Questions tagged [differential-geometry]
The differential-geometry tag has no usage guidance.
110
questions
1
vote
0answers
24 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 ...
2
votes
1answer
157 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
0answers
35 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\...
2
votes
1answer
73 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 ...
6
votes
2answers
238 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
0answers
65 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 ...
1
vote
1answer
83 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
3answers
365 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 ...
1
vote
3answers
109 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
0answers
88 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
3answers
322 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 ...
2
votes
1answer
77 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.
...
4
votes
1answer
159 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^...
1
vote
0answers
68 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
1answer
115 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
1answer
159 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:
$$ds^2=dr^2+\sqrt{\lvert k\rvert}^{-1}\sinh\left(r\sqrt{\lvert k\rvert}\right)\left( d\...
1
vote
1answer
99 views
Frenet frame in Pseudo Galilean Space [duplicate]
If I have the Frenet frames for an admissible curve say $\alpha$ in a pseudo Galilean space, which is given by: $ t'(x) = \kappa(x) n(x)$, $n'(x) = \tau(x) b(x)$ and $b'(x) = tau(x) n(x)$, where tau ...
2
votes
1answer
62 views
I need help solving this hyperbolic equation [duplicate]
I have some data and I'd like to calculate the radius of curvature. The formula is:
$$R_{oc}\space Sinh\left[\frac{D_{LSS}}{R_{OC}}\right]=\frac{s_*}{\theta_*}$$
Noting that $s_*$ is sh, $\theta_*$ ...
4
votes
1answer
274 views
Find geodesics given two points
Suppose I want to find the geodesic on a paraboloid going through two points located on the paraboloid, using Mathematica 12.
I chose the paraboloid parametrization as follows:
...
0
votes
0answers
73 views
How to compute trajectories normal to field lines?
I have function [Psi][r,z] found from solution of a Grad-Shafranov equation. Magnetic field is expressed through [Psi][r,z] as follows
...
0
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0answers
41 views
Correct Inputs for Normal Curvature of a surface
I am having multiple issues with a Mathematica assignment for my differential geometry course. I understand the concepts but seem unable to format things correctly in Mathematica. I've attached the ...
1
vote
0answers
70 views
Dirac Operator in de Sitter background
I am attempting to write the Dirac operator in a curved background, and eventually solve the equation as a second order PDE, since I am attempting to bring it into Klein-Gordon form. Essentially what ...
3
votes
1answer
142 views
Difficulties on Mathematica code to solve Christoffel Symbols of a particular metric
I) The Problem
There's a particular metric $[1],[2]$ in general relativity which is written as:
$$ds^{2} = -[c^2-v_{s}^2f(r_{s})^2]dt^2+v_{s}f(r_{s})dtdx+v_{s}f(r_{s})dxdt+ dy^2+dz^2 \tag{1}$$
So ...
4
votes
1answer
329 views
Fourth Order Tensor rotation [duplicate]
What is the easiest way to perform Rotation for Higher Order Tensors in Mathematica ? For Instance 4th order tensor
$C_{ijkl} = \lambda_{im}\lambda_{jn}\lambda_{ko}\lambda_{lp} C_{mnop}$
4
votes
3answers
491 views
It seems Eigensystem[m] returns vectors that are not eigenvectors
I am new to here so please forgive me if I do something wrong carelessly. I have faced a serious problem in eigensystem method, or more particular, eigenvalue. It seems that the following codes that ...
4
votes
1answer
186 views
Archimedean spiral from curvature
I am trying to reconstruct an Archimedean spiral from its curvature
$$\kappa (\text{s$\_$})\text{:=}\frac{s^2+2}{\left(s^2+1\right)^{3/2}};$$
eqns:
$$\left(
\begin{array}{c}
t'(s)=\frac{\left(s^2+...
0
votes
0answers
43 views
How to eliminate the error in the code of calculating geodesic curve
I find the code to calculate the geodesic on a general surface from here.
But there is a mistake in calculating the geodesic between point {0, 2 ,f[0,2]} and point ...
asked Feb 10 '20 at 2:53
A little mouse on the pampas
4,3582 gold badges9 silver badges27 bronze badges
9
votes
2answers
946 views
The time-like geodesics (orbits) in the Schwarzschild spacetime
I am trying to plot Schwarzschild's orbit without invoking the geodesic equation.
As a reference I am using Chandrasekhar's Book (The Mathematical theory of Black Holes, Oxford University Press). In ...
0
votes
1answer
43 views
How to plot parametric time dependent plot with different parameters taking average of time?
I am trying to plot the solution given in the code with respect to "delc". Now the problem is that it can be plot for a particular value of "t" like t=10,20,50,60,.. upto 100 but what i need is mean ...
