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Forma canónica/ x^2+y^2+z^2-x*y+x*z+y*z+3*x+3*y-3*z

x^2+y^2+z^2-x*y+x*z+y*z+3*x+3*y-3*z forma canónica

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Solución

Ha introducido [src] 
 2    2    2                                        
x  + y  + z  - 3*z + 3*x + 3*y + x*z + y*z - x*y = 0
x2−xy+xz+3x+y2+yz+3y+z2−3z=0x^{2} - x y + x z + 3 x + y^{2} + y z + 3 y + z^{2} - 3 z = 0 
x^2 - x*y + x*z + 3*x + y^2 + y*z + 3*y + z^2 - 3*z = 0
Método de invariantes
Se da la ecuación de superficie de 2 grado:
x2−xy+xz+3x+y2+yz+3y+z2−3z=0x^{2} - x y + x z + 3 x + y^{2} + y z + 3 y + z^{2} - 3 z = 0
Esta ecuación tiene la forma:
a11x2+2a12xy+2a13xz+2a14x+a22y2+2a23yz+2a24y+a33z2+2a34z+a44=0a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y z + 2 a_{24} y + a_{33} z^{2} + 2 a_{34} z + a_{44} = 0
donde
a11=1a_{11} = 1
a12=−12a_{12} = - \frac{1}{2}
a13=12a_{13} = \frac{1}{2}
a14=32a_{14} = \frac{3}{2}
a22=1a_{22} = 1
a23=12a_{23} = \frac{1}{2}
a24=32a_{24} = \frac{3}{2}
a33=1a_{33} = 1
a34=−32a_{34} = - \frac{3}{2}
a44=0a_{44} = 0
Las invariantes de esta ecuación al transformar las coordenadas son los determinantes:
I1=a11+a22+a33I_{1} = a_{11} + a_{22} + a_{33}
     |a11  a12|   |a22  a23|   |a11  a13|
I2 = |        | + |        | + |        |
     |a12  a22|   |a23  a33|   |a13  a33|

I3=∣a11a12a13a12a22a23a13a23a33∣I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|
I4=∣a11a12a13a14a12a22a23a24a13a23a33a34a14a24a34a44∣I_{4} = \left|\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\a_{12} & a_{22} & a_{23} & a_{24}\\a_{13} & a_{23} & a_{33} & a_{34}\\a_{14} & a_{24} & a_{34} & a_{44}\end{matrix}\right|
I(λ)=∣a11−λa12a13a12a22−λa23a13a23a33−λ∣I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12} & a_{13}\\a_{12} & a_{22} - \lambda & a_{23}\\a_{13} & a_{23} & a_{33} - \lambda\end{matrix}\right|
     |a11  a14|   |a22  a24|   |a33  a34|
K2 = |        | + |        | + |        |
     |a14  a44|   |a24  a44|   |a34  a44|

     |a11  a12  a14|   |a22  a23  a24|   |a11  a13  a14|
     |             |   |             |   |             |
K3 = |a12  a22  a24| + |a23  a33  a34| + |a13  a33  a34|
     |             |   |             |   |             |
     |a14  a24  a44|   |a24  a34  a44|   |a14  a34  a44|

sustituimos coeficientes
I1=3I_{1} = 3
     | 1    -1/2|   | 1   1/2|   | 1   1/2|
I2 = |          | + |        | + |        |
     |-1/2   1  |   |1/2   1 |   |1/2   1 |

I3=∣1−1212−1211212121∣I_{3} = \left|\begin{matrix}1 & - \frac{1}{2} & \frac{1}{2}\\- \frac{1}{2} & 1 & \frac{1}{2}\\\frac{1}{2} & \frac{1}{2} & 1\end{matrix}\right|
I4=∣1−121232−121123212121−323232−320∣I_{4} = \left|\begin{matrix}1 & - \frac{1}{2} & \frac{1}{2} & \frac{3}{2}\\- \frac{1}{2} & 1 & \frac{1}{2} & \frac{3}{2}\\\frac{1}{2} & \frac{1}{2} & 1 & - \frac{3}{2}\\\frac{3}{2} & \frac{3}{2} & - \frac{3}{2} & 0\end{matrix}\right|
I(λ)=∣1−λ−1212−121−λ1212121−λ∣I{\left(\lambda \right)} = \left|\begin{matrix}1 - \lambda & - \frac{1}{2} & \frac{1}{2}\\- \frac{1}{2} & 1 - \lambda & \frac{1}{2}\\\frac{1}{2} & \frac{1}{2} & 1 - \lambda\end{matrix}\right|
     | 1   3/2|   | 1   3/2|   | 1    -3/2|
K2 = |        | + |        | + |          |
     |3/2   0 |   |3/2   0 |   |-3/2   0  |

     | 1    -1/2  3/2|   | 1   1/2   3/2 |   | 1   1/2   3/2 |
     |               |   |               |   |               |
K3 = |-1/2   1    3/2| + |1/2   1    -3/2| + |1/2   1    -3/2|
     |               |   |               |   |               |
     |3/2   3/2    0 |   |3/2  -3/2   0  |   |3/2  -3/2   0  |

I1=3I_{1} = 3
I2=94I_{2} = \frac{9}{4}
I3=0I_{3} = 0
I4=−24316I_{4} = - \frac{243}{16}
I(λ)=−λ3+3λ2−9λ4I{\left(\lambda \right)} = - \lambda^{3} + 3 \lambda^{2} - \frac{9 \lambda}{4}
K2=−274K_{2} = - \frac{27}{4}
K3=−814K_{3} = - \frac{81}{4}
Como
I3=0∧I2≠0∧I4≠0I_{3} = 0 \wedge I_{2} \neq 0 \wedge I_{4} \neq 0
entonces por razón de tipos de rectas:
hay que
Formulamos la ecuación característica para nuestra superficie:
−I1λ2+I2λ−I3+λ3=0- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0
o
λ3−3λ2+9λ4=0\lambda^{3} - 3 \lambda^{2} + \frac{9 \lambda}{4} = 0
λ1=32\lambda_{1} = \frac{3}{2}
λ2=32\lambda_{2} = \frac{3}{2}
λ3=0\lambda_{3} = 0
entonces la forma canónica de la ecuación será
z~2(−1)I4I2+(x~2λ1+y~2λ2)=0\tilde z 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0
y
−z~2(−1)I4I2+(x~2λ1+y~2λ2)=0- \tilde z 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0
3x~22+3y~22+33z~=0\frac{3 \tilde x^{2}}{2} + \frac{3 \tilde y^{2}}{2} + 3 \sqrt{3} \tilde z = 0
y
3x~22+3y~22−33z~=0\frac{3 \tilde x^{2}}{2} + \frac{3 \tilde y^{2}}{2} - 3 \sqrt{3} \tilde z = 0
2z~+(x~23+y~23)=02 \tilde z + \left(\frac{\tilde x^{2}}{\sqrt{3}} + \frac{\tilde y^{2}}{\sqrt{3}}\right) = 0
y
−2z~+(x~23+y~23)=0- 2 \tilde z + \left(\frac{\tilde x^{2}}{\sqrt{3}} + \frac{\tilde y^{2}}{\sqrt{3}}\right) = 0
es la ecuación para el tipo paraboloide elíptico
- está reducida a la forma canónica
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