Questions tagged [differential-geometry]

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45
votes
6answers
35k views

How to calculate scalar curvature, Ricci tensor and Christoffel symbols in Mathematica?

I am seeking a convenient and effective way to calculate such geometric quantities. I've used packages like TensoriaCalc, but they don't work at all time. Sometimes,...
39
votes
3answers
13k views

Differential geometry add-ons for Mathematica

I need a differential geometry tool for Mathematica. Is there any add-on? Unfortunately, Mathematica does not include such functionality or I can not find it.
42
votes
5answers
3k 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 ...
29
votes
4answers
2k 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 ...
7
votes
2answers
3k 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 ...
7
votes
1answer
1k views

Visualization of Gaussian Curvature

I need to visualize Gaussian Curvature of a parametric surface. There is a solution in this math.SE post. However, I'm not sure its working because when I draw a sphere it's all white, but it should ...
19
votes
2answers
3k 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). ...
12
votes
3answers
5k views

Computing Gaussian curvature

Can Gaussian curvature $K$ be computed from WolframAlpha or any other available Mathematica program? Please indicate the program or its reference. If input ...
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 ...
11
votes
2answers
2k views

Geodesics on a torus

Is it possible to draw geodesics between the points in a path on a torus - toroidal surface? geodesics: generalization of the notion of a "straight line" to "curved spaces" ...
4
votes
1answer
370 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
3answers
335 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. ...
6
votes
2answers
384 views

Looking for a package for Cartan formalism in Mathematica

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 ...
24
votes
1answer
5k views

Solving Killing equations

Is it possible to solve Killing equations in Mathematica for a general vector? I am looking for a way to create Killing equations and then find what the vectors are, but I have a problem with this. ...
18
votes
3answers
415 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 ...
12
votes
1answer
302 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
0answers
511 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 ...
8
votes
3answers
359 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 ...
5
votes
1answer
773 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 ...
2
votes
1answer
610 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
355 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 ...
8
votes
2answers
227 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 ...
8
votes
1answer
280 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
1answer
149 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: ...
3
votes
2answers
157 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 ...
3
votes
1answer
829 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 ...
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. ...