Questions tagged [differential-geometry]
The differential-geometry tag has no usage guidance.
146
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2
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How to use MMA to solve the minimal surface?
There is a great old post, but since MMA greatly improves the ability of solving differential equations, especially the Region can be used to define the range of variables. So I ask it again. As the ...
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0
votes
2
answers
87
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How to get the normal vector of a planar curve pointing "outside" (its circle of curvature)?
I have the following implicit function, for which I need to pick up a point $(x0, y0)$ on it and determine the expression of its normal vector pointing outside. "Outside" here means that the ...
- 695
1
vote
1
answer
67
views
TensoriaCalc does not display the correct output
I am trying to use TensoriaCalc to calculate the components of the Ricci and the Riemann tensor of the following metric: $R^{2} \left(d\theta^{2} + \sin^{2}\left(\theta \right)d\phi^{2} \right)$;
...
- 111
1
vote
0
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44
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Vector Potential Function of Vector Field with DSolve?
(This is a redux of this.)
Why doesn't this work? I have a vector field F1 whose div is 0 with a known vector potential function A2, and I try to get DSolve to solve the differential equation for A2 ...
- 263
0
votes
0
answers
83
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Find Vector Potential from Vector Field with Div = 0?
This works to compute the scalar potential function of a vector field whose curl is 0 (using the DifferentialForms.m package):
...
- 263
1
vote
0
answers
66
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Integral of (Elementary) Volume Form over Orientable Surface of Genus 2?
Per this post on math.stackexchange.com, I am told it is impossible to find an elementary formula for a coordinate patch covering of the orientable surface of genus 2 (or higher genus) (or the non-...
- 263
0
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0
answers
34
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Defining a differential form in RGTC
I am using RGTC and following one of the tutorials (from RGTC.m - example 4 calculating Killing vectors). I am struggling with fundamental understanding how does the package keeps track of which ...
3
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3
answers
227
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1
vote
0
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37
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Integrate Differential Form with Parameter with DifferentialForms.m?
In this post, it is shown how to integrate a function with parameter(s) and output a function of the parameter(s).
I am trying to do something similar with the DifferentialForms.m package. I have ...
- 263
0
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0
answers
24
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HomotopyOperator in DifferentialForms.m Not Working
I'm trying to implement and check Faraday's tensor with the DifferentialForms.m package. I code this to implement the Faraday 2-form
...
- 263
-2
votes
2
answers
157
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Plotting surface [closed]
Does anyone know how to plot this image? I am writing an example for Kenmotsu's representation theorem , for $\varphi :%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
-\{0\}\...
- 47
2
votes
1
answer
116
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Calculating distance between a set of random points in hyperbolic space
Given the metric of the Poincaré upper half-plane model
$$(ds)^2 = \frac{(dx)^2 + (dy)^2}{y^2}$$
and two known points $(x_1, y_1)$ and $(x_2, y_2)$ in the corresponding hyperbolic space $\mathbb{H} = \...
- 1,619
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0
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43
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Output of expressions unable to have operations be performed on the expression
I am currently trying to use the code linked in the catalog of spacetimes pdf to calculate chirstoffel symbols for a metric which follows as
n := 4
...
- 119
4
votes
1
answer
224
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Curvature and torsion of this curve
i was trying to get curvature and torsion of curve in mathematica. of this curve
...
- 41
0
votes
1
answer
187
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Covariant Derivative
I'm trying to create a function which receives as inputs a coordinate system $x^\mu$, metric tensor $g_{\mu\nu}$, a general tensor $T$ of rank $n$, and a list $I$ of length $n$ that looks like this:
$$...
- 123
0
votes
2
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125
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Is it how you do a projection? Is it a Projection? I don’t know what this is
I don't really know, if this is a projection. Maybe you can tell me. . . From the parametric equations
...
- 27
1
vote
1
answer
128
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How to compute the divergence of a four-vector?
I have a quadri-vector which is given by
u = {(E^(-φ0[r]))*(1 - ε δφ[t,
r]), (E^(-φ0[r])) D[ε ξ[t, r], t], 0,
0}
and a quantity n which is given by
<...
0
votes
1
answer
259
views
How to solve the einstein field equation symbolically? [closed]
How can I use Mathematica to solve the disturbed Einstein field equation? Is there any notebook that introduces this subject or package like diffgeo?
1
vote
2
answers
693
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How do I get the Schwarzschild solution from the Einstein Equations?
Consider the Schwartzchild-like metric $$ds^2=-A(r)dt^2+B(r)dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2).$$ The Einstein field equations for this metric reduce to $$R_{\mu\nu}=0,$$ which is also known as ...
- 119
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1
answer
488
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How to do Einstein Summation in Mathematica?
I am working with the de Sitter metric which takes the form
$$g=\tau^{-2}\left(-\left(\Lambda / 3-\tau^{2}\right)^{-1} d \tau^{2}+\left(\Lambda / 3-\tau^{2}\right) d t^{2}+g_{\mathbb{S}^{2}}\right)$$
...
- 93
1
vote
3
answers
144
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What is the most efficient way to turn a metric formula into a metric tensor?
