tom · @tomcuchta
112 followers · 202 posts · Server mathstodon.xyz
Erwin Schrödinger Institute · @ESIVienna
124 followers · 64 posts · Server mathstodon.xyz
Erwin Schrödinger Institute · @ESIVienna
119 followers · 57 posts · Server mathstodon.xyz
Seth Axen 🪓 :julia: 🦖 · @sethaxen
712 followers · 483 posts · Server bayes.club

I've worked out that the injectivity radius under the Euclidean metric for the group U(n) is π and for real and special subgroups O(n), SO(n), and SU(n) is π√2.

This seems like a pretty basic property, but I can't find a single reference that gives the injectivity radii for any of these groups. Anyone know of one?

#unitary #differentialgeometry #liegroups #manifolds

Last updated 2 years ago

GENERALIZED STOKES THEOREM:
The integral of a differential form \(\omega\) over the boundary \(\partial\Omega\) of some orientable manifold \(\Omega\) is equal to the integral of its exterior derivative \(d\omega\) over the whole of \(\Omega\).
\[\displaystyle\int_{\partial\Omega}\omega=\int_\Omega d\omega\]

#Stokes #exteriorderivative #boundary #manifold #fundamentaltheorem #generalizedstokestheorem #stokestheorem #calculus #multivariatecalculus #differentialgeometry #vectorcalculus

Last updated 2 years ago

GENERALIZED STOKES THEOREM:

The integral of a differential form \(\omega\) over the boundary \(\partial\Omega\) of some orientable manifold \(\Omega\) is equal to the integral of its exterior derivative \(d\omega\) over the whole of \(\Omega\).
\[\displaystyle\int_{\partial\Omega}\omega=\int_\Omega d\omega\]

#Stokes #exteriorderivative #boundary #manifold #fundamentaltheorem #generalizedstokestheorem #stokestheorem #calculus #multivariatecalculus #differentialgeometry #vectorcalculus

Last updated 2 years ago

GENERALIZED STOKES THEOREM:
The integral of a differential form \(\omega\) over the boundary \(\partial\Omega\) of some orientable manifold \(\Omega\) is equal to the integral of its exterior derivative \(d\omega\) over the whole of \(\Omega\).
\[\displaystyle\int_{\partial\Omega}\omega=\int_\Omega d\omega\]

#Stokes #exteriorderivative #boundary #manifold #fundamentaltheorem #generalizedstokestheorem #stokestheorem #calculus #multivariatecalculus #differentialgeometry #vectorcalculus

Last updated 2 years ago

Björn Gohla · @6d03
48 followers · 649 posts · Server mathstodon.xyz

Conjecture: Monoids in the category of smooth manifolds are groups.

Conjecture: A monoid in the category of smooth manifolds with boundary has non-invertible elements iff the boundary is non-empty. In that case the boundary is the maximal subgroup; in particular, the monoid unit lies on the boundary.

#showerthought #differentialgeometry

Last updated 2 years ago

Spacetime is a four-dimensional topological manifold with a smooth atlas carrying a torsion-free connection compatible with a Lorentzian metric and a time orientation satisfying the Einstein field equations (EFE).
\[\underbrace{(\mathcal{M},\mathcal{O},\mathcal{A},\nabla,\text{g},\mathcal{T})}_{\text{Relativistic spacetime}}\]

#Lorentz #einstein #theoreticalphysics #manifold #differentialgeometry #specialrelativity #generalrelativity #relativity #physics #time #space #spacetime

Last updated 2 years ago

Geodesic equation using Einstein summation convention:
\[\boxed{\ddot{\gamma}^\mu+\Gamma^\mu_{\alpha\beta}\dot\gamma^\alpha\dot\gamma^\beta=0}\]
which are the Euler-Lagrange equations of motion for the energy functional, and \(\Gamma^\mu_{\alpha\beta}\) are the Christoffel symbols. In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime.

#differentialequations #physics #differentialgeometry #geodesic #mathematicalphysics #spacetime #generalrelativity

Last updated 2 years ago

· @j_bertolotti
1301 followers · 341 posts · Server mathstodon.xyz


Field lines are a convenient way to visualize vector fields, and are defined to be tangent to them at each point.
Due to inertia, field lines do not represent the trajectory that a test mass would follow in a force field.

#visualization #differentialgeometry #vectorfields #physicsfactlet

Last updated 2 years ago

Quiet_Guy · @quiet
37 followers · 146 posts · Server infosec.exchange

Has it ever bothered any of you nerds that in "normal" and "tangent" mean the opposite of their use outside of math?

Rhetorically, a tangent is a digression and an orthogonal concept isn't normal.

#math #differentialgeometry #randomthoughts

Last updated 2 years ago

Ulises Rayon · @uncrayon
0 followers · 1 posts · Server masto.es

What's the best textbook on Differential Geometry for undergraduates?

#math #undergraduates #differentialgeometry #geometry

Last updated 2 years ago

latent-variable · @bobflagg
11 followers · 4 posts · Server mathstodon.xyz
SciHi Blog · @scihiblog
117 followers · 28 posts · Server fedihum.org

On December 8, 1865, French mathematician Jacques Salomon Hadamard was born. Hadamard made major contributions in number theory, complex function theory, differential geometry and partial differential equations. Moreover, he is also known for his description of the mathematical though process in his book Psychology of Invention in the Mathematical Field.

scihi.org/jacques-hadamard/


#maths #historyofscience #otd #numbertheory #differentialgeometry #complexfunction

Last updated 2 years ago

Jon Awbrey · @Inquiry
85 followers · 291 posts · Server mathstodon.xyz

@charlottekl

Hi Charlotte,

I think a lot about analogies between and physics, or more generally. This led me to see the need for the “missing grape” of analogous to . A lot of it amounts to differential geometry over \(\mathbb{B} = \mathbb{F}_2 = \mathrm{GF}(2)\) and of course Char 2 makes things a little bit weird and degenerate to a degree but it can be worked out.

#differentialgeometry #DifferentialLogic #systemstheory #logic

Last updated 2 years ago

Junghyeon Park · @j824h
162 followers · 341 posts · Server mathstodon.xyz

@mrdk @narain On the other hand, a parallelogram in the etymological sense, both pair of opposite edges being parallel (their tangents in a parallel transport by other two lines?), is broken as soon as non-zero Riemann curvature is introduced.

In that sense, we might instead want something like a quadrilateral with as many parallel edges as possible. Turns out such shapes have a name.
en.m.wikipedia.org/wiki/Levi-C

#differentialgeometry

Last updated 2 years ago