# Questions tagged [differential-geometry]

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### 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 ...
100 views

### Solving a System of Differential Equations for Pendant Drop Application [duplicate]

So I'm trying to solve the system of differential equations describing a pendant drop. The system is as follows: ...
49 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: ...
124 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 ...
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 ...
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### 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 ...
306 views

### Prove $R^\top R = I_3$ and find skew-symmetric matrix $R^\top \dot R$

Consider the following operator defined over unit vectors of $\mathbb{R}^3$: $$R(u,v) = (u\cdot v)I_3 + hat(u\times v) + \dfrac{(u\times v)\otimes (u\times v)}{1+u\cdot v}$$ ...
35 views

### Partial differentiation second order

I have some rules for differentiation: ...
135 views

### 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 ...
91 views

### 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 ...
771 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 ...
34k 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,...
156 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 ...
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 ...
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### 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: ...
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 ...
358 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 ...
101 views

### 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 ...
73 views

### 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 ...
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