Questions tagged [differential-geometry]

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3 votes
3 answers
215 views

Length of a toroidal helix

For the toroidal helix defined by ...
1 vote
0 answers
34 views

Integrate Differential Form with Parameter with DifferentialForms.m?

In this post, it is shown how to integrate a function with parameter(s) and output a function of the parameter(s). I am trying to do something similar with the DifferentialForms.m package. I have ...
0 votes
0 answers
23 views

HomotopyOperator in DifferentialForms.m Not Working

I'm trying to implement and check Faraday's tensor with the DifferentialForms.m package. I code this to implement the Faraday 2-form ...
-2 votes
2 answers
154 views

Plotting surface [closed]

Does anyone know how to plot this image? I am writing an example for Kenmotsu's representation theorem , for $\varphi :% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion -\{0\}\...
0 votes
0 answers
56 views

Implementing a 10-dimensional metric using diffgeo

I am trying to implement a 10-dimensional metric, namely a solution of type IIB supergravity equations of motion, using diffgeo by Matthew Headrick. Physics The metric is $$ds_{10}^2=\sqrt{\frac{...
2 votes
1 answer
97 views

Calculating distance between a set of random points in hyperbolic space

Given the metric of the Poincaré upper half-plane model $$(ds)^2 = \frac{(dx)^2 + (dy)^2}{y^2}$$ and two known points $(x_1, y_1)$ and $(x_2, y_2)$ in the corresponding hyperbolic space $\mathbb{H} = \...
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0 votes
0 answers
41 views

Output of expressions unable to have operations be performed on the expression

I am currently trying to use the code linked in the catalog of spacetimes pdf to calculate chirstoffel symbols for a metric which follows as n := 4 ...
  • 119
3 votes
1 answer
156 views

Curvature and torsion of this curve

i was trying to get curvature and torsion of curve in mathematica. of this curve ...
0 votes
1 answer
121 views

Covariant Derivative

I'm trying to create a function which receives as inputs a coordinate system $x^\mu$, metric tensor $g_{\mu\nu}$, a general tensor $T$ of rank $n$, and a list $I$ of length $n$ that looks like this: $$...
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0 votes
2 answers
120 views

Is it how you do a projection? Is it a Projection? I don’t know what this is

I don't really know, if this is a projection. Maybe you can tell me. . . From the parametric equations ...
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1 vote
1 answer
101 views

How to compute the divergence of a four-vector?

I have a quadri-vector which is given by u = {(E^(-φ0[r]))*(1 - ε δφ[t, r]), (E^(-φ0[r])) D[ε ξ[t, r], t], 0, 0} and a quantity n which is given by <...
0 votes
1 answer
160 views

How to solve the einstein field equation symbolically? [closed]

How can I use Mathematica to solve the disturbed Einstein field equation? Is there any notebook that introduces this subject or package like diffgeo?
0 votes
0 answers
52 views

How to compute the covariant derivative of a vector? [duplicate]

I want to compute a derivative of a vector, but I'm new to mathematica (I programmed in other languages). I was advised to work with the diffgeo.m package. What I want to do is calculate using ...
1 vote
2 answers
453 views

How do I get the Schwarzschild solution from the Einstein Equations?

Consider the Schwartzchild-like metric $$ds^2=-A(r)dt^2+B(r)dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2).$$ The Einstein field equations for this metric reduce to $$R_{\mu\nu}=0,$$ which is also known as ...
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0 votes
1 answer
375 views

How to do Einstein Summation in Mathematica?

I am working with the de Sitter metric which takes the form $$g=\tau^{-2}\left(-\left(\Lambda / 3-\tau^{2}\right)^{-1} d \tau^{2}+\left(\Lambda / 3-\tau^{2}\right) d t^{2}+g_{\mathbb{S}^{2}}\right)$$ ...
  • 91
1 vote
3 answers
111 views

What is the most efficient way to turn a metric formula into a metric tensor?

