Questions tagged [differential-geometry]

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24 questions with no upvoted or accepted answers
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Higher order Laplacian flows

Given disjoint surfaces $q_i$ in 3D and their 1D boundary curves $\partial q_i = \gamma_i$, I seek a surface $p$ that joins the $q_i$, where $p \cup q_i$ forms a (piecewise) $C^k$ surface that ...
6
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0answers
510 views

How to fit B-splines to unstructured grids (triangulated surfaces)?

As a continuation of trying to calculate curvature tensors on triangulated surfaces (here), I am interested in trying other methods. One approach is to use NURBS. To be more precise I would like to be ...
4
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0answers
91 views

Computing differentials on a manifold

Consider $\phi:SO(3) \to \mathbb{R}^3$, $R \mapsto (R^\top e_3)\times e_3$ where $R$ is a real $3\times 3$ orthogonal matrix and $e_3 = [0\ 0\ 1]^\top$. Can Mathematica compute the differential of $\...
4
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93 views

Lie-algebra valued non-abelian differential forms in Mathematica

I am trying to implement wedge product of Lie-algebra valued differential forms in the non-abelian case. Concretely, I am intereste in 1- and 2-forms, that is, $A$ and $F$. Is there a way of doing ...
3
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0answers
857 views

Visualising Special Relativity

Mathematica newbie learning by doing (or not so far, in this case). Problem: Special Relativity - consider two inertial frames $A$ & $A'$ in relative motion. At a time $t0$ as measured in $S$, ...
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0answers
2k views

Calculating the mean curvature of a surface - suggestions

I am attempting to calculate the mean curvature [1, 2] of a surface defined by a function of x and y. My function is rather ...
2
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0answers
155 views

Phase portrait of n-dimensional state-space system

It is usually not difficult to study the state space for n = 2,3 variable states. What if these variables are more than 3, for example 4,5 or 6? There is a rule according to which the dynamic features ...
2
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0answers
275 views

Exterior products of differential forms

In $ \mathbb{R}^4 $ I have the forms $ \omega_1=z\;\mathrm dx+t\;\mathrm dy+x\;\mathrm dz+y\;\mathrm dt $ and $ \omega_2=t\;\mathrm dx+z\;\mathrm dy+y\;\mathrm dz+x\;\mathrm dt$. I want to compute ...
2
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334 views

Calculating Gaussian Curvature for an irregular surface

Below is an irregular surface generated from large data and plotted with ListSurfacePlot3D from this data file using columns {3,4,2}. I would like to calculate the ...
1
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0answers
60 views

Product of manifolds with non−zero non−diagonal boxes in the metric

I'm trying to construct in xAct a metric like this where $g_{\mu\nu}$ is 4-dimentional, and $g_{MN}$ - 5-dimentional, $A_\mu$ - 4-vector and $\phi$ is a scalar field. I already tried to do it like ...
1
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73 views

Problem with the analytical study of transient processes in nonlinear systems with linear dynamic links

I ask for advice and help. I am having difficulties of this nature. There is a nonlinear system of the following type: I need to analyze analytically the transient process in such a system. The ...
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0answers
98 views

How Can I draw a curve on a sphere in Minkowski 3 space?

There is a curve, which evolves with time, in the the Euclidean space. And the solution of its evolution equations has been used to draw it using this code: ...
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0answers
107 views

Computing wedge product of vector fields

This is a question of actually linear algebra. Say I have a vector spaces spanned on x3,x4,..., x9 I am trying to find wedge product of two elements of this vector space. My numbers a3, ..., a9 are ...
1
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0answers
91 views

Dirac Operator in de Sitter background

I am attempting to write the Dirac operator in a curved background, and eventually solve the equation as a second order PDE, since I am attempting to bring it into Klein-Gordon form. Essentially what ...
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0answers
112 views

Affine connection with torsion using xAct

I'm working with xAct and I need to obtain the affinne connection with torsion. Without torsion it is easy: ...
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0answers
116 views

Visualizing a manifold created using DimensionReduce

Apologies if the question is too vague, please delete if so. I am trying to understand Manifolds in general (if there is any recommended text?) and how to use the function ...
1
vote
1answer
192 views

Coding the Gibbons-Hawking metric

I am studying the Gibbons-Hawking metric, which is $ g= U^{-1}(d\tau + \omega.dx)^2 + U.dx.dx$ where $U = \sum_{s=1}^n \frac{1}{|x-P_n|}$. It is a family of metrics defined on a four-dimensional ...
0
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0answers
73 views

Computations on differential geometry

I have to do some really long computations of basic riemannian geometry. I am trying to use Mathematica to check them. This is the kind of computations I want to do. Let $M$ be a riemannian manifold ...
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0answers
47 views

How do I interpret this table of Christoffel symbols?

So I found a code that allows me to compute the covariant derivative of some vector, here it is: ...
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0answers
45 views

Parametrization of twisted pseudospheres types 2 and 3

Among the three types of rotationally symmetric Pseudospherical surfaces $K=-1$ (Beltrami central, type 2 and type 3) (http://xahlee.info/surface/gallery.html) we have Dini twist addition term $ b\...
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0answers
80 views

How to compute trajectories normal to field lines?

I have function [Psi][r,z] found from solution of a Grad-Shafranov equation. Magnetic field is expressed through [Psi][r,z] as follows ...
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59 views

General coordinate transfomation

I am looking for a (free) package or snippet that makes the transformation of tensors between spacetimes easy. What do you suggest?
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159 views

Constant positive and negative Gaussian curvature $K$ meridians as orthogonal trajectories

The plot code below depicts two point through which profiles of constant $K$ are drawn positive and negative. ...
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72 views

How get Mathematica to recognize a simple trigonometric identity

I am experimenting with simple differential geometry problem, computing the Frenet apparatus. ...