# Questions tagged [curves]

For questions about or involving curves.

1,839 questions
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### Shortest distance between two points on the surface of a closed cylinder

What is the shortest distance between two points on the surface of a closed cylinder? I understand simple euclidean distance will work if both points are on curved surface, but I am looking for a ...
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### Prove all the integral curves of a vector fld given by $\phi_1(x,y,z) = c_1$ $\phi_2(x,y,z) = c_2$, where $\phi_1, \phi_2$ are first integrals of $V$

In a PDE book, I've come across the following theorem; Let $\phi_1$ and $\phi_2$ be two functionally independent (i.e $\nabla\phi_1 \times \nabla \phi_2 \ne 0$ on the given domain) first ...
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### Curves intersecting level surfaces at right angles

I've been trying to solve a question in my textbook for a little while now, and I can't seem to get the reasoning behind the answer. I'll quote the question, followed by my attempted solution, then ...
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### Scale invariant curvature (plane curve)

Is there a form of curvature for a plane curve which is invariant under uniform scaling? Ideally, I am looking for a way to characterize the effective 'local eccentricity' of a plane curve so that [...
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### Show that the following map is surjective

Situation: Consider the curves $C_1 = x^2 + y^2 +x^2y^2 = 1$ and $C_2 = w^2 = 1 - z^4 .$ Define a map $\lambda$ from $C_1$ to $C_2$ with $\lambda(x,y) = (x,(1+x^2)y)$. This map should be ...