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Questions tagged [curvature]

In differential geometry, the term curvature tensor may refer to the Riemann curvature tensor of a Riemannian manifold, the curvature of an affine connection or covariant derivative (on tensors), or the curvature form of an Ehresmann connection. (Def: http://en.m.wikipedia.org/wiki/Curvature_tensor)

In differential geometry, the term curvature tensor may refer to the Riemann curvature tensor of a Riemannian manifold, the curvature of an affine connection or covariant derivative (on tensors), or the curvature form of an Ehresmann connection. Reference: Wikipedia.

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds. Reference: Wikipedia.

1913 questions
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Curvature and plane curves

In multi variable calculus I learned the definition of curvature for a plane curve. I used the definition to find a function describing the curvature at various points. What I would like to do is to construct a plane curve given a curvature…
Bryhed
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How to evaluate the curvature by using normal gradient of a function?

The gradient of the function $\phi$ is: $$ \nabla\phi =(\frac{\partial\phi}{\partial x},\frac{\partial\phi}{\partial y},\frac{\partial\phi}{\partial z}) $$ and the unit normal is: $$ \vec{N}=\frac{\nabla\phi}{|\nabla\phi|} $$ while the curvature can…
withparadox2
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Circumference of a circle in hyperbolic space

I've read that in a negatively curved space (hyperbolic space), the measured circumference of a circle is greater then the expected circumference. But I can't just imagine that virtually. Can anyone help?
astronomy olampiadist
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In rectangular Cartesian coordinates, $\rho=\frac{(1+y_1^2)^{3/2}}{y_2}$ and ($y_2\neq 0$).

I was reading about curvatures. There was a topic about Cartesian Equation (Explicit function). It was given in the book that: In rectangular Cartesian coordinates , we have $\tan\psi=dy/dx=y_1$ and, therefore, $\sec^2\psi…
Arthur
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Why does a constant positive Gaussian curvature imply a sphere?

The Gaussian curvature value $G$ is the product of the two principal curvatures, which are the largest, $\kappa_1$, and the smallest (most downwards-curving), $\kappa_2$, normal curvatures through a point): $$G=\kappa_1\kappa_2$$ For a mountain top…
Steeven
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Name for the expression $f''/f'$ that *doesn't* mention risk aversion

$-f''(x)/f'(x)$ is known to economists as the Arrow Pratt measure of absolute risk aversion. We want to use it in a paper to capture a scale invariant notion of concavity while specifcally avoiding any term that suggests we are talking about…
Leo Simon
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Is this the correct curvature of the space curve?

So lets say we have a space curve $r=\langle\cos t,\sin t,3t\rangle$ and we need to find the curvature at the point $P(1,0,0)$ $r'(t)=\langle-\sin t,\cos t,3\rangle$ $r''(t)=\langle-\cos t,-\sin t,0\rangle$ $||r'(t)||=\sqrt{(-\sin t)^2+(\cos…
A Row
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Surface curvature for cylindrical jet for determining surface tension

Hi, I have this problem of the stability of a cylindrical jet. The things I do not understand is the expression for the surface curvature for the cylindrical jet, which I need in order to find the difference in pressure across the interface caused…
Stephen Kwan
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Are the curvature value of a straight line zero(parametrized curves)

$$k = \frac{y''x' - x''y'}{(x'^{2}+y'^{2})^{\frac{3}{2}}}$$ Are the curvature value of a straight line zero by using this formula?How to prove it?
jack fukey
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Why are points of zero Gaussian curvature called parabolic?

The sign of the Gaussian curvature can be used to classify points as elliptic, hyperbolic, and parabolic. Wikipedia has this image with example surfaces: I see how a hyperboloid surface has hyperbolic points, and respectively a sphere or ellipsoid…
Andre
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How to proof curvature and torsion are independent

As we know that curvature describes the change of curve in tangent and normal plane, while torsion describes the change the curve in binomial and normal plane. Assume we have a trajectory with length of $T$, then we can compute its curvature and…
Ben
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Determine angle from radius of curvature

This is another grade school problem that's giving me trouble (posting on someone else's behalf). I can see that a 36 inch semi-circumference yields a radius of 36/Pi or about 11.46 inches. However, I can't see how to use this information to…
user2469
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Curvature at $t=0$ of epicycloid (parametrization included in body of question)

The epicycloid is constructed from a circle rolling around a stationary circle Let $R$ be the stationary circle's radius Let $r$ be the rolling circle's radius Let $a$ be the ratio $\frac{R}{r}$ Let $d$ be how far to the left of the rolling circle's…
Simon M
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For the equiangular spiral $r=ae^{\theta \cot\alpha}$. Prove that the radius of curvature subtends a right angle at the pole .

For the equiangular spiral $r=ae^{\theta \cot\alpha}$. Prove that the radius of curvature subtends a right angle at the pole . The solution given in the book is as follows: We may easily deduce ,$\phi=\alpha$ and hence $\psi =\theta+\alpha$ so…
Arthur
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Mean curvature of a level set

On page 256 http://zakuski.utsa.edu/~jagy/papers/Michigan_1991.pdf there is a formula for the mean curvature of a level set of a function. Im interested in the case n=3. How do you prove this formula? I can prove it for the case of a graph, so i…
user78410
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