Questions tagged [differential-geometry]
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120
questions
0
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1answer
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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:
...
1
vote
2answers
116 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
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0answers
69 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
1answer
35 views
8
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2answers
224 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
1answer
88 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 ...
3
votes
2answers
153 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 ...
4
votes
1answer
87 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
1answer
132 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
...
9
votes
2answers
300 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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0answers
46 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
1answer
126 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
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0answers
51 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
221 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?
...
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0answers
41 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
152 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 ...
7
votes
4answers
329 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
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0answers
72 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
96 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
406 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
135 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
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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:
...
8
votes
3answers
350 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
111 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
342 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
100 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
419 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
168 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:
$$ds^2=dr^2+\sqrt{\lvert k\rvert}^{-1}\sinh\left(r\sqrt{\lvert k\rvert}\right)\left( d\...
1
vote
1answer
119 views
Frenet frame in Pseudo Galilean Space [duplicate]
If I have the Frenet frames for an admissible curve say $\alpha$ in a pseudo Galilean space, which is given by: $ t'(x) = \kappa(x) n(x)$, $n'(x) = \tau(x) b(x)$ and $b'(x) = tau(x) n(x)$, where tau ...
2
votes
1answer
66 views
I need help solving this hyperbolic equation [duplicate]
I have some data and I'd like to calculate the radius of curvature. The formula is:
$$R_{oc}\space Sinh\left[\frac{D_{LSS}}{R_{OC}}\right]=\frac{s_*}{\theta_*}$$
Noting that $s_*$ is sh, $\theta_*$ ...
4
votes
1answer
348 views
Find geodesics given two points
Suppose I want to find the geodesic on a paraboloid going through two points located on the paraboloid, using Mathematica 12.
I chose the paraboloid parametrization as follows:
...
0
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0answers
79 views
How to compute trajectories normal to field lines?
I have function [Psi][r,z] found from solution of a Grad-Shafranov equation. Magnetic field is expressed through [Psi][r,z] as follows
...
1
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0answers
82 views
Dirac Operator in de Sitter background
I am attempting to write the Dirac operator in a curved background, and eventually solve the equation as a second order PDE, since I am attempting to bring it into Klein-Gordon form. Essentially what ...
3
votes
1answer
305 views
Difficulties on Mathematica code to solve Christoffel Symbols of a particular metric
I) The Problem
There's a particular metric $[1],[2]$ in general relativity which is written as:
$$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}$$
So ...
4
votes
1answer
484 views
Fourth Order Tensor rotation [duplicate]
What is the easiest way to perform Rotation for Higher Order Tensors in Mathematica ? For Instance 4th order tensor
$C_{ijkl} = \lambda_{im}\lambda_{jn}\lambda_{ko}\lambda_{lp} C_{mnop}$
4
votes
3answers
555 views
It seems Eigensystem[m] returns vectors that are not eigenvectors
I am new to here so please forgive me if I do something wrong carelessly. I have faced a serious problem in eigensystem method, or more particular, eigenvalue. It seems that the following codes that ...
5
votes
1answer
225 views
Archimedean spiral from curvature
I am trying to reconstruct an Archimedean spiral from its curvature
$$\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+...
8
votes
2answers
1k 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
48 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
65 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
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0
votes
0answers
44 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
5,2002 gold badges10 silver badges28 bronze badges
2
votes
1answer
243 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:
$$\frac{d^2r}{d\lambda^2}-r\...
0
votes
1answer
59 views
How do you typeset a fraction as a variable?
I want to keep track of a large number of variables, and the only way to express them properly is as a fraction. Is there a way to typeset a variable as a fraction? For instance, I would like to do ...
5
votes
2answers
392 views
Can Mathematica solve nonlinear, coupled differential equations?
I've got two equations that describe a Geodesic on a sphere.
$$
\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
171 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
48 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
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1answer
40 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
461 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
149 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
94 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 (...