I have a metric formula:
ds=(-dt^2)*(c3+a3*t)^2+(dxC^2*(t0^2+t1^2)^2)/(4*t0^4)+
(dxM^2*(t0^2+t1^2)^2)/(4*t0^4)+(dxY^2*(t0^2+t1^2)^2)/(4*t0^4)
How do I turn this ...
- 1,521
0
votes
0
answers
102
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Christoffel symbol of the first kind [duplicate]
Suppose that we are given a metric $$ds^2=-\left(1-\frac{r_s}{r}\right)c^2dt^2+\left(1-\frac{r_s}{r}\right)^{-1}dr^2+r^2d \theta^2+r^2\sin^2(\theta)d \phi^2.$$ Given the Catalogs of Spacetimes pdf we ...
- 119
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0
answers
74
views
Finding transformation matrix for a coordinate transform
I have a matrix in two different bases. Suppose they are called $g_{\mu \nu}$ in one basis and $\eta_{ab}$ in the other basis. If the transformation between the bases is represented by $T_{\mu}^{\text{...
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1
answer
99
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8
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0
answers
266
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Shortest Distance between two points on a 2D surface
I am working on an interesting problem, that consists of trying to compute the shortest distance between two points on a 2D surface. This 2D surface can be represented in a 3D space by the function $f(...
- 397
0
votes
1
answer
76
views
Define Matrix Function in a For loop
Let $g\in\mathcal R^{3\times 3}$ be a given known rank 2 tensor function, $gInv$ its inverse, and $dg\in\mathcal R^{3\times 3\times 3}$ its gradiant:
...
- 1
2
votes
2
answers
242
views
Plot or draw a differential geometry graph in Mathematica?
I want to draw a differential geometry diagram, but I don’t know how to draw it. I can only draw graphics like the first picture. There are many differential geometry diagrams in textbooks. What tools ...
- 21
0
votes
0
answers
117
views
Computations on differential geometry
I have to do some really long computations of basic riemannian geometry. I am trying to use Mathematica to check them. This is the kind of computations I want to do. Let $M$ be a riemannian manifold ...
- 131
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1
answer
51
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8
votes
2
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296
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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 ...
- 147
1
vote
1
answer
165
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Einstein field equations for Bondi-Sachs formalism
I'm trying to re-derive the results of Bondi-Sachs formalism. The metric is given in the form
\begin{array}{c}g_{a b} d x^{a} d x^{b}=-\frac{V}{r} e^{2 \beta} d u^{2}-2 e^{2 \beta} d u d r+r^{2} h_{A ...
- 13
3
votes
2
answers
243
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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 ...
- 473
4
votes
1
answer
145
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Differentiation by indexed variable in equation of Christoffel Symbols
I am very new to Mathematica, and am trying to use it to compute Christoffel symbols for a certain manifold. All this requires is taking some indexed sums of derivatives, but it builds up on several ...
5
votes
1
answer
247
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Finding unit tangent, normal, and binormal vectors for interpolated function
As an extension of this question, is it possible to find the unit tangent, normal, and binormal vectors for an interpolated function? eg
...
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9
votes
2
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334
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Prove $R^\top R = I_3$ and find skew-symmetric matrix $R^\top \dot R$
Consider the following operator defined over unit vectors of $\mathbb{R}^3$:
$$R(u,v) = (u\cdot v)I_3 + hat(u\times v) + \dfrac{(u\times v)\otimes (u\times v)}{1+u\cdot v}$$
...
- 14k
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0
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65
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How do I interpret this table of Christoffel symbols?
So I found a code that allows me to compute the covariant derivative of some vector, here it is:
...
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3
votes
1
answer
250
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Covariant derivative of a vector [duplicate]
I have a code that gives me the Christoffel symbols of a metric. How do I take the covariant derivative of a vector?
It does not necessarily have to build upon my code, but this is what I have used so ...
- 135
1
vote
0
answers
106
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Product of manifolds with non−zero non−diagonal boxes in the metric
I'm trying to construct in xAct a metric like this
where $g_{\mu\nu}$ is 4-dimentional, and $g_{MN}$ - 5-dimentional, $A_\mu$ - 4-vector and $\phi$ is a scalar field.
I already tried to do it like ...
- 11
2
votes
1
answer
375
views
How can I draw a sphere in the Minkowski space?
If the equation of the circle in the Minkowski 3 space is given as
$S_1^2 = \{x \in E_1^3:- x_1^2 + x_2^2 + x_3^2 \}$, how can I replace it in the following code?
...
- 53
0
votes
0
answers
55
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Parametrization of twisted pseudospheres types 2 and 3
Among the three types of rotationally symmetric Pseudospherical surfaces $K=-1$ (Beltrami central, type 2 and type 3)
(http://xahlee.info/surface/gallery.html)
we have Dini twist addition term $ b\...
- 2,964
3
votes
1
answer
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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 ...
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4
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369
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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 ...
- 531
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0
answers
94
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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 ...
- 2,304
1
vote
1
answer
158
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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
...
- 53
19
votes
3
answers
464
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 ...
- 14k
1
vote
3
answers
172
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:
...
- 113
1
vote
0
answers
117
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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:
...
- 53
8
votes
3
answers
455
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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 ...
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2
votes
1
answer
232
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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.
...
- 163
4
votes
1
answer
680
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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^...
- 163