I have a metric formula: ds=(-dt^2)*(c3+a3*t)^2+(dxC^2*(t0^2+t1^2)^2)/(4*t0^4)+ (dxM^2*(t0^2+t1^2)^2)/(4*t0^4)+(dxY^2*(t0^2+t1^2)^2)/(4*t0^4) How do I turn this ...
  • 1,521
0 votes
0 answers
100 views

Christoffel symbol of the first kind [duplicate]

Suppose that we are given a metric $$ds^2=-\left(1-\frac{r_s}{r}\right)c^2dt^2+\left(1-\frac{r_s}{r}\right)^{-1}dr^2+r^2d \theta^2+r^2\sin^2(\theta)d \phi^2.$$ Given the Catalogs of Spacetimes pdf we ...
  • 119
0 votes
0 answers
56 views

Finding transformation matrix for a coordinate transform

I have a matrix in two different bases. Suppose they are called $g_{\mu \nu}$ in one basis and $\eta_{ab}$ in the other basis. If the transformation between the bases is represented by $T_{\mu}^{\text{...
2 votes
1 answer
92 views

Problem with ArcCurvature

When comparing the outputs of ...
  • 2,030
8 votes
0 answers
240 views

Shortest Distance between two points on a 2D surface

I am working on an interesting problem, that consists of trying to compute the shortest distance between two points on a 2D surface. This 2D surface can be represented in a 3D space by the function $f(...
  • 397
0 votes
1 answer
62 views

Define Matrix Function in a For loop

Let $g\in\mathcal R^{3\times 3}$ be a given known rank 2 tensor function, $gInv$ its inverse, and $dg\in\mathcal R^{3\times 3\times 3}$ its gradiant: ...
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1 vote
2 answers
193 views

Plot or draw a differential geometry graph in Mathematica?

I want to draw a differential geometry diagram, but I don’t know how to draw it. I can only draw graphics like the first picture. There are many differential geometry diagrams in textbooks. What tools ...
0 votes
0 answers
102 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 ...
1 vote
1 answer
41 views

Partial differentiation second order

I have some rules for differentiation: ...
  • 1,215
8 votes
2 answers
265 views

What is the best source to learn how to use tensor operations (exterior algebra) in Mathematica?

I'm specifically interested in the TensorProduct,TensorWedge, HodgeDual and certain build in functions to do tensor arithmetic like TensorReduce, TensorExpand. I would like to do exterior algebra ...
1 vote
1 answer
134 views

Einstein field equations for Bondi-Sachs formalism

I'm trying to re-derive the results of Bondi-Sachs formalism. The metric is given in the form \begin{array}{c}g_{a b} d x^{a} d x^{b}=-\frac{V}{r} e^{2 \beta} d u^{2}-2 e^{2 \beta} d u d r+r^{2} h_{A ...
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3 votes
2 answers
207 views

Calculate surface normals at the boundary of a Graphics3D object

How do I go about calculating and plotting the surface normals at the boundary of a Graphics3D object? For example, consider this custom-defined ...
  • 473
4 votes
1 answer
117 views

Differentiation by indexed variable in equation of Christoffel Symbols

I am very new to Mathematica, and am trying to use it to compute Christoffel symbols for a certain manifold. All this requires is taking some indexed sums of derivatives, but it builds up on several ...
5 votes
1 answer
206 views

Finding unit tangent, normal, and binormal vectors for interpolated function

As an extension of this question, is it possible to find the unit tangent, normal, and binormal vectors for an interpolated function? eg ...
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9 votes
2 answers
326 views

Prove $R^\top R = I_3$ and find skew-symmetric matrix $R^\top \dot R$

Consider the following operator defined over unit vectors of $\mathbb{R}^3$: $$R(u,v) = (u\cdot v)I_3 + hat(u\times v) + \dfrac{(u\times v)\otimes (u\times v)}{1+u\cdot v}$$ ...
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0 votes
0 answers
57 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: ...
3 votes
1 answer
210 views

Covariant derivative of a vector [duplicate]

I have a code that gives me the Christoffel symbols of a metric. How do I take the covariant derivative of a vector? It does not necessarily have to build upon my code, but this is what I have used so ...
1 vote
0 answers
92 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 ...
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2 votes
1 answer
313 views

How can I draw a sphere in the Minkowski space?

