Questions tagged [differential-geometry]
The differential-geometry tag has no usage guidance.
98
questions
1
vote
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
votes
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:
$$ds^2=dr^2+\sqrt{\lvert k\rvert}^{-1}\sinh\left(r\sqrt{\lvert k\rvert}\right)\left( d\...
1
vote
1answer
70 views
Frenet frame in Pseudo Galilean Space [duplicate]
If I have the Frenet frames for an admissible curve say $\alpha$ in a pseudo Galilean space, which is given by: $ t'(x) = \kappa(x) n(x)$, $n'(x) = \tau(x) b(x)$ and $b'(x) = tau(x) n(x)$, where tau ...
2
votes
1answer
52 views
I need help solving this hyperbolic equation [duplicate]
I have some data and I'd like to calculate the radius of curvature. The formula is:
$$R_{oc}\space Sinh\left[\frac{D_{LSS}}{R_{OC}}\right]=\frac{s_*}{\theta_*}$$
Noting that $s_*$ is sh, $\theta_*$ ...
4
votes
1answer
146 views
Find geodesics given two points
Suppose I want to find the geodesic on a paraboloid going through two points located on the paraboloid, using Mathematica 12.
I chose the paraboloid parametrization as follows:
...
0
votes
0answers
67 views
How to compute trajectories normal to field lines?
I have function [Psi][r,z] found from solution of a Grad-Shafranov equation. Magnetic field is expressed through [Psi][r,z] as follows
...
0
votes
0answers
33 views
Correct Inputs for Normal Curvature of a surface
I am having multiple issues with a Mathematica assignment for my differential geometry course. I understand the concepts but seem unable to format things correctly in Mathematica. I've attached the ...
1
vote
0answers
60 views
Dirac Operator in de Sitter background
I am attempting to write the Dirac operator in a curved background, and eventually solve the equation as a second order PDE, since I am attempting to bring it into Klein-Gordon form. Essentially what ...
2
votes
1answer
83 views
Difficulties on Mathematica code to solve Christoffel Symbols of a particular metric
I) The Problem
There's a particular metric $[1],[2]$ in general relativity which is written as:
$$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}$$
So ...
5
votes
1answer
110 views
Fourth Order Tensor rotation [duplicate]
What is the easiest way to perform Rotation for Higher Order Tensors in Mathematica ? For Instance 4th order tensor
$C_{ijkl} = \lambda_{im}\lambda_{jn}\lambda_{ko}\lambda_{lp} C_{mnop}$
4
votes
3answers
466 views
It seems Eigensystem[m] returns vectors that are not eigenvectors
I am new to here so please forgive me if I do something wrong carelessly. I have faced a serious problem in eigensystem method, or more particular, eigenvalue. It seems that the following codes that ...
4
votes
1answer
181 views
Archimedean spiral from curvature
I am trying to reconstruct an Archimedean spiral from its curvature
$$\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+...
1
vote
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 ...
asked Feb 10 at 2:53
Please Correct GrammarMistakes
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10
votes
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
votes
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
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.
...
asked Jan 30 at 1:47
Please Correct GrammarMistakes
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0
votes
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:
...
asked Jan 29 at 8:40
Please Correct GrammarMistakes
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2
votes
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:
$$\frac{d^2r}{d\lambda^2}-r\...
0
votes
1answer
53 views
How do you typeset a fraction as a variable?
I want to keep track of a large number of variables, and the only way to express them properly is as a fraction. Is there a way to typeset a variable as a fraction? For instance, I would like to do ...
5
votes
2answers
363 views
Can Mathematica solve nonlinear, coupled differential equations?
I've got two equations that describe a Geodesic on a sphere.
$$
\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 \\
\...
0
votes
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
vote
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
votes
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
votes
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 ...
0
votes
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
vote
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 (...
-1
votes
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}} ) \...
0
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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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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.
...
0
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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
votes
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
votes
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:
$$\theta(t,k_{x0},r) = k_{x_0} + r \cos(t);$$
$$\phi(t,k_{...
0
votes
1answer
160 views
How can I make a convex catenoid (minimum surface of revolution that closes on -1 and 1? [closed]
In Wolfram MathWorld I see the catenoid (minimum surface of revolution which is concave and open ended, but I want the one where the sides are convex and close on the long axis (say z) at -1 and 1.
I ...
0
votes
1answer
86 views
Putting first solution of three NDsolve into an array and plotting
Suppose I have three differential equations systems, each one of them has 4 equations.
I find the 4 solutions of each one, let's call them x,y,z,w. Now, I want to take $x_1$,$x_2$,$x_3$ and put them ...
2
votes
0answers
157 views
Exterior products of differential forms
In $ \mathbb{R}^4 $ I have the forms $ \omega_1=z\;\mathrm dx+t\;\mathrm dy+x\;\mathrm dz+y\;\mathrm dt $ and $ \omega_2=t\;\mathrm dx+z\;\mathrm dy+y\;\mathrm dz+x\;\mathrm dt$.
I want to compute ...
1
vote
1answer
110 views
Solution of differential equation and then draw a graph
I have two differential equations:
$da/dt = a (.3 a^{-3} + .7)^{1/2}$ and $d \tau /dt = 1/a$. The initial conditions are $t = 0$; $a = 1$ and $\tau = 0$, respectively.
How can I solve the ...
5
votes
3answers
385 views
Calculating curvature of a contour
I have an equation of a scalar field in the form
$$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
votes
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
votes
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
votes
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:
...
1
vote
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:
...
