In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.
Every plane section of a paraboloid by a plane parallel to the axis of symmetry is a parabola. The paraboloid is hyperbolic if every other plane section is either a hyperbola, or two crossing lines (in the case of a section by a tangent plane). The paraboloid is elliptic if every other nonempty plane section is either an ellipse, or a single point (in the case of a section by a tangent plane). A paraboloid is either elliptic or hyperbolic.
Equivalently, a paraboloid may be defined as a quadric surface that is not a cylinder, and has an implicit equation whose part of degree two may be factored over the complex numbers into two different linear factors. The paraboloid is hyperbolic if the factors are real; elliptic if the factors are complex conjugate.
An elliptic paraboloid is shaped like an oval cup and has a maximum or minimum point when its axis is vertical. In a suitable coordinate system with three axes x, y, and z, it can be represented by the equation
z
=
x
2
a
2
+
y
2
b
2
.
{\displaystyle z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}.}
where a and b are constants that dictate the level of curvature in the xz and yz planes respectively. In this position, the elliptic paraboloid opens upward.
A hyperbolic paraboloid (not to be confused with a hyperboloid) is a doubly ruled surface shaped like a saddle. In a suitable coordinate system, a hyperbolic paraboloid can be represented by the equation
z
=
y
2
b
2
−
x
2
a
2
.
{\displaystyle z={\frac {y^{2}}{b^{2}}}-{\frac {x^{2}}{a^{2}}}.}
In this position, the hyperbolic paraboloid opens downward along the x-axis and upward along the y-axis (that is, the parabola in the plane x = 0 opens upward and the parabola in the plane y = 0 opens downward).
Any paraboloid (elliptic or hyperbolic) is a translation surface, as it can be generated by a moving parabola directed by a second parabola.
A question of sign. Is the curvature of Flamm's paraboloid positive or negative? If I've gotten the signs correct, it's a negative curvature. This is the opposite of the positive curvature of a sphere, and it implies that that geodesics drawn on Flamm's parabaloid should diverge. I think...
Good day
while solving some integral I met with the following equation
z=sqrt(2-x^-y^2) that looks like a paraboloid?!
I thought first that it might be a cone!
any insights?
thank you!
the equation of a parabola that is obtained by taking a cross-section passing through the center of the paraboloid is ##y = ax^2##
breaking the paraboloid into cylinders of height ##(dy)## the volume of each tiny cylinder is given by ##\pi x^2 dy##
since ##y = ax^2## we have ##\pi (y/a) dy##...
How to prove that every quadric surface can be translated and/or rotated so that its equation matches one of the six types of quadric surfaces namely 1) Ellipsoid
2)Hyperboloid of one sheet
3) Hyperboloid of two sheet 4)Elliptic Paraboloid
5) Elliptic Cone 6) Hyperbolic Paraboloid
The...
I am checking the divergence theorem for the vector field:
$$v = 9y\hat{i} + 9xy\hat{j} -6z\hat{k}$$
The region is inside the cylinder ##x^2 + y^2 = 4## and between ##z = 0## and ##z = x^2 + y^2##
This is my set up for the integral of the derivative (##\nabla \cdot v##) over the region...
I am trying to piece together how the parabolic mirror manages to reflect the "red dot" from the focal point to the eye without distortion.
I compare this with a conventional car headlight, which operates almost exactly the same way, except it has a non-transparent backing. Why does the ret dot...
Homework Statement
A bead slides under the influence of gravity on the frictionless interior surface of the paraboloid of revolution z = (x^2+y^2)/2a = r^2/2a Find the speed v_0 at which the bead will move in a horizontal circle of radius r_0 Find the frequency of small radial...
Hello. I solve this problem:
1. Homework Statement
The particles of mass m moves without friction on the inner wall of the axially symmetric vessel with the equation of the rotational paraboloid:
where b>0.
a) The particle moves along the circular trajectory at a height of z = z(0)...