1
vote
1answer
61 views
How to avoid mistakes when drawing Gaussian curvature image of explicit function?
I've got a way to calculate Gaussian curvature from here (which was written by J. M.).
But when I applied it to the following function, I got a lot of error messages.
...
asked Jan 30 '20 at 1:47
A little mouse on the pampas
4,3582 gold badges9 silver badges27 bronze badges
0
votes
0answers
41 views
How to calculate the Gauss curvature of any point of function with two variables [duplicate]
I see many methods to calculate the Gaussian curvature of parametric surfaces in SE.But how to calculate the Gaussian curvature of any point of the following explicit function:
...
asked Jan 29 '20 at 8:40
A little mouse on the pampas
4,3582 gold badges9 silver badges27 bronze badges
2
votes
1answer
167 views
Please explain what's going on with this Geodesic Equation of a Sphere
I'm using the Christoffel Symbols found on this link to generate a set of three coupled differential equations as solutions to the Geodesic Equation of a Sphere, I have:
$$\frac{d^2r}{d\lambda^2}-r\...
0
votes
1answer
55 views
How do you typeset a fraction as a variable?
I want to keep track of a large number of variables, and the only way to express them properly is as a fraction. Is there a way to typeset a variable as a fraction? For instance, I would like to do ...
5
votes
2answers
374 views
Can Mathematica solve nonlinear, coupled differential equations?
I've got two equations that describe a Geodesic on a sphere.
$$
\frac{\mathrm d^2 u}{\mathrm d\lambda^2} - \cos u \sin u \frac{\mathrm dv}{\mathrm d\lambda} \frac{\mathrm dv}{\mathrm d\lambda} = 0 \\
\...
0
votes
1answer
107 views
How do you use DSolve with Vector?
I'm trying to create an ODE for motion in 2 dimensions. What I have so far is:
x0 = {1.5, -4.};
v = {0, 8};
DSolve[{x'[t] == v, x[0] == x0}, x[t], t]
This ...
1
vote
1answer
43 views
How do I get this to reduce/simplify further?
These equations set up an operation I'm trying to do to calculate the Christoffel Symbol:
...
0
votes
1answer
37 views
How do you reduce an equation involving Trig identities in Mathematica?
I'm trying to automatically generate the Christoffel Symbol in Mathematica. I'm starting with the formulas:
...
5
votes
2answers
340 views
Adding a bar legend to a 3D plot indicating surface curvature by color
I want to color surfaces according to its Gaussian curvature, but the color bar is not consistent with the color.
How can I improve it?
Color a HyperbolicParaboloid according to its Gaussian ...
2
votes
0answers
98 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 ...
1
vote
1answer
91 views
Plotting a function in 3 dimensions within a domain
In this paper:
https://www.staff.science.uu.nl/~beuke106/HypergeometricFunctions/COGP.pdf
Any help how to reproduce the plot in Figure:7
It’s the leading order of the complex function, Equation (...
-1
votes
1answer
130 views
Can I numerically solve these equation in Mathematica? [closed]
I have this couple of equations :
$ \partial_\mu \partial^\mu z^i + G^{i\bar{p}} (\partial_j G_{k\bar{p}} ) \partial_\mu z^j \partial^\mu z^k + G^{i\bar{j}} (\partial_{\bar{j}} G_{k\bar{l}} ) \...
1
vote
1answer
120 views
Experiments with Feedback Linearization and StateTransformLinearization in Mathematica
I have some nonlinear system, and i have three big question:
Non-linear ODE from closed-loop system and Response
1. how correctly use such terms, like "Feedback Linearization and StateTransform ...
3
votes
1answer
126 views
Constructing a 2D curve from a curvature function dynamically
I want to plot a curve starting from its curvature function and some initial conditions. This code generates a 2D curve with a given curvature (fun) and some initial conditions:
...
4
votes
0answers
78 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 $\...
2
votes
1answer
72 views
Solving a System of Differential Equations for Pendant Drop Application
So I'm trying to solve the system of differential equations describing a pendant drop. The system is as follows:
...
0
votes
0answers
56 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?
3
votes
0answers
75 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 ...
2
votes
2answers
192 views
Plotting an osculating circle at the leading edge of a developing Cornu spiral
I need to plot an interactive Cornu spiral like so:
...
0
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0answers
142 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.
...
0
votes
0answers
42 views
Finding tangents and curvature of a parametric curve in 3D [duplicate]
I am in a calculus 3 class and cannot figure out how to get Mathematica to solve for unit tangent, normal tangent, binormal tangent, and curvature without getting a supper messy result. This is the ...
