Questions tagged [differential-geometry]

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2
votes
1answer
40 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 ...
4
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1answer
182 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
60 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
79 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
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3answers
346 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
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3answers
80 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
63 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
308 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
72 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
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1answer
125 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
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0answers
56 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
60 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
154 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
84 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
54 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
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1answer
160 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
70 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
33 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
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0answers
65 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
111 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
223 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
476 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
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1answer
182 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
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0answers
42 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 ...
9
votes
2answers
888 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
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1answer
42 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
60 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. ...
0
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0answers
39 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: ...
2
votes
1answer
143 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
371 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
87 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
42 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
304 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
86 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 ...
0
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0answers
85 views

I need a starter kit for Differential Geometry

In Modern Cosmology, the Geodesic Equation is described as:$$\frac {d^2x'^2}{dt}+\left[ \left( \{\frac{\partial x}{\partial x'}\}^{-1} \right)^l_i \frac{\partial^2 x^i}{\partial x'^j \partial x'^k} \...
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
126 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
116 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
122 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
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1answer
69 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
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0answers
55 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
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0answers
67 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
169 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
135 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
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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 ...
8
votes
1answer
146 views

How to calculate specific area on surface of sphere?

I am trying to calculate the solid angle subtended by arbitrary-shaped loops on a sphere's surface. First, I parametrize circular loops by: $$\theta(t,k_{x0},r) = k_{x_0} + r \cos(t);$$ $$\phi(t,k_{...
3
votes
3answers
268 views

How to calculate arbitrary area on surface of sphere?

I am trying to calculate the solid angle subtended by arbitrary-shaped loops on a sphere's surface. First, I parametrize circular loops by: $$\theta(t,k_{x0},r) = k_{x_0} + r \cos(t);$$ $$\phi(t,k_{...