New answers tagged surfaces
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Find the volume of the figure given below..
The issue is that you have a wrong picture. Your problem looks more like this:
This is the "front" view of the tank. Point $V$ is in the direction perpendicular to this figure. Think of it ...
- 39k
1
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For a fixed surface area, what is optimal shape of a boat so that it can carry the most weight?
There is a simple trick to arrive at the maximum capacity of a vessel of a given surface area that is open at the top.
Suppose you have two such vessels, one a mirror image of the other. Now invert ...
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-1
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For a fixed surface area, what is optimal shape of a boat so that it can carry the most weight?
Even though a sphere technically holds maximum volume for a given surface area, it can not be used as such practically or one needs to cut it somewhere to move something in or out or you need an ...
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How can I determine if a vector is pointing inwards or outwards a surface?
Let's define a family of closed surfaces indexed by $t$,
$$\Psi(u, v, t) = t \phi(u, v)$$
The 'surface' described by $\Psi(u, v, 0)$ encloses zero volume as it has been contracted to a single point at ...
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3
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Accepted
How can I determine if a vector is pointing inwards or outwards a surface?
If you look at the point $A$ in your original drawing, the vector $\hat{n}(u, v)$ (for $u = \pi/4$) points in the direction up-and-to-the-right. That's an arrow into the unbounded component of the ...
- 97k
1
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Accepted
The tangent plane of Roman surface at any point intersects the surface in a pair of conics
Since a fourth degree curve with four singular points is reducible it suffices to prove that $\cal C$ has four singular points, one of which is $(x_0,y_0,z_0)$, the other three singular points are the ...
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In $\mathbb{R}^3$, does $dS = \sqrt{dx^2dy^2+dy^2dz^2+dx^2dz^2}$ hold in surface integrals?
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In $\mathbb{R}^3$, does $dS = \sqrt{dx^2dy^2+dy^2dz^2+dx^2dz^2}$ hold in surface integrals?
Not a rigorous proof, but an indication of the geometry behind the proposed area differential.
Think of a triangle in three-dimensional space having vertices at $(a,0,0),(0,b,0),(0,0,c)$. We work out ...
- 44k
1
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Accepted
Two normal vector fields of a connected surface are equal or opposite
Given $p \in S$, let $T_p S$ the tangent plane of $S$ at $p$. Then, its orthogonal complement, call it $V_p$, is one-dimensional (then is a line) . By hypothesis: $$N_1(p), N_2 (p) \in V_p \implies \...
- 496
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Accepted
How many retractions can a projective surface have to a curve?
Here are some examples: Let $Y, Z$ be nonsingular complex projective curves with $Y$ having infinite automorphism group (i.e. genus 0 or 1). Hence, there are infinitely many nonconstant morphisms $f: ...
- 105k
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A surface with a local max is tangent to a sphere
I’m assuming by tangent you mean that there is an open ball around $p$ in which the surface $S$ is always on one side of $S_R(O)$ and they intersect in at least one point. Let’s prove this.
Since $p$ ...
- 3,693
2
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Accepted
Proof that the second fundamental form vanishes in a direction, when a line in that direction is contained in a surface
Your title and text are highly confusing (title subsequently revised). The second fundamental form does not vanish. You are saying that the normal curvature in the direction of the line vanishes. The ...
- 121k
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