Questions tagged [differential-geometry]

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1answer
37 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
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1answer
146 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
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1answer
70 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
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1answer
52 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
146 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: ...
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0answers
67 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
60 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 ...
2
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1answer
83 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 ...
5
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1answer
110 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
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3answers
466 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
181 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+...
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0answers
37 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 ...
10
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2answers
764 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
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1answer
54 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. ...
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0answers
37 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
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1answer
130 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\...
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1answer
53 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
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2answers
363 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 \\ \...
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1answer
79 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
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1answer
40 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
36 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
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2answers
268 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
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0answers
83 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 ...
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0answers
71 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
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1answer
90 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 (...
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1answer
122 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}} ) \...
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0answers
66 views

Solving differential forms equations

I have this couple of equations in differential forms language: $ \Delta z^i \star {\bf{1}}+ G^{i\bar{p}} (\partial_j G_{k\bar{p}} ) dz^j \wedge \star dz^k + G^{i\bar{j}} (\partial_{\bar{j}} G_{k\bar{...
1
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1answer
107 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
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1answer
119 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
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0answers
75 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
64 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
54 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
65 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
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2answers
145 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
129 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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0answers
41 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 ...
4
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0answers
96 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
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3answers
213 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_{...
0
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1answer
160 views

How can I make a convex catenoid (minimum surface of revolution that closes on -1 and 1? [closed]

In Wolfram MathWorld I see the catenoid (minimum surface of revolution which is concave and open ended, but I want the one where the sides are convex and close on the long axis (say z) at -1 and 1. I ...
0
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1answer
86 views

Putting first solution of three NDsolve into an array and plotting

Suppose I have three differential equations systems, each one of them has 4 equations. I find the 4 solutions of each one, let's call them x,y,z,w. Now, I want to take $x_1$,$x_2$,$x_3$ and put them ...
2
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0answers
157 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 ...
1
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1answer
110 views

Solution of differential equation and then draw a graph

I have two differential equations: $da/dt = a (.3 a^{-3} + .7)^{1/2}$ and $d \tau /dt = 1/a$. The initial conditions are $t = 0$; $a = 1$ and $\tau = 0$, respectively. How can I solve the ...
5
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3answers
385 views

Calculating curvature of a contour

I have an equation of a scalar field in the form $$f(x, y) = x^2 + y^2 + xy + c$$ I want to find the curvature of the contour of the curve at $f_c = f(0.5, 0.5)$. So I need to calculate the ...
1
vote
1answer
353 views

Solving the Dirac equation in an arbitrary metric [closed]

I want to solve Dirac equation in a metric like $ds^2=g(u,v)\,du\,dv$. The relations of $u$ and $v$ with Minkowski coordinates $t$ and $x$ are given by functions $A$ and $B$, $t=A(u,v)$ and $x=B(u,v)$....
4
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1answer
102 views

Question regarding exterior products and differential forms

I'm trying to compute the following differential form $\omega = x(dy\wedge dz) + y(dx \wedge dz) + z(dx \wedge dy)$ but using a change of coordinates into spherical coords. So far, this is my code: ...
7
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0answers
104 views

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 ...
0
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1answer
123 views

Find initial surface to minimize between two close curves

I have two close curves in space defined by $g$ and $h$ with: ...
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0answers
63 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: ...