Questions tagged [differential-geometry]
The differential-geometry tag has no usage guidance.
3
votes
0answers
46 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 ...
0
votes
0answers
28 views
Can't get TensoriaCalc to work
So I'm trying to use TensoriaCalc, and it won't work. I put it this
...
0
votes
0answers
94 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
39 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
58 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:
$$\theta(t,k_{x0},r) = k_{x_0} + r \cos(t);$$
$$\phi(t,k_{...
1
vote
2answers
100 views
How to calculate arbitrary area on surface of sphere?
I am trying to calculate the solid angle subtended by arbitrary-shaped loops on a sphere's surface.
First, I parametrize circular loops by:
$$\theta(t,k_{x0},r) = k_{x_0} + r \cos(t);$$
$$\phi(t,k_{...
0
votes
1answer
65 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
91 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
86 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
319 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
192 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 $x=B(u,v)$....
3
votes
1answer
79 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
83 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
116 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
40 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:
...
0
votes
0answers
65 views
How get Mathematica to recognize a simple trigonometric identity
I am experimenting with simple differential geometry problem, computing the Frenet apparatus.
...
1
vote
0answers
134 views
Wedge product of one-forms [closed]
I want to check my hand calculations of wedge products of 1-forms such as
yz dx and sin(z) dx
I see that
Wedge
has no ...
0
votes
0answers
99 views
Ricci tensor with mathematica for arbitrary dimension
How to calculate scalar curvature Ricci tensor and Christoffel symbols in Mathematica for arbitrary $n$ dimensions instead of just $4$?
1
vote
0answers
42 views
Need a package for tetrad formalism [duplicate]
Does anyone know a package for tetrad formalism?
want to convert a gravity action in terms of differential forms to tensorial expressions. A procedure known as tetrad formalism/Cartan formalism
Is ...
1
vote
0answers
78 views
Visualizing a manifold created using DimensionReduce
Apologies if the question is too vague, please delete if so.
I am trying to understand Manifolds in general (if there is any recommended text?) and how to use the function ...
1
vote
1answer
187 views
Dynamics of space curves using frenet serret formulas [duplicate]
I am trying to animate the dynamics of a space curve for which the curvature and torsion are given as a function of time. Using Frenet Serret equations, I want to find out the dynamics of the curve. ...
0
votes
1answer
84 views
Finding a set of characteristics of a system of polynomial equations
we have an ODE system as following
...
1
vote
1answer
143 views
Gauss Map of a Hyperboloid of Two Sheets
I am currently working through the textbook Modern Differential Geometry of Curves and Surfaces with Mathematica. I am new to Mathematica and would like some help with this notebook file.
I was able ...
1
vote
1answer
126 views
Principal Curvature of a monkey saddle using ParametricPlot3D but graph is showing blank
Edit: I'm working through a textbook by Alfred Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica.
I'm trying to plot the principal curvature of a monkey saddle, which I've ...
3
votes
0answers
92 views
Looking for a package for Cartan formalism in Mathematiça
I want to convert a gravity action in terms of differential forms to tensorial expressions. A procedure known as tetrad formalism/Cartan formalism like Palatini action in this page: CARTAN FORMALISM
...
1
vote
2answers
194 views
How can I use procedure, circle3D, to make an animation of the osculating circle of a parametric curve?
I try to use circle3D procedure, shown below, to make an animation of the osculating circle of a parametric curve.
...
1
vote
1answer
267 views
Trying to plot a wormhole; getting bad results
I have been attempting to plot a "drainhole" wormhole with the metric
$\qquad ds^2 = dt^2-dr^2-(r^2+a^2)(d\theta^2+Sin[\theta]^2d\phi^2)$
I set $a=0$, and after converting from spherical to ...
2
votes
2answers
240 views
Trying to define the Lie bracket of two vector fields
I am trying to define in the simplest possible way (only one coordinate system, no checking that variables are vectors, etc.) the Lie bracket of two vector fields in 3-space.
What is wrong with the ...
1
vote
1answer
214 views
Vector analysis in curvilinear coordinates
It is known that vector calculus in 3D takes quite a simple form when one uses orthonormal curvilinear coordinates (check out, for example, here, in section "Differentiation"). Is there any ...
4
votes
1answer
319 views
Expand wedge product
How can I force mathematica to expand for example this expression
$$(\cos (\theta ) dr-r d\theta \sin (\theta ))\wedge (\sin (\theta ) dr+r d\theta \cos (\theta ))$$
into what is should be, that ...