I would like to know if there exist any equations in Cartesian coordinates that describe the shape in three dimensions of Flamm´s paraboloid and if you can write them to me because I have searched for them but I can’t find any specific equations of what I want. I suppose that this shape would...
Hi everyone!I'm trying to obtain the natural and dual basis of a circular paraboloid parametrized by:
$$x = \sqrt U cos(V)$$
$$y = \sqrt U sen(V)$$
$$z = U$$
with the inverse relationship:
$$V = \arctan \frac{y}{x}$$
$$U = z$$
The natural basis is:
$$e_U = \frac{\partial \overrightarrow{r}}...
Homework Statement
To generate a elliptical paraboloid antenna surface in MATLAB using the given F/D ratio (= 0.3)
F- focal length
D- Diameter = 50 m
Homework Equations
## \frac {F}{D} =\frac {1}{4tan(\theta/2)} ##
## F = \frac {D^2}{16H} ##
H = height of the paraboloid
Equation of a...
Homework Statement
Sketch the surface of a paraboloid z=9-x2 -92 in 3-dimensional xyz-space
Homework Equations
I assume partial derivatives are involved in some manner
The Attempt at a Solution
[/B]
I attempted to solve by making each variable equal to zero... That didn't work xD. I would...
Homework Statement
Let S be the portion of the paraboloid ##z = 4 - x^2 - y^2 ## that lies above the plane ##z = 0## and let ##\vec F = < z-y, x+z, -e^{ xyz }cos y >##. Use Stoke's Theorem to find the surface integral ##\iint_S (\nabla × \vec F) ⋅ \vec n \,dS##.
Homework Equations
##\iint_S...
The shape of the Einstein-Rosen bridge is often visualized/modelized with the Flamm's paraboloid, and many other references have also stated clearly that it's a "surface of revolution of a parabola".
But as far as I can see, when we rotate the parabola w^2 = 8M(r-2M) (in natural units c=G=1)...
Homework Statement
[/B]
The diameter and depth of an antenna that is
shaped like a paraboloid are ##d = 2.0m## and ##s = 0.5m## respectively. The antenna is set up so that its axis of symmetry is at an angle ##\theta = 30^{\circ}## from it's usual vertical orientation. How much can the antenna...
Homework Statement
So I am trying to accomplish the above by using spherical coordinates, I am aware the problem may be solved using dv=dxdydz= zdxdy were z is known but I would like to try it using a different approach (using spherical coordinates). Any help would be greatly appreciated...
Homework Statement
I am only currently in multivariate calculus, so i haven't even touched differential geometry yet, but a question that i had while learning about gradients came up and it led me to the topic of geodesics and differential geometry, so here goes:
Class problem: Find the...
Homework Statement
The angular velocity is ω, R is the radius of the vessel. at rest the water has depth H.
The face of the water form a paraboloid y=Ax2. find R for which the maximum height h of the water above the bottom doesn't depend on ω.
Homework Equations
Centripetal force: ##F=m...
Evaluate the triple integral ∫∫∫E 5x dV, where E is bounded by the paraboloid
x = 5y2+ 5z2 and the plane x = 5.
My work so far:
Since it's a paraboloid, where each cross section parallel to the plane x = 5 is a circle, cylindrical polars is what I used, so my bounds are 5y2+5z2 ≤x ≤ 5 ----->...
Homework Statement
[/B]
Find the coordinates of the point P on the surface of the paraboloid z=6x2+6y2-(35/6) where the normal line to the surface passes through the point (25/6, (25√22)/6, -4). Note that a graphing calculator may be used to solve the resulting cubic equation.
Homework...
Homework Statement
Let F = <x, z, xz> evaluate ∫∫F⋅dS for the following region:
x2+y2≤z≤1 and x≥0
Homework Equations
Gauss Theorem
∫∫∫(∇⋅F)dV = ∫∫F⋅dS
The Attempt at a Solution
This is the graph of the entire function:
Thank you Wolfram Alpha.
But my surface is just the half of this...
