + (LaTeX[ymax]) + "} {\int_{" + (LaTeX[xmin]) + "}^{" + (LaTeX[xmax]) + "  }{(" + (LaTeX[f]) + ") \; dx}  \; dy}\;= \;" + (LaTeX[Volumen]) + "}"
Text26: "\left(  " + (LaTeX[TableText[{{"f_x", "f_y", "0"}}, "v"]]) + " \right) "
Text18: "\scalebox{1.1}{f_{xy}(x,y) = f_{yx}(x,y) =  " + (LaTeX[f_{xy}]) + "}\\
\scalebox{1.0} { (Satz  \;von \; Schwarz)} "
Text25: "\left(   " + (LaTeX[TableText[{{m_x, m_y, 0}}, "v"]]) + " \right)"
Text21: "Gradient = " + (LaTeX[Text25]) + "=" + (LaTeX[Text26]) + ""
Text27: "\left( " + (LaTeX[TableText[{{"-f_x", "-f_y", "1"}}, "v"]]) + "   \right)"
Text28: "\left( " + (LaTeX[TableText[{{Normale_x, Normale_y, 1}}, "v"]]) + "   \right)"
Text29: "\left( " + (LaTeX[TableText[{{1, 0, "f_x"}}, "v"]]) + "   \right)"
Text30: "\left( " + (LaTeX[TableText[{{0, 1, "f_y"}}, "v"]]) + "   \right)"
Text20: "\scalebox{0.95} {Normale =" + (LaTeX[Text29]) + " x " + (LaTeX[Text30]) + "= " + (LaTeX[Text27]) + "=" + (LaTeX[Text28]) + " }"

Tool: Raumpunkt
Input: a, b, c, Dummy
Output: A_1

Dummy: {{-0.6819983600624986, 0.7313537016191705, 0}, {0.49878202512991193, 0.46512176312793724, 0.7313537016191707}}
P: {{a}, {b}, {c}}
p1: (Dummy * P)
p_1: (Element[Element[p1, 1], 1], Element[Element[p1, 2], 1])
A_1: p_1