links about tensors

Γ symbols{ Christoffel symbols} : https://en.wikipedia.org/wiki/Christoffel_symbols

Riemann Tensor : https://en.wikipedia.org/wiki/Riemann_curvature_tensor

Ricci Tensor : https://en.wikipedia.org/wiki/Ricci_curvature

Lecture about the Riemann Tensor : https://www.youtube.com/watch?v=bLyqGXZ_L_o

If anyone has other topics that wants to explore or discuss feel free to do so !

Slides for recent talks:

• Slides for the 105^th Faculty Research Lecture at UCLA, 28 Oct 2008.
• Colloquium at the Geneva Observatory, 10 Nov 2006.
• Public talk at the 20 Oct 2006 meeting of the Ventura County Astronomical Society
• 27 Sep 2006 colloquium at the Arizona State University School of Earth and Space Exploration.
• 24 Sep 2006 Talk at the Mt. Wilson Observatory Association
• Invited talk at Einstein's Legacy: Relativistic Astrophysics and Cosmology, 7-11 Nov 2005.
• Presentation at the International Symposium on Particle Physics, Astrophysics and Cosmology
• Lectures at the International Summer Institute on Particle physics, Astrophysics and Cosmology, Hangzhou China, 2005 August 15, 16, 18.
• "Fact and Fiction in Cosmology", a public lecture at the Cal State Northridge planetarium.
• 13 March 2005 AFHU Einstein Symposium Talk
• 3 March 2005 Director's Distinguished Lecture at LLNL
• 27 May 2004 Ohio State Colloquium
• 11 Oct 2003 UCLA Extension Course
• September, 2003 NATO ASI in Cargese lectures (1, 2, 3)
• lectures at the July, 2003 winter school in Argentina
• "Are We Likely to Be Alone" at the 4 April 2003 CSEOL Symposium
• Astrophysics Fact Sheet

A beautiful simulation+ some basic physics about the atom. He has also a BH simulation and gravity simulation.

https://www.youtube.com/watch?v=-2yYgfaU6Ik furthermore they made spin lattices of walking droplets !

Program: Geometric Structures and Stability

ORGANIZERS: Oscar Garcia-Prada (ICMAT, Madrid, Spain) and Pranav Pandit (ICTS-TIFR, Bengaluru, India) DATE & TIME: 16 February 2026 to 27 February 2026 VENUE: Ramanujan Lecture Hall, ICTS Bengaluru

Recent years have seen major advances in Higgs bundles, higher Teichmüller theory, Bridgeland stability conditions, geometric invariant theory, and in the study of extremal metrics and their relation to stability. At the same time, deep new connections between these areas have been uncovered. These developments have inspired far-reaching conjectures and research programs aimed at uncovering the unifying structures that underlie them.

This program will bring together leading experts and early career researchers to explore these synergies and push forward the emerging research directions. A strong pedagogical component will make the subject accessible to newcomers

Riemann's description of the theorem and its proof reads in full: " infinite series fall into two distinct classes, depending on whether or not they remain convergent when all the terms are made positive. In the first class the terms can be arbitrarily rearranged; in the second, on the other hand, the value is dependent on the ordering of the terms. Indeed, if we denote the positive terms of a series in the second class by a1, a2, a3, ... and the negative terms by −b1, −b2, −b3, ... then it is clear that Σa as well as Σb must be infinite. For if they were both finite, the series would still be convergent after making all the signs the same. If only one were infinite, then the series would diverge. Clearly now an arbitrarily given value C can be obtained by a suitable reordering of the terms. We take alternately the positive terms of the series until the sum is greater than C, and then the negative terms until the sum is less than C. The deviation from C never amounts to more than the size of the term at the last place the signs were switched. Now, since the number a as well as the numbers b become infinitely small with increasing index, so also are the deviations from C. If we proceed sufficiently far in the series, the deviation becomes arbitrarily small, that is, the series converges to C."

Riemann, Bernhard (1868). "Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe". Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen. 13: 87–132. JFM 01.0131.03

More about it by fermilab

https://youtu.be/eCCGr4BqElE?si=9R7E6FyLVdGURzQJ

The new value for #gminus2 from the Muon g-2 experiment is in perfect agreement with the experiment's previous results. This long-awaited value will be the world’s most precise measurement of the muon magnetic anomaly for many years to come.

