# Questions tagged [differential-geometry]

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### 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: ...
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### 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 ...
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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 ...
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### Partial differentiation second order

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

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

Consider the following operator defined over unit vectors of $\mathbb{R}^3$: $$R(u,v) = (u\cdot v)I_3 + hat(u\times v) + \dfrac{(u\times v)\otimes (u\times v)}{1+u\cdot v}$$ ...
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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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### 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 ...
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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 ...
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### 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? ...
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### Computing wedge product of vector fields

This is a question of actually linear algebra. Say I have a vector spaces spanned on x3,x4,..., x9 I am trying to find wedge product of two elements of this vector space. My numbers a3, ..., a9 are ...
419 views

### Parametric Plot 3D: A curve and a surface plot together

I am new to Mathematica and I tried to make a superimposed plot of a curve on a half sphere and need help with some graphics. This is the code line that i used to generate the following graphs <...
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### 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 ...
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### How to plot parametric time dependent plot with different parameters taking average of time?

I am trying to plot the solution given in the code with respect to "delc". Now the problem is that it can be plot for a particular value of "t" like t=10,20,50,60,.. upto 100 but what i need is mean ...
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### 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. ...
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### How to calculate the Gauss curvature of any point of function with two variables [duplicate]

I see many methods to calculate the Gaussian curvature of parametric surfaces in SE.But how to calculate the Gaussian curvature of any point of the following explicit function: ...
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### 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 == x0}, x[t], t] This ...
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### 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: ...
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### 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: ...
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### Adding a bar legend to a 3D plot indicating surface curvature by color

I want to color surfaces according to its Gaussian curvature, but the color bar is not consistent with the color. How can I improve it? Color a HyperbolicParaboloid according to its Gaussian ...