# Tagged Questions

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

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### How to calculate Euler characteristic of surfaces $K$ and $P$?

The book Introduction to Topology by C. Adams and R. Franzosa says : From the triangulations in Figure 14.8, we see that $\chi(S^2) = 2$, $\chi(T^2) = 0$, $\chi(K) = 0$ and $\chi(P) = 1$. And ...
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### Prove that $S$ is colorable if and only if it is orientable

I am taking a course on algebraic topology and I am trying to prove the following exercise: Let $S$ be a differentiable surface in $\mathbb{R}^3$. Prove that $S$ is colorable (you can paint one ...
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### Can any surface be described by an equation?

There seems to be two definitions of a surface: The set $S$ of points $(x,y,z)$ satisfying the equation $f(x,y,z)=0$ for some smooth/differentiable function $f:E^{3} \to R$ with $\nabla f \neq 0$ on ...
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### Is a point on a plane part of a face on the plane?

There is a line and a face in $\Bbb R^3$, does the line inersect the face? I have a plane (infinite area) in $\Bbb R^3$ defined by a point $(x_0,y_0,z_0)$ and its normal $n$. The plane contains a ...
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### Relation between curves on a surface and divisors

Given a projective surface $S$, the irreducible curves contained in $S$ are exactly, by definition, the prime (Weil) divisors of $S$. I was wondering what are reducible curves on $S$ in terms of ...
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### Is the principal curvature of a cylinder positive or negative according to the second fundamental form?

First off, what is the name of the tensor associated with the second fundamental form? For the first fundamental form, I believe we call the associated tensor, "the metric tensor." Principal ...
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### Points on ellipsoid with maximum Gaussian curvature/mean curvature.

Find the points on the ellipsoid $$x^2/a^2+y^2/b^2+z^2/c^2=1$$ with maximum Gaussian curvature and mean curvature respectively. I parametrized it as $(a\sin u\cos v,b\sin u\sin v, c\cos v)$ and ...
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### Surface area common to two perpendicularly intersecting cylinders

I need help to calculate the following surface area: the surface area common to the two cylinders $x^2 + y^2 = a^2$ and $x^2 + z^2 = a^2$ using surface integrals essentially. My attempt: Let ...
I am a high school student, so I know how to derive the volume $V=\dfrac{4}{3}\pi r^3$ using calculus, but I am unable to derive its surface area. However, I notice that we can approximate the ...