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6

Yes, such curves are algebraic. There are several ways to see this via abstract algebra (field extensions, transcendence degrees, etc.); an explicit equation can be found by elimination theory, specifically by writing $x=p(t)$, $y=q(t)$ as polynomial relations between $t$ and $x,y$ respectively, and computing the resultant of these two polynomials with ...

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If the focus is on the existence of an arclength parametrization then, for a first step, it is convenient to assume that $\phi :[a,b]\rightarrow M$ is a differentiable curve with nonvanishing derivative. The length of that curve, restricted to $[a,t]$ is then given by $$\ell(t) =\int_a^t |\phi^\prime(s)|\,ds$$ This is, by the assumption $\phi^\prime \neq0$, ...

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First, if $S$ is a regular surface and $c\colon I\subset \mathbb{R}\longrightarrow S$ is a geodesic then $$N (c(s))\parallel \vec{n}(s),\ \ \forall s\in I,\ \ \ \ \ \ \ \ (\dagger)$$ where $N$ is the unit normal of $S$ and $\vec{n}$ the first normal of the curve $c$. Thus, in your problem you are interested in finding the parallels of your surface of ...

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I think you are right that the example needs some work. A regular plane curve could have figure $8$ intersections with itself. In such cases the exterior and interior of the curve can be confusing. After rotation it is not hard to imagine the resulting surface, which tends to have a cusp-type singularity. It is very unlikely De Carmo meant this when writing ...

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I am not sure if I understood you properly, ... whether self intersection is for the 3D curves on surface of revolution or for the meridian itself.By generating curve do you mean the meridian or the non-planar 3D space curve written on it? (f(v),g(v)) is a parameterization determining a single unique point on a meridian through which any number of curves ...

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Hint: You are essentially integrating over a triangle. I think that it will be better to visualize these areas rather than write inequalities. Imagine the region, it is triangle. Pick any variable you like(however, note that sometimes you don't have that liberty). Say you chose x. Then for any value of x, what height or value of y will you get? It will be ...

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The image illustrate the domain of integration, that is a triangle with sides on the $x$ axis, the $y$ axis and the straight line $x+y=1$. For a point $P$ on such line, given $x \in [0,1]$ we have $y=1-x$ and given $y \in[0,1]$ we have $x=1-y$, so integrating in the order $y,x$ we have $$\int _0^1 \int_0^{1-x} xydydx$$ and in the order $x,y$  \int ...

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You have $\gamma(t)=5(\cos at,\sin at)$, where $t$ is the time. Since $\gamma(2)=\gamma(0)=(5,0)$, then $a=\pi$. Thus $\gamma(t)=5(\cos \pi t,\sin \pi t)$. Thus $\gamma^{\prime\prime}(t)=-5\pi^2(\cos\pi t,\sin\pi t)$. Now, for point $(3,4)$, we have $\cos\pi t_1=3/5$ and $\sin\pi t_1=4/5$ for some time $t_1\in[0,2]$. Then ...

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Quoting from wikipedia, this is the definition of asymptote. In analytic geometry, an asymptote (/ˈæsɪmptoʊt/) of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity Consider the curve $\frac{\sin x}{x}$. It's asymptote simply is $x = 0$, yet the asymptote does cross the curve infinitely many ...

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