What is Paraboloid: Definition and 83 Discussions

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.

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1. A Sign of Curvature in Flamm's Paraboloid: Negative?

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...
2. Solving an Equation: Is it a Paraboloid or Cone?

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!
3. B Volume of paraboloid in a cylinder

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##...
4. MHB How is z=2xy a Hyperbolic Paraboloid in the rotated 45° in the xy-plane?

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...
5. Checking divergence theorem inside a cylinder and under a paraboloid

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...
6. Collimated light from red dot scopes?

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...
7. What is the Speed and Frequency of a Bead Sliding Inside a Paraboloid?

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...
8. Lagrangian equations of particle in rotational paraboloid

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)...
9. V

I Can Flamm's Paraboloid be described by Cartesian equations?

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...
10. Natural basis and dual basis of a circular paraboloid

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}}...
11. Elliptical paraboloid surface in matlab using F/D = 0.3

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...
12. Sketch the surface of a paraboloid

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...
13. Find the Surface integral of a Paraboloid using Stoke's Theorem

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...
14. I Is Flamm's paraboloid a paraboloid?

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)...
15. The volume of water gathered by a tilted paraboloid antenna

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...
16. Volume of a solid bounded by a paraboloid and the x-y plane

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...
17. Shortest path between two points on a paraboloid

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...
18. Can with water rotates -- the water forms a paraboloid

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...
19. Is There a Mistake in the Hyperbolic Paraboloid Curve Demonstration?

then look at : the 2 curves are nearly the same while the equations are not, is there anything wrong ?
20. Evaluate the triple integral paraboloid

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 ----->...
21. Finding point on paraboloid surface given normal line point

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...
22. Integrating F over a Paraboloid Region

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...
23. Force of Constraint for Particle in a Paraboloid

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...
24. Evaluate integral for surface of a paraboloid

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...
25. The curve formed by the intersection of paraboloid and ellipsoid

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...
26. Point mass in a (non-hyperbolic) paraboloid.

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...
27. Surface Integral involving Paraboloid

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...
28. Volume of paraboloid using divergence theorem (gives zero)

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...
29. MHB The equation of a hyperbolic paraboloid to derive the corner points of rectangle

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...
30. Finding The Distance From A Paraboloid To A Plane.

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...
31. Hyperbolic Paraboloid and Isometry

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...
32. Stokes Theorem paraboloid intersecting with cylinder

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...
33. Find volume of solid elliptic paraboloid using polar coordinates

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...
34. Prove cross-section of elliptic paraboloid is a ellipse

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...
35. Verify Stokes' Theorem for F across a paraboloid

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...
36. Equation of a circular paraboloid

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...
37. Moment of Inertia of a non-uniform density paraboloid

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...
38. Flux of a Paraboloid without Parametrization

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...
39. Volume Between A Circular Paraboloid and a Plane

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...
40. Intersection of paraboloid and normal line

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.
41. Parametric representation of paraboloid cylinder

The equation is z = y^3. I know how to do normal planes and spheres, but I don't know what to set for r(u,v) when it comes to paraboloid cylinders.
42. Surface Area of Paraboloid Limited by Plane

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...
43. Finding volume bounded by paraboloid and cylinder

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...
44. Flux through a paraboloid? The Divergence Theorem and Integration Error

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...
45. Can a paraboloid become cone under limiting conditions?

What will be the limiting conditions?
46. Parametric Paraboloid In Polar Coordinates

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?
47. Sphere Intersecting a Paraboloid

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...
48. Finding the Volume of a Solid Below a Plane and Above a Paraboloid

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.
49. Optimizing Paraboloid Bounds for Triple Integral

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...
50. Finding the equation of a paraboloid

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...