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paraboloid

paraboloid, n. (a.) Geom. (pəˈræbəlɔɪd) Also 7 -oeides, -oeid, 8–9 -oide. [In form, ad. Gr. παραβολοειδής a. (in a different sense), whence in 17th c. use paraboloeides: see parabola and -oid, and cf. F. paraboloïde.] † 1. A parabola of a higher degree: = parabola b.1656 Hobbes Six Lessons Wks. 1845... Oxford English Dictionary

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Paraboloid

A paraboloid is either elliptic or hyperbolic. paraboloid is a circular paraboloid or paraboloid of revolution. wikipedia.org

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A paraboloid is a surface. $$z=x^2+y^2$$ gives paraboloid. Let $\sigma$ be a surface patch for a paraboloid defined by $$\sigma (u,v)= (u,v, u^2+v^2)$$ I want to show that this is a surface. To show that $\sigma...

You are right, proving that $\sigma$ is continuous with continuous inverse is sufficient. Also $\\{ \sigma \\}$ gives an atlas consisting of only one chart.

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Surface area of an elliptic paraboloid The elliptic paraboloid has height h, and two semiaxes a, b. How to find its surface area? Does it possible to use a direct formula without integrals?

The elliptic paraboloid is represented parametrically as follows: $$x=a \sqrt{u} \cos{v}$$ $$y=b \sqrt{u} \sin{v}$$ $$z=u$$ The surface area of this

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Volume of a cut paraboloid I have a cut paraboloid made from parabola $y(x)=c+x-ax^2$ and $x = 0$ line. How do I compute volume of this cut paraboloid? I researched on Wolfram. see formula 16 and 17

It may help to take a step back and look at the basic definitions. The area and centroid are given by $$ A=\int\\!\\!\\!\int dy~dx=\int y(x)~dx\\\ R_x=\frac{\int\\!\\!\\!\int x~dy~dx}{\int\\!\\!\\!\int dy~dx}=\frac{1}{A}\int x~y(x)~dx\\\ R_y=\frac{\int\\!\\!\\!\int y~dy~dx}{\int\\!\\!\\!\int dy~dx}=...

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Spherical coordinates and unregular paraboloid If we have the following equation of a paraboloid: $z=4-x^2-y^2$ and we have the region in space bounded by this paraboloid from above and by the $xy$-plane from below; w...

From: $$x^2+y^2=4-z$$ we obtain: $$\rho^2 \sin^2\phi=4-\rho \cos \phi\implies\rho^2 \sin^2\phi+\rho \cos \phi-4=0$$ and thus $$\rho=\frac{-cos \phi\pm \sqrt{\cos^2 \phi+16\sin^2 \phi}}{2\sin^2 \phi}\implies \rho=\frac{-cos \phi+ \sqrt{\cos^2 \phi+16\sin^2 \phi}}{2\sin^2 \phi}>0$$ > $$ \iiint \; dV =...

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Paramaterization of paraboloid and plane. Consider the paraboloid $z=x^2+y^2$. The plane $2x-4y+z-6=0$ cuts the paraboloid, its intersection being a curve. Find "the natural" parameterization of this curve. I have se...

Plugging $z=x^2+y^2$ into $2x-4y+z-6=0$ we get \begin{align*} 2x-4y+x^2+y^2-6&=0\\\ x^2+2x+\color{red}{1}+y^2-4y+\color{red}{4}&=6+\color{red}{1+4}\\\ (x+1)^2+(y-2)^2&=11 \end{align*} Then, a parameterization for the curve is $$\begin{cases}x&=\sqrt{11}\cos t-1 \\\ y&=\sqrt{11}\sin t+2\\\z&=2\sqrt{1...

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拋物面 Paraboloid: 最新的百科全書、新聞、評論和研究

類似地，拋物面可以定義為具有隱式方程的非圓柱二次曲面，其二次部分可以將復數分解為兩個不同的線性因子。如果因子是真實的，拋物線就變成雙曲面。如果因子是複共軛，則為橢圓。橢圓拋物線的形狀像橢圓杯，當軸線垂直時具有最大值或最小值。

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surface area of the part of the circular paraboloid Find the surface area of the part of the circular paraboloid!enter image description here that lies inside the cylinder !enter image description here

You do this exactly the same way as your other surface area problem. Recall the surface area formula: !integral $$f_y=2y, f_x=2x.$$ So we have $$S=\int_{-5}^5\int_{-\sqrt{25-x^2}}^{\sqrt{25-x^2}}\sqrt{(2x)^2+(2y)^2+1}dydx$$ Can you take it from there? Alternatively, you could do this using polar coo...

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Area of a Paraboloid inside a Cylinder Find the area of the part of the paraboloid $x=y^2+z^2$ that is inside the cylinder $y^2+z^2=9$. I'm not sure how to set up the integral to compute this. Thanks.

The paraboloid and cylinder intersect at $x=9$, so the height of the paraboloid is $h=9$ then find the surface area by integrating: $$ A=\int\int \sqrt

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Why this is a Elliptic Paraboloid?? Let $x^2+2z^2-6x-y+10=0$ and we know that $z/c=x^2/a^2+y^2/b^2$ shows a Elliptic Paraboloid. if we compelete the square we gonna have $(y-1)=(x-3)^2+2z^2.$ Book says we can see that...