1
vote
1answer
58 views
Binormal vector not plotting as expected
I cannot see why the binormal vector is not plotting orthogonal to T and N. Surely I am overlooking something. Do you see what it might be?
...
2
votes
1answer
389 views
Covariant derivative given Christoffel symbols
I've been trying to take covariant derivative of various quantities along various surfaces in Mathematica, but I keep running into issues, frequently I get a tensor with the wrong dimension. Using the ...
2
votes
0answers
224 views
Calculating Gaussian Curvature for an irregular surface
Below is an irregular surface generated from large data and plotted with ListSurfacePlot3D from this data file using columns {3,4,2}.
I would like to calculate the ...
1
vote
1answer
729 views
Tangent Plane and Normal Vector
I have this code that shows that the derivative is vertical to the surface. I need to change the point to an arrow that is vertical and moves as the point moves
...
2
votes
1answer
534 views
Computing Christoffel symbols of the first kind
I am trying to compute the Christoffel symbols of the first kind. To keep the question short I am only computing the second term:
$$\Gamma _{\text{ijk}}=\partial \text{Mg}_{\text{ik}}/\partial q_j $$ ...
6
votes
1answer
100 views
Gauss Bonnet failure
This is a follow-up to this question:The OP asks how to compute the genus (or Euler characteristic) of a parametrized surface in $\mathbb{R}^3.$ One of the commenters recommends the Gauss-Bonnet ...
1
vote
1answer
542 views
Computing the metric tensor under a coordinate transformation
I am try to compute the metric tensor under under a coordinate transformation, using the formula:
\begin{align} g_{\mu'\nu'} = \frac{\partial x^{\mu}}{\partial x^{\mu'}} \frac{\partial x^{\nu}}{\...
4
votes
0answers
283 views
How to fit B-splines to unstructured grids (triangulated surfaces)?
As a continuation of trying to calculate curvature tensors on triangulated surfaces (here), I am interested in trying other methods. One approach is to use NURBS. To be more precise I would like to be ...
0
votes
2answers
271 views
Solving a system of first-order partial differential equations (Analytically and Numerically)
I am trying to use Mathematica 10 to solve a system of partial differential equations but I could not.
This system has an exact solution and my question is: How do I solve it exactly and numerically ...
6
votes
2answers
2k views
Compute covariant derivative in Mathematica
I need to compute covariant derivatives in Mathematica. Searching online I just found the package "Ricci" which only does symbolic computations: I instead need to do actual computations.
This is ...
4
votes
1answer
245 views
Plot curvature comb
Does anyone here knows how to create a curvature comb as in the picture but for a 3D curve?
For example if I use the 3D curve as below:
...
33
votes
5answers
2k views
How to estimate geodesics on discrete surfaces?
Continuing with my interest on curvature of discrete surfaces here and here, I would like to also calculate and plot geodesics on discretised (triangulated) surfaces. Basically, my long-term idea ...
17
votes
2answers
2k views
Estimating Principal Curvature Directions on Discrete Surfaces
I am continuing with working on implementing curvature estimations on triangulated surfaces (See: here), using the algorithm of : Meyer, M., Desbrun, M., Schröder, P., & Barr, A. H. (2003). ...
23
votes
3answers
1k views
How to speed up estimation of Mean and Gaussian curvatures on triangular meshes?
I am trying to estimate curvatures on a triangulated surface/manifold using the algorithm of : Meyer, M., Desbrun, M., Schröder, P., & Barr, A. H. (2003). Discrete differential-geometry operators ...
0
votes
1answer
163 views
13
votes
1answer
2k views
How does a Pringle lose its curvature?
Nom!
As part of a bigger project, I've was writing some code to calculate the scalar curvature of surfaces of the form $z = f(x,y)$. This uses a general calculation of the scalar curvature to produce ...
1
vote
1answer
171 views
Extract coefficient list of a differential form
I know may be this could be a duplicate of Extract coefficients of differential form in package RGTC, but the question was not solved and in a sense different from mine. I am going to explain better:
...
7
votes
1answer
180 views
12
votes
1answer
241 views
RegionNearest and neighborhoods
In this previous question we see that RegionNearest isn't quite as 'capable' as Nearest for some things.
Similarly, I need to ...
6
votes
1answer
999 views
Solving the Frenet Serret equations for non-constant curvature and torsion, obtaining parametric equations [duplicate]
I wish to solve for the curvature and torsion functions $k_1 = \dfrac{1}{1+s^2}, k_2 = \dfrac{s}{1+s^2}$ using the Frenet Serret system and obtain the parametric equations for the curve. I need the ...
