Questions tagged [parametrization]

For questions on parametrization, the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation.

672 questions
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Parameterization of a part of a cylinder

I have to find the parameterization of the surface that is part of the cylinder $$y^{2}=2-x$$ for $x\geq0$, bounded by the cylinders: $$y^{2}=z\quad\text{and}\quad y=z^{3}$$ for $0\leq y\leq1$. I ...
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Intersection of a sphere and a plane.

I have a curve: $\gamma =:[ x^2+y^2+z^2 = 1, x+y+z=1]$ How I can parametrize it? Or write it out in explicit form? I need this for compute the integral $\iint_{S}dS$, where $S$ is a surface of ...
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Relationship between tangent normal of parametric curve and range of parameter

It is intuitive that by the right-hand rule one can choose direction of motion along a parametric curve by selecting one of the unit normal vectors. However, the relationship to the parameter is less ...
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Can you give a piecewise smooth parametrization of the astroid

The astroid is given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$. It is not smooth as the derivative of the function at $0, \frac{\pi}{2}, \frac{3\pi}{2}$ and $2\pi$ is $0$. However, is it ...
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Question on equivalent parametric curves

I'm confused about the concept of equivalent parametric curves. Based on my understanding, two parametric curves, $\phi$ and $\psi$, are equivalent, if there is a strictly monotonically increasing ...
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Finding a path of a particle given that it moves parallel to $-\nabla f$

I am told that a particle starting off at $(-1,1)$ moves parallel to $-\nabla f$ where $f(x,y)=x^2-3y^2$. So $-\nabla f=(-2x,6y)$. The question wants the path of the particle and I considered ...
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What is the parametrisation of the flat torus: $\mathbb{T}^2 = \mathbb{E}^2/\mathbb{Z}^2$?

Is the parametrization of the flat torus $$\mathbb{T}^2 = \mathbb{E}^2/\mathbb{Z}^2$$ just the set of charts $$(\pi,U_{ij})\quad \pi(x)=x\quad U_{ij}=(i,i+1)\times(j,j+1)\quad i,j\in\mathbb{Z}^2$$ ...
Prove that $$\int_0^{\infty} {e^{-t^{2}}}\cos(t^2) \, dt = \frac{1}{4} \sqrt{\pi} \sqrt{1+\sqrt{2}}$$ by integrating the function ${e^{-z^{2}}}$ in the counterclockwise direction around the boundary ...