Mister Exam
Other calculators
- Canonical form of a elliptical paraboloid
- Canonical form of a double hyperboloid
- Canonical form of a imaginary ellipsoid
- Canonical form of a degenerate ellipse
- Canonical form of a parabola
-
How to use it?
- Canonical form:
- -2x^2+12xy+7y^2-8sqrt(5)x-6sqrt(5)y-5=0
- 6x^2+4xy+6y^2+12x-4y-23=0
- y^2-z^2+2*x+(z-y)^2=0
- -z=2x^2+18y^2
- Plot:
-
2*x^3-3*x^2*y+4*y^3=1
- x*y=28
- 10*x+5*y+15=0
- (y-x)*x^4+8=0
- Integral of d{x}:
- -z
- Derivative of:
-
-z
-
Identical expressions
- -z= two x^ two +18y^2
- minus z equally 2x squared plus 18y squared
- minus z equally two x to the power of two plus 18y squared
- -z=2x2+18y2
- -z=2x²+18y²
- -z=2x to the power of 2+18y to the power of 2
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Similar expressions
- z=2x^2+18y^2
- -z=2x^2-18y^2
-z=2x^2+18y^2 canonical form
The teacher will be very surprised to see your correct solution 😉
The solution
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2 2 -z - 18*y - 2*x = 0
$$- 2 x^{2} - 18 y^{2} - z = 0$$
-2*x^2 - 18*y^2 - z = 0
Invariants method
Given equation of the surface of 2-order:
$$- 2 x^{2} - 18 y^{2} - z = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y z + 2 a_{24} y + a_{33} z^{2} + 2 a_{34} z + a_{44} = 0$$
where
$$a_{11} = -2$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{14} = 0$$
$$a_{22} = -18$$
$$a_{23} = 0$$
$$a_{24} = 0$$
$$a_{33} = 0$$
$$a_{34} = - \frac{1}{2}$$
$$a_{44} = 0$$
The invariants of the equation when converting coordinates are determinants:
$$I_{1} = a_{11} + a_{22} + a_{33}$$
$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\a_{12} & a_{22} & a_{23} & a_{24}\\a_{13} & a_{23} & a_{33} & a_{34}\\a_{14} & a_{24} & a_{34} & a_{44}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12} & a_{13}\\a_{12} & a_{22} - \lambda & a_{23}\\a_{13} & a_{23} & a_{33} - \lambda\end{matrix}\right|$$
substitute coefficients
$$I_{1} = -20$$
$$I_{3} = \left|\begin{matrix}-2 & 0 & 0\\0 & -18 & 0\\0 & 0 & 0\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}-2 & 0 & 0 & 0\\0 & -18 & 0 & 0\\0 & 0 & 0 & - \frac{1}{2}\\0 & 0 & - \frac{1}{2} & 0\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}- \lambda - 2 & 0 & 0\\0 & - \lambda - 18 & 0\\0 & 0 & - \lambda\end{matrix}\right|$$
$$I_{1} = -20$$
$$I_{2} = 36$$
$$I_{3} = 0$$
$$I_{4} = -9$$
$$I{\left(\lambda \right)} = - \lambda^{3} - 20 \lambda^{2} - 36 \lambda$$
$$K_{2} = - \frac{1}{4}$$
$$K_{3} = 5$$
Because
$$I_{3} = 0 \wedge I_{2} \neq 0 \wedge I_{4} \neq 0$$
then by type of surface:
you need to
Make the characteristic equation for the surface:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
or
$$\lambda^{3} + 20 \lambda^{2} + 36 \lambda = 0$$
$$\lambda_{1} = -2$$
$$\lambda_{2} = -18$$
$$\lambda_{3} = 0$$
then the canonical form of the equation will be
$$\tilde z 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
and
$$- \tilde z 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
$$- 2 \tilde x^{2} - 18 \tilde y^{2} + \tilde z = 0$$
and
$$- 2 \tilde x^{2} - 18 \tilde y^{2} - \tilde z = 0$$
and
this equation is fora type elliptical paraboloid
- reduced to canonical form
$$- 2 x^{2} - 18 y^{2} - z = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y z + 2 a_{24} y + a_{33} z^{2} + 2 a_{34} z + a_{44} = 0$$
where
$$a_{11} = -2$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{14} = 0$$
$$a_{22} = -18$$
$$a_{23} = 0$$
$$a_{24} = 0$$
$$a_{33} = 0$$
$$a_{34} = - \frac{1}{2}$$
$$a_{44} = 0$$
The invariants of the equation when converting coordinates are determinants:
$$I_{1} = a_{11} + a_{22} + a_{33}$$
|a11 a12| |a22 a23| |a11 a13|
I2 = | | + | | + | |
|a12 a22| |a23 a33| |a13 a33|$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\a_{12} & a_{22} & a_{23} & a_{24}\\a_{13} & a_{23} & a_{33} & a_{34}\\a_{14} & a_{24} & a_{34} & a_{44}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12} & a_{13}\\a_{12} & a_{22} - \lambda & a_{23}\\a_{13} & a_{23} & a_{33} - \lambda\end{matrix}\right|$$
|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|substitute coefficients
$$I_{1} = -20$$
|-2 0 | |-18 0| |-2 0|
I2 = | | + | | + | |
|0 -18| | 0 0| |0 0|$$I_{3} = \left|\begin{matrix}-2 & 0 & 0\\0 & -18 & 0\\0 & 0 & 0\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}-2 & 0 & 0 & 0\\0 & -18 & 0 & 0\\0 & 0 & 0 & - \frac{1}{2}\\0 & 0 & - \frac{1}{2} & 0\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}- \lambda - 2 & 0 & 0\\0 & - \lambda - 18 & 0\\0 & 0 & - \lambda\end{matrix}\right|$$
|-2 0| |-18 0| | 0 -1/2|
K2 = | | + | | + | |
|0 0| | 0 0| |-1/2 0 | |-2 0 0| |-18 0 0 | |-2 0 0 |
| | | | | |
K3 = |0 -18 0| + | 0 0 -1/2| + |0 0 -1/2|
| | | | | |
|0 0 0| | 0 -1/2 0 | |0 -1/2 0 |$$I_{1} = -20$$
$$I_{2} = 36$$
$$I_{3} = 0$$
$$I_{4} = -9$$
$$I{\left(\lambda \right)} = - \lambda^{3} - 20 \lambda^{2} - 36 \lambda$$
$$K_{2} = - \frac{1}{4}$$
$$K_{3} = 5$$
Because
$$I_{3} = 0 \wedge I_{2} \neq 0 \wedge I_{4} \neq 0$$
then by type of surface:
you need to
Make the characteristic equation for the surface:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
or
$$\lambda^{3} + 20 \lambda^{2} + 36 \lambda = 0$$
$$\lambda_{1} = -2$$
$$\lambda_{2} = -18$$
$$\lambda_{3} = 0$$
then the canonical form of the equation will be
$$\tilde z 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
and
$$- \tilde z 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
$$- 2 \tilde x^{2} - 18 \tilde y^{2} + \tilde z = 0$$
and
$$- 2 \tilde x^{2} - 18 \tilde y^{2} - \tilde z = 0$$
2 2 \tilde x \tilde y --------- + --------- - 2*\tilde z = 0 1/4 1/36
and
2 2 \tilde x \tilde y --------- + --------- + 2*\tilde z = 0 1/4 1/36
this equation is fora type elliptical paraboloid
- reduced to canonical form