Surface is a two-dimensional submanifold of three-dimensional Euclidean space.

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Square surface with four fixed points

I'm looking for a function of two variables, $f(x, y)$, that satisfies the following constraints: $f(0, 0) = z_1$ $f(0, 1) = z_2$ $f(1, 0) = z_3$ $f(1, 1) = z_4$ and within the unit square, it ...
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70 views

Curve Orientation on a Surface

Let $S\subset\mathbb{R}^3$ be an orientable surface, but not oriented (yet). Let $X:(u_1,u_2)\in U\subset \mathbb{R}^2\longrightarrow X(U)\subset S$ be a local parametrization of the surface $S$. We ...
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finding a point on a surface? the surface is an ellipsoid

I have drawn the cross-sections of the surface $2(x-1)^2 + (y+2)^2 +z^2 = 2$ for the given planes, but am now asked to write down a point which is on the surface. I have no idea how to go about this, ...
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238 views

Why is area of a surface of revolution integral $2\pi y~ds$? not '$dx$'?

For me, intuitively, integral $2\pi y~dx$ make more sense. I know intuition can not be proof, but by far, most part of math I've learned does match with my intuition. So, I think this one should 'make ...
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124 views

Building a tube around torus knot

I am currently trying to parametrize a surface constructed by thickening a rather complicated curve, defining its normal, binormal and tangent vectors. Even using Mathematica simplification, the ...
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229 views

How to mathematically formulate the surface of a spring?

I would like to mathematically map the surface of a cylinder constructed like a coil pot (or compressed spring), where the surface area and height of the pot is a function of the length of the coil, ...
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53 views

Derivative of the Gauss map is zero

If the derivative of the Gauss map is zero in every point in the image of a given local chart, can I conclude that the normal vector is constant and such image is contained in a plane? Edit: The ...
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368 views

How to find percentage of one rectangles area based on another rectangles area

I know I might sound the dumbest person in the galaxy, but I just wanted to make sure I am doing this right. I have a rectangle say [R1] placed inside a bigger rectangle [R2]. R1 will always be <= ...
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130 views

How to calculate radius of a spherical surface having four circles touching one another?

There are four circles having radii $r_1, r_2, r_3 $ and $r_4$ touching one another on a spherical surface of radius $R$ (as shown in the picture below, four colored circles touching one another at 6 ...
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25 views

Surfaces described using norms and dot products

Okay this seems like a simple question, I think I'm just missing something obvious... The question asks to identify surfaces from the following formulae: $\vert{\bf r}\vert = a$, ${\bf r}\cdot{\bf ...
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65 views

Extending the metric of a hyperbolic surface with boundary to its double

Let $M$ be a hyperbolic surface with totally geodesic boundary. Taking the double $DM$ of $M$, it is easy to see using Euler characteristic that $DM$ is itself a hyperbolic surface (without boundary). ...
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Pappus theorem and area of a revolution surface.

Let $y=f(z)$ be a function. How can I calculate the area of the surface obtained rotating the function along the $z$ axis, where y is the revolution torus? Is it possible to do it using Pappus formula ...
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Arithmetic picard rank of smooth cubic surfaces

Assume a smooth cubic surface is defined over a field $k$ characteristic $0$, that it has line defined over $k$ and that its arithmetic Picard rank over $k$ is maximal i.e. $7$. Does this imply that ...
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56 views

What is a parametrized surface? How is it different from a surface? (Multivariable Calculus)

My textbook defines it like this: Let F be a continuous function from a subset D(F) R2 into Rq. Suppose that D(F) is pathwise connected, and that every point in D(F) is either an interior point of D(...
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315 views

What equation will create a 3D rose curve?

The parametric equation $x=a\cos(bt)\cos(t)$, $y=a\cos(bt)\sin(t)$ where $a$ & $b$ are constants and $t$ is parameter gives a rose curve which looks like, On a similar basis, is there a equation ...
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55 views

What is the probability of selecting a random point q within x sq units of a random point p?

I was using an online app called "GeoGuessr" where you guess the location on a world map of a given Google Street view image. On one certain attempt, I picked a location within 92.7 square miles of ...
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181 views

Tangential space to the rational normal curve

Exercise 15.5 (Harris, Algebraic Geometry: A First Course): Describe the tangential surface to the twisted cubic curve $C \subset \mathbb P^3$. In particular, show that it is a quartic surface. What ...
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291 views

surface area of 'cylinder' with the top cut at an angle

I don't know what the name for this shape is, so in essence it is a cylinder, radius at base $r$, which has had a wedge of the top cut off at an angle so that rather than a circle the upper face is an ...
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40 views

How do I incorporate into the definition of a function that its output is constant?