Homework Statement
A particle is sliding inside a frictionless paraboloid defined by r^2 = az with no gravity. We must show that the force of constraint is proportional to (1+4r^2/a^2)^{-3/2}
Homework Equations
f(r,z) = r^2-az = 0
F_r = \lambda \frac{\partial f}{\partial r} (and similarly for...
Homework Statement
Evaluate s∫∫ lxyzl dS, where S is part of the surface of paraboloid
z = x2 + y2, lying below the plane z = 1Homework Equations
The Attempt at a Solution
since z=1 and x2+y2=z,
therefore integral becomes 0∫^1 0∫^(1-x2) xyz dy dx
which solves to 1/8. Apprently this is...
I will state the specifics to this problem if necessary.
I need to find the parametric equations for the the tan line at point, P(x1,y1,z1) on the curve formed from paraboloid intersection with ellipsoid.
The parametric equations for the level surfaces that make up paraboloid and ellipsoid...
Need a 2nd opinion on my solution.
Homework Statement
A point mass moves frictionlessly in a circle inside a parabolic cup, with the radius at the top being R. The particle's position vector makes an angle theta wrt the center of symmetry (generatrix going from -z to +z).Homework Equations...
Homework Statement
Evaluate the surface integral:
∫∫s y dS
S is the part of the paraboloid y= x2 + z2 that lies inside the cylinder x2 + z2 =4.Homework Equations
∫∫sf(x,y,z)dS = ∫∫Df(r(u,v))*|ru x rv|dAThe Attempt at a Solution
I've drawn the region D in the xz-plane as a circle with...
Homework Statement
A surface S in three dimensional space may be speciﬁed by the equation
f(x, y, z) = 0, where f(x, y, z) is a real function. Show that a unit vector nˆ normal to
the surface at point (x0, y0, z0) is given by
Homework Equations
The Attempt at a Solution
r...
Hi Folks,I have come across some text where f(x,y)=c_1+c_2x+c_3y+c_4xy is used to define the corner pointsf_1=f(0,0)=c_1
f_2=f(a,0)=c_1+c_2a
f_3=f(a,b)=c_1+c_2a+c_3b+c_4ab
f_4=f(0,b)=c_1+c_3bHow are these equations determined? F_1 to F_4 starts at bottom left hand corner and rotates counter...
Homework Statement
Find the distance from the paraboloid z = X2 + 2Y2 to the plane
2X + 8Y + Z = -8.
Homework Equations
The partial derivatives with respect to X, And Y for the paraboloid.
The Attempt at a Solution
My professor said we need to find the point where the...
If the hyperbolic paraboloid z=(x/a)^2 - (y/b)^2
is rotated by an angle of π/4 in the +z direction (according to the right hand rule), the result is the surface
z=(1/2)(x^2 + y^2) ((1/a^2)-((1/b^2)) + xy((1/a^2)-((1/b^2))
and if a= b then this simplifies to
z=2/(a^2) (xy)
suppose...
Homework Statement
Use stokes theorem to elaluate to integral \int\int_{s} curlF.dS where F(x,y,z)= x^2 z^2 i + y^2 z^2 j + xyz k and s is the part of the paraboliod z=x^2+ y^2 that lies inside the cylinder x^2 +y^2 =4 and is orientated upwards
Homework Equations
The Attempt at a...
Homework Statement
a elliptic paraboloid is x^2/a^2+y^2/b^2<=(h-z)/h, 0<=z<=h. Its apex occurs at the point (0,0,h). Suppose a>=b. Calculate the volume of that part of the paraboloid that lies above the disc x^2+y^2<=b^2.:confused:
2. The attempt at a solution
We normally do the...
Homework Statement
a elliptic paraboloid is x^2/a^2+y^2/b^2<=(h-z)/h, 0<=z<=h, show that the horizontal cross-section at height z, is an ellipse
Homework Equations
The Attempt at a Solution
i don't know how to prove this, i only know that the standard ellipse is...
Homework Statement
Verify Stokes' Theorem for F(x,y,z)=(3y,4z,-6x) where S is part of the paraboloid z=9-x2-y2 that lies above the xy-plane, oriented upward.
