# Tagged Questions

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### 3D Vectors and Geometry [on hold]

With the points A = (2, 0, 3), B = (1, 1, 1), C = (0, 1, −1) and D = (1, −1, 2)and using five subtractions, two cross products and one dot products find the distance from point D to the plane through ...
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### Computing the unit vector for a generalised helix

The space curve $$\mathbf x (t) = \begin{pmatrix} \cosh t \\ \sinh t \\ t \end{pmatrix}$$ is an example of a generalized helix, meaning that its tangent vector makes a constant angle $\theta$ with a ...
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### Surface area of intersection of two cylinders

Let $$R=\{(x,y,z):y^2+z^2\leq 1\,\, \text{and}\,\, x^2+z^2\leq 1\}.$$ Compute the volume of $R$. Compute the area of its boundary $\partial R$. I'm fine with #1. For #2, I have a ...
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### Question about spherical curvature ( binormal, tangent vectors)

Let $a:I\mapsto R^3$ be a unit speed curve. if $\rho^2+(\rho'\sigma)^2$ is constant and equal to $r^2$, show that a curve is on a sphere with r radius. answer is: define $\gamma:I\mapsto R^3$ ...
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### The curvature and torsion of the tangent indicatrix

Let $\alpha$ be a unit speed curve. Its tangent indicatrix $\sigma$ is defined by $\sigma(t)=T(t)$. Find torsion and curvature of $\sigma$ with respect to the torsion and curvature of $\alpha$. ...
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### Change of Variables from Sphere to Plane

Say we are in the space $\mathbb{R}^{n}$. Consider the $n-1$ dimensional surface measure $dS$ on the boundary of the upper half sphere. We can define the coordinates on this sphere by ...
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### Calculate normal vector to $2$-face of polytope in $\Bbb R^n$

I am trying to work through a divergence theorem application for a function integrated over an $n$-dimensional convex polytope, but I can't seem to figure out how to properly calculate the normal ...
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### Stokes' Theorem: line integrals around 2-faces of n-dimensional surface?

Suppose we have a convex polytope in $n$ dimensions and are trying to calculate the surface integral (over this polytope) of some scalar function $f:R^n \rightarrow R$. Suppose all edges and vertices ...
### Volume of Generalized Tetrahedron in $R^n$
I'm having difficulty finding the volume of a tetrahedron in $\mathbb{R}^n$. Find the volume of a generalized tetrahedron in $\mathbb{R}^n$ bounded by the coordinate hyperplanes and the hyperplane ...