It'a an elliptic paraboloid pointing in the $y$-direction. The standard formula is pointing in the $z$-direction.

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The volume of a classic paraboloid at $h$ So, I have an object $z=x^2+y^2$ filled with water at $h=50$ where $h=0$ is at the deepest spot of the paraboloid. With this information I am to find out the volume of the w...

Hint: The paraboloid as the $z$ axis as axis of symmetry, so you can find the volume as the solid of revolution of the parabola $z=x^2$ around the axis

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Parallel surface to paraboloid I have the equation for a paraboloid $z=x^2+y^2$ I need to obtain a parallel surface, Does anyone know what is the procedure to determine it?

Just add a normal vector of desired length. The equation of the surface is $$g\colon (x,y)\mapsto \bigl(x,y,f(x,y)\bigr),$$ the normal vector is the cross product of the derivatives in respect to $x$ and $y$, namely $$n=\begin{pmatrix}1\\\0\\\f_x\end{pmatrix} \times\begin{pmatrix}0\\\1\\\f_y\end{pma...

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I have to find the surface area of a paraboloid within a cylinder. I have to find the surface area of a paraboloid within a cylinder. The paraboloid is $x = y^2 + z^2$ and the cylinder is $y^2 + z^2 = 4$, and I know ...

Let $g(y,z)= y^2 + z^2.$ As I understand your question, you are to find the surface area for the cylindrical region $y^2+z^2\leq 4.$ Then $$ A = \int\int_{y^2+z^2\leq 4} \sqrt{1 + (g_y)^2 + (g_z)^2} \ dy \, dz. $$ Then convert to polar coordinates: $$ A = \int^{2\pi}_0 \int^2_0 \sqrt{1 + 4r^2} \ r \...

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Showing the parametrically representation of hyperbolic paraboloid. And how to find the curves $u$ and $v$ be constant. Show that the hyperbolic paraboloid can be represented parametrically as $$r(u,v)=\langle a(u+v),...

You've written $$r(u,v)=\langle a(u+v), b(u-v), uv\rangle$$ and that's the same thing as saying $\langle x,y,z\rangle = \langle a(u+v), b(u-v), uv\rangle$, so that $$ \begin{align} x & = a(u+v), \\\ y & = b(u-v), \\\ z & = uv. \end{align} $$ So $$ \frac{y^2}{b^2} - \frac{x^2}{a^2} = \frac{(b(u-v))^2...

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paraboloid

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paraboloid

Paraboloid

A paraboloid is a surface. $$z=x^2+y^2$$ gives paraboloid. Let $\sigma$ be a surface patch for a paraboloid defined by $$\sigma (u,v)= (u,v, u^2+v^2)$$ I want to show that this is a surface. To show that $\sigma...

Surface area of an elliptic paraboloid The elliptic paraboloid has height h, and two semiaxes a, b. How to find its surface area? Does it possible to use a direct formula without integrals?

Volume of a cut paraboloid I have a cut paraboloid made from parabola $y(x)=c+x-ax^2$ and $x = 0$ line. How do I compute volume of this cut paraboloid? I researched on Wolfram. see formula 16 and 17

Spherical coordinates and unregular paraboloid If we have the following equation of a paraboloid: $z=4-x^2-y^2$ and we have the region in space bounded by this paraboloid from above and by the $xy$-plane from below; w...

Paramaterization of paraboloid and plane. Consider the paraboloid $z=x^2+y^2$. The plane $2x-4y+z-6=0$ cuts the paraboloid, its intersection being a curve. Find "the natural" parameterization of this curve. I have se...

拋物面 Paraboloid: 最新的百科全書、新聞、評論和研究

surface area of the part of the circular paraboloid Find the surface area of the part of the circular paraboloid!enter image description here that lies inside the cylinder !enter image description here

Area of a Paraboloid inside a Cylinder Find the area of the part of the paraboloid $x=y^2+z^2$ that is inside the cylinder $y^2+z^2=9$. I'm not sure how to set up the integral to compute this. Thanks.

Why this is a Elliptic Paraboloid?? Let $x^2+2z^2-6x-y+10=0$ and we know that $z/c=x^2/a^2+y^2/b^2$ shows a Elliptic Paraboloid. if we compelete the square we gonna have $(y-1)=(x-3)^2+2z^2.$ Book says we can see that...

The volume of a classic paraboloid at $h$ So, I have an object $z=x^2+y^2$ filled with water at $h=50$ where $h=0$ is at the deepest spot of the paraboloid. With this information I am to find out the volume of the w...

Parallel surface to paraboloid I have the equation for a paraboloid $z=x^2+y^2$ I need to obtain a parallel surface, Does anyone know what is the procedure to determine it?

I have to find the surface area of a paraboloid within a cylinder. I have to find the surface area of a paraboloid within a cylinder. The paraboloid is $x = y^2 + z^2$ and the cylinder is $y^2 + z^2 = 4$, and I know ...

Showing the parametrically representation of hyperbolic paraboloid. And how to find the curves $u$ and $v$ be constant. Show that the hyperbolic paraboloid can be represented parametrically as $$r(u,v)=\langle a(u+v),...