Homework Equations
Stokes' Theorem is ∫F*ds=∫∫scurl(F)*ds
Where curl(F)=∇*F
The Attempt at a Solution
I got...
Homework Statement
Find the equation of the surface that is equidistant from the plane x=1, and the point (-1,0,0).
The Attempt at a Solution
Okay, if I set the distance from the surface to the point, and the distance from the surface to the plane as being equal, I should have the...
Homework Statement
Find the moment of inertia of a paraboloid f(x,y)=x^2+y^2 whose density function is ρ(r)=cr=dm/dv. use mass M and height H to express your answer
2. The attempt at a solution
I took the double integral ∫∫r^2 ρ r dr dθ to find the I of a single disk as a function of r...
Homework Statement
Find the outward flux of F = <x + z, y + z, xy> through the surface of the paraboloid z = x^2 + y^2, 0 ≤ z ≤ 4, including its top disk.
Homework Equations
double integral (-P(∂f/∂x) - Q(∂f/∂y) + R)dA
where the vector F(x,y) = <P, Q, R>
and where z = f(x,y) <-- f(x,y) is the...
Homework Statement
Find the volume of the solid E bounded by z = 3+x2 +y2 and z = 6.
Homework Equations
The Attempt at a Solution
I'm going to use cylindrical coordinates. So, I have,
z = 3 + r2
Clearly, my bounds on z are 3 and 6. If I project the intersection of the...
Homework Statement
Where does the normal line to the paraboloid z=x^2+y^2 at the point (1,1,2) intersect the paraboloid a second time?
Homework Equations
The Attempt at a Solution
I found the normal line to be 0=2x+2y-1, but I'm not sure what to do next.
Hi there, I have to compute the surface area for
V:\{ -2(x+y)\leq{}z\leq{}4-x^2-y^2 \}
I have a problem on finding the surface area for the paraboloid limited by the plane. I've parametrized the plane in polar coordinates, I thought it would be easier this way, but also tried in cartesian...
Homework Statement
Find the volume bounded by the paraboloid z= 2x2+y2 and the cylinder z=4-y2. Diagram is included that shows the shapes overlaying one another, with coordinates at intersections. (Will be given if necessary)
Homework Equations
double integral? function1-function2...
The Problem: I have a paraboloid open along the positive z-axis, starting at the origin and ending at z = 100. At z=100, the horizontal surface is a circle with a radius of 20. Water is flowing through the paraboloid with the velocity F = 2xzi - (1100 + xe^-x^2)j + z(1100 - z)k. I'm asked to...
I just want to see if my logic is sound here. If we have the paraboloid z=x2+y2 from z=0 to z=1, and I wanted a parametric form of that I think this should work for polar coordinates:
\vec{r}(u,v)=(vcosu,vsinu,v^{2})
u:[0..2\pi],v:[0..1]
Does this make sense?
Homework Statement
Hi, I am trying to solve the following problem, and seem to just be going in circles.
A sphere of radius=4 is "dropped" into a paraboloid with equation z=(x^2)+(y^2).
Find the distance "a" from the origin to the center of the sphere at the point where it will
"get stuck" or...
Homework Statement
The volume of the solid below the plane: z=2x and above the paraboloid z=x^2 + y^2.
I need help setting this one up, I can handle the evaluating.
The Attempt at a Solution
I just don't know.
Homework Statement
Evaluate the triple integral xdV where E is the solid bounded by the paraboloid x= 2y^2 + 2z^2 and x=2.
The Attempt at a Solution
The bounds I got are
for z
-sqrt(1-y^2) <= y <= sqrt(1-y^2)
for y
-1 <= y <= 1
for x
2y^2 + 2z^2 <= x <= 2
are these...
Homework Statement
Find an equation of the form Ax2+By2+Cz2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0 Satisfied by the set of all points in space, (x,y,z), whose distance to the origin is equal to their distance to the plane x+y+z=3. Based on what you know about parabolas, what does this collection of points...