Say I have a surface, x-y+xy+yz+z^2 = 0 (There is nothing particulary special about this expression. I've just made it too complicated to be simplified so that the variables are independent of one ...
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44 views

The normal curvature is bounded by the principal curvatures.

Let the inclusion $i:S\subset\mathbb R^3$ be an immersion of a surface $S$, and let $N:S\to \mathbb R^3$ be a local Gauss map. Let $a:I\to S$ be an arc length parametrized curve, with $a(0)=p$ and $a'(...
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167 views

Normal of a coons patch at a given point

Disclamer: Rendering the Coons patch is part of 3D Graphics homework, but finding the normals at a given point isn't. Just curious. Here's what I got so far: It's a Coons patch defined by four ...
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41 views

Determinant of this coherent sheaf on a surface $S$

If $C$ is a curve on a surface $S$, i.e. $i:C\subset S$, and $G$ is a line bundle on $C$, then $G|_U\cong \mathcal{O}_C$ where $U$ is an open subset of $S$, that is, $G$ is trivial on the complement ...
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If $\dfrac{\mathrm {circumference}}{\mathrm {diameter}}$ is the same for all circles, does the surface have to be flat?

Given a two dimensional Riemannian manifold with the property that the ratio of the circumference and the diameter is the same for all circles. What can be said about it? Does it have to be the ...
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Convexity of a subset of $R^3$ based on its bounding surface

Let D be a closed region in $R^3$ whose boundary S is a closed, smooth, orientable surface. The tangent plane at every point on S intersects it in a connected set $M$. Is D (the interior of S) a ...
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a subspace of $\mathbb R^3$ with $\pi_1=\mathbb Z_2$

I've been wondering about such problems. It is well known that $\mathbb{RP}^2$ cannot be realized as a subspace of $\mathbb R^3$. But does there exist a space $X\subset\mathbb R^3$ (maybe even $...
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145 views

A question about the Möbius Strip and the Projective Plane

I know that both the Möbius Strip and the Projective Plane are both 2-manifolds. I try to prove that they are locally homeomorphic to $\mathbb{R}^2$ and Hausdorff. It seems easy to see that the ...
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53 views

surfaces, curves and lines

Could someone please assist with the following questions: Consider $f(x,y) = x^{\frac{1}{3}}y^{\frac{1}{3}}$ and take $C$ to be the curve of intersection of $z = f(x,y)$ with the plane $y=x$. Show ...
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148 views

Using Principal Directions and Curvatures to Find Point On Surface

Given the principal directions (max and min), principal curvatures, and normal of a surface at point n, how would you go about looking for a point on the surface at a given vector distance from n? I'...
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115 views

Describe the graph of the function: Level Sets, Intersections

Describe the graph of the function $$f: \mathbb{R}^2 \rightarrow \mathbb{R}, (x, y) \rightarrow |y|$$ computing some level sets and some intersections. I have done the following: The level curves ...
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67 views

Level set for the function

Draw or describe the level surface and an intersection of the graph for the function $$f: \mathbb{R}^3 \rightarrow \mathbb{R}, (x, y, z) \rightarrow x^2+y^2$$ I have done the following: The level ...
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37 views

Compute surface area about the $x$-axis

Compute the surface area swept out if the graph $y = e^{-x}$, for $0 \leq x \leq 1$, is revolved about the $x$-axis. The formula for the surface area is $$S = 2\pi\int_{a}^{b} f(x) \sqrt{1 + f'(...
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What is the co-kernel of the morphism of vector bundles?

Let $X$ be a surface, and $i:C\subset X$ be a smooth curve. Let $A$ be a line bundle on $C$, and $E$ be a vector bundle of rank $r$ on $X$. Suppose there is a surjection: $E\longrightarrow i_*A\...
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123 views

Isothermal parameterization, Inverse of the Gauss Map

This problem is from Do Carmo's Differential Geometry of Curves and Surfaces. It is question 13 from chapter 3.5, to be specific. Suppose that S is a minimal surface without any umbilical points (...
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How to prove that dim$|L|=D^/2+1$ on a $K3$ surface?

Let $D$ be an irreducible curve on a $K3$ surface $S$, and let $L=\mathcal{O}_S(D)$. The Riemann Roch formula on the $K3$ surface is given by : $\qquad\qquad\qquad\qquad\qquad h^0(S,L)+h^0(S,L^{-1})=...
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Existence of periodic orbits (non-linear systems)

I'm trying to solve the following problem: Use the Poincaré-Bendixson's criterion to show that the system has a periodic orbit $$ \dot{x}_1 =x_2 \\ \dot{x}_2=-x_1+x_2-2(x_1+2x_2)x_2^2 $$ The unique ...
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Why is $h^1(S,\mathcal{O}_S(-D))=h^0(D,\mathcal{O}_D)-1$ on a $K3$ surface?