If the equation of the circle in the Minkowski 3 space is given as $S_1^2 = \{x \in E_1^3:- x_1^2 + x_2^2 + x_3^2 \}$, how can I replace it in the following code? ...
0 votes
0 answers
53 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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2 votes
1 answer
240 views

A doubt on ParametricPlot3D, RevolutionPlot3D, ListPlots and NIntegrate: can I build an "RevolutionListPlot3D"?

First of all: this is question lies in the context of Surfaces and Embbedings on differential geometry. More precisely in the context of Kruskal coordinates and how to plot a 3D dynamical ...
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7 votes
4 answers
358 views

Why do isolated large values of WorkingPrecision fail in NDSolve?

Executive Summary: Getting an accurate answer often depends on setting the WorkingPrecision high enough. Once it is high enough, though, I would expect that its ...
1 vote
0 answers
89 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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1 vote
1 answer
129 views

DSolve does not work

I am trying to solve this coupled nonlinear pdes for $\kappa(x,t)$ and $\tau(x,t)$: where $\zeta_1 = \kappa(x,t)$, $\zeta_2 = 0$. I used this code ...
18 votes
3 answers
453 views

Coordinate-free derivative

Given the function \begin{align*} f \colon \mathbb{R}^n &\to \mathbb{R}^n\\ v&\mapsto \dfrac{v}{\|v\|}, \end{align*} I would like to compute the derivative of $f$, that is $df(v)$. It is ...
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1 vote
3 answers
158 views

Cannot derive `Norm` or `Normalize` when recreating Frenet Serret equations

I'm trying to calculate the torsion of a curve at a point using the following code: ...
1 vote
0 answers
108 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: ...
8 votes
3 answers
410 views

How is Grad defined for array particularly in non-Cartesian coordinates?

This question can be viewed as a follow-up of What is the definition of Curl in Mathematica? First argument of Grad can be an array, but what definition does ...
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2 votes
1 answer
179 views

How to identify specific Christoffel symbols and Riemann Tensor components from a general solution [closed]

I want to learn how to identify specific Christoffel Symbols and Riemman Tensor components from the general solution provided by Mathematica. Let us work out an example to see what I mean clearly. ...
  • 163
4 votes
1 answer
565 views

Computing Christoffel symbols of the second kind [duplicate]

I want to compute the Christoffel-symbol for a given metric. I am using the code here, but I am missing something. The Chrisfoffel-symbol formula is $\Gamma^{\mu}_{\phantom{\mu}\nu\sigma}=\frac{1}{2}g^...
  • 163
1 vote
0 answers
124 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 vote
1 answer
707 views

Parametric Plot 3D: A curve and a surface plot together

I am new to Mathematica and I tried to make a superimposed plot of a curve on a half sphere and need help with some graphics. This is the code line that i used to generate the following graphs <...
2 votes
1 answer
196 views

How can I create this graph in Mathematica?

I'm interested in illustrating a hyperbolic plane for a report I'm writing. Here's the metric that I'm using: $$ds^2=dr^2+\sqrt{\lvert k\rvert}^{-1}\sinh\left(r\sqrt{\lvert k\rvert}\right)\left( d\...
  • 1,521
1 vote
1 answer
129 views

Frenet frame in Pseudo Galilean Space [duplicate]

If I have the Frenet frames for an admissible curve say $\alpha$ in a pseudo Galilean space, which is given by: $ t'(x) = \kappa(x) n(x)$, $n'(x) = \tau(x) b(x)$ and $b'(x) = tau(x) n(x)$, where tau ...
2 votes
1 answer
66 views

I need help solving this hyperbolic equation [duplicate]

I have some data and I'd like to calculate the radius of curvature. The formula is: $$R_{oc}\space Sinh\left[\frac{D_{LSS}}{R_{OC}}\right]=\frac{s_*}{\theta_*}$$ Noting that $s_*$ is sh, $\theta_*$ ...
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