Questions tagged [differential-geometry]

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2
votes
1answer
103 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
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1answer
47 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 ...
4
votes
1answer
264 views

Can Mathematica solve nonlinear, coupled differential equations?

I've got two equations that describe a Geodesic on a sphere. $$\frac{d^2 u}{d\lambda^2}-Cos[u]Sin[u]\frac{dv}{d\lambda}\frac{dv}{d\lambda}=0$$ $$\frac{d^2 v}{d\lambda^2}-2Cot[u]\frac{du}{d\lambda}\...
0
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1answer
72 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
39 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
33 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: ...
1
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0answers
65 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
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0answers
57 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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0answers
60 views

How to find curvature and torsion of a list of points?

I have list of points. The list represent around 1000 points of a curve in $\mathbb{R}^3$. I want to find the curvture and torsion of this curve. How to do this?
1
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1answer
83 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
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1answer
119 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
51 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
98 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
95 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
73 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
58 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
52 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
58 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
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0answers
40 views

Can't get TensoriaCalc to work

So I'm trying to use TensoriaCalc, and it won't work. I put it this ...
2
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2answers
126 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
125 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
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
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0answers
77 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
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2answers
130 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
76 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
132 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
102 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
339 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
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1answer
288 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
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1answer
94 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
96 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
122 views

Find initial surface to minimize between two close curves

I have two close curves in space defined by $g$ and $h$ with: ...
1
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0answers
52 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
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0answers
68 views

How get Mathematica to recognize a simple trigonometric identity

I am experimenting with simple differential geometry problem, computing the Frenet apparatus. ...
1
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0answers
170 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 ...
1
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0answers
63 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
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0answers
92 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
241 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
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1answer
90 views

Finding a set of characteristics of a system of polynomial equations

we have an ODE system as following ...
1
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1answer
189 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
164 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 ...
6
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2answers
194 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 ...
1
vote
2answers
229 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
316 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
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2answers
303 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
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1answer
253 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
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1answer
414 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
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1answer
61 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
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1answer
504 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
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0answers
273 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 ...