Let $D$ be a divisor on a $K3$ surface $S$. We have an exact sequence : $\qquad\qquad\qquad\qquad\qquad0\longrightarrow\mathcal{O}_S(-D)\longrightarrow\mathcal{O}_S\longrightarrow\mathcal{O}_D\...
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Why is this divisor on a K3-surface not effective?

Let $D$ be a divisor on a $K3$ surface and set $L:=\mathcal{O}_S(D)$. Riemann Roch theorem : $\chi(L)=\chi(\mathcal{O}_S)+\frac{1}{2}D.(D-K)$ reduces to $h^0(S,L)+h^0(S,L^{-1})=2+\frac{1}{2}D^2+h^1(S,...
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Pyramid Surface Area

Square based pyramid has a side length of $220$ (b) and a height of $105.$ Find the surface area. I tried by "doing" Pythagorean theorem $110^2+105^2=s$ then i did the equation for surface area (...
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92 views

Connected Sum of Surfaces

I am trying to prove that the connected sum of surfaces is a surface. My definition of surface is: A topological space locally homeomorphic to $\mathbb{R}^2$, second countable, Hausdorff and ...
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36 views

Cylindrical coordinates - Surfaces

I found the following: Cylindrical coordinates $(\rho , \theta , z)$. This system consists of the following coordinate surfaces: Cylinders with common $z-$axis: $\rho=\sqrt{x^2+y^2}=\text{ ...
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168 views

Linking surface integral of a gradient field to a contour integral [duplicate]

I have a vector field $F$ deriving from a scalar potential $f$, i.e. $F=\text{grad}(f)$. I want to compute the integral of $F$ over a surface (To evaluate the flux of $F$). I think there exists a ...
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From a family of projective curves to a surface

Suppose that $$\mathcal F:\;(X^3+Y^3+Z^3)\lambda+Z^2X\mu=0$$ is a family of projective plane curves parameterized by $(\lambda:\mu)\in\mathbb P^1(\mathbb C)$. This family of curves forms a surface $...
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427 views

Parametrization where coordinates lines are lines of curvature

I am asked to prove that given a surface $S$ and a point $p\in S$ non-umbilical, then there exists $U$ open in $\mathbb{R}^2$, there exists $Y:U\subset \mathbb{R}^2\longrightarrow \mathbb{R}^3$ a ...
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236 views

Find the surface area generated when the curve is revolved around the x-axis

Find the surface area generated when the curve is revolved around the x-axis $y=\frac{x^3}{10}$ on $[0,\sqrt{10}]$ This is what I have so far: $$f'(x)=\frac{3x^2}{10}$$ $$f'(x)^2=\frac{9x^4}{100}$$ ...
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147 views

Get 4 points lying on the plane by given normal

I would like to create plane using 4 points (which I need to find out), when I know the intersection point of the 2 diagonals in the plane. Next thing I know, that the Y coord of 2 bottom points will ...
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1answer
48 views

Evaluate a double integral over a domain

I'm asked to evaluate $$\iint(x^3+y)$$ over the ellipse on the xy plane such that $2x^2+y^2<2y$ I figured that the ellipse can be parametrized by $$\vec r(t)=\left(\frac{\cos t}{\sqrt2};1-\sin ...
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50 views

Surfaces of constant curvature using the conformal method

I'm doing a study of surfaces with constant curvature which leads to solving the equation: $$\Delta\phi = -e^{2\phi}K_0$$ for a 2-dimensional metric with constant curvature such that rotation around ...
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42 views

How to find the side for the biggest area?

Let $x$ be one side of a rectangle and $a$ its perimeter. We know that it's area is given by: $$ S = x\cdot\left(\frac{a}{2}-x\right). $$ $$ S=-x^2+ax/2$$ where a=-1, b= a/2 and c= 0 $$D=a^2/4$$ $$n=-...
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227 views

Can someone clarify the definition of flux?

I am confused by the concept of flux as used in vector calculus. Suppose I have a sphere. On the inside of this sphere is a spherically symmetric electric charge distribution. Now I want to find the ...
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44 views

Canonical map from fundamental group to Fuchsian group?

Suppose we have a Riemann Surface $S$ of constant negative curvature $-1$. What is the canonical map from the fundamental group $\pi_1(S)$ to the discrete subgroup $\Delta \subset PSL_2(\mathbb{R})$ ...