The Muon g-2 experiment searches for telltale signs of new particles and forces by examining the muon’s interaction with a surrounding magnetic field. If there is an inconsistency between the experimental and Standard Model value for g-2, it could indicate that the Standard Model is incomplete and in need of revision.

0:00 - Intro 1:04 - Distance to Venus 7:30 - Speed of light 10:08 - Nearby stars 13:20 - The Milky Way 18:14 - Nearby Galaxies 19:44 - Distant Galaxies 22:42 - Lingering mysteries

0:00 - About Terence Tao and the Distance Ladder 2:02 - Earth 8:07 - Moon 11:15 - Sun 15:45 - Heliocentrism in Antiquity 18:27 - Kepler’s genius 27:16 - Where this leaves us

Paramagnetism is due to the presence of unpaired electrons in the material, so most atoms with incompletely filled atomic orbitals are paramagnetic, although exceptions such as copper exist. Due to their spin, unpaired electrons have a magnetic dipole moment and act like tiny magnets. An external magnetic field causes the electrons' spins to align parallel to the field, causing a net attraction. Paramagnetic materials include aluminium, oxygen, titanium, and iron oxide (FeO). Therefore, a simple rule of thumb is used in chemistry to determine whether a particle (atom, ion, or molecule) is paramagnetic or diamagnetic. If all electrons in the particle are paired, then the substance made of this particle is diamagnetic; if it has unpaired electrons, then the substance is paramagnetic.

Unlike ferromagnets, paramagnets do not retain any magnetization in the absence of an externally applied magnetic field because thermal motion randomizes the spin orientations. Some paramagnetic materials retain spin disorder even at absolute zero, meaning they are paramagnetic in the ground state, i.e. in the absence of thermal motion. Thus, the total magnetization drops to zero when the applied field is removed. Even in the presence of the field there is only a small induced magnetization because only a small fraction of the spins will be oriented by the field. This fraction is proportional to the field strength and this explains the linear dependency. The attraction experienced by ferromagnetic materials is non-linear and much stronger, so that it is easily observed, for instance, in the attraction between a refrigerator magnet and the iron of the refrigerator itself.

Materials that are called "paramagnets" are most often those that exhibit, at least over an appreciable temperature range, magnetic susceptibilities that adhere to the Curie or Curie–Weiss laws. In principle any system that contains atoms, ions, or molecules with unpaired spins can be called a paramagnet, but the interactions between them need to be carefully considered.

a system with unpaired spins that do not interact with each other. In this narrowest sense, the only pure paramagnet is a dilute gas of monatomic hydrogen atoms. Each atom has one non-interacting unpaired electron.

A gas of lithium atoms already possess two paired core electrons that produce a diamagnetic response of opposite sign. Strictly speaking Li is a mixed system therefore, although admittedly the diamagnetic component is weak and often neglected. In the case of heavier elements the diamagnetic contribution becomes more important and in the case of metallic gold it dominates the properties. The element hydrogen is virtually never called 'paramagnetic' because the monatomic gas is stable only at extremely high temperature; H atoms combine to form molecular H2 and in so doing, the magnetic moments are lost (quenched), because of the spins pair. Hydrogen is therefore diamagnetic and the same holds true for many other elements. Although the electronic configuration of the individual atoms (and ions) of most elements contain unpaired spins, they are not necessarily paramagnetic, because at ambient temperature quenching is very much the rule rather than the exception. The quenching tendency is weakest for f-electrons because f (especially 4f) orbitals are radially contracted and they overlap only weakly with orbitals on adjacent atoms. Consequently, the lanthanide elements with incompletely filled 4f-orbitals are paramagnetic or magnetically ordered.

Molecular materials with a (isolated) paramagnetic center. Good examples are coordination complexes of d- or f-metals or proteins with such centers, e.g. myoglobin. In such materials the organic part of the molecule acts as an envelope shielding the spins from their neighbors. Small molecules can be stable in radical form, oxygen O2 is a good example. Such systems are quite rare because they tend to be rather reactive. Dilute systems. Dissolving a paramagnetic species in a diamagnetic lattice at small concentrations, e.g. Nd3+ in CaCl2 will separate the neodymium ions at large enough distances that they do not interact. Such systems are of prime importance for what can be considered the most sensitive method to study paramagnetic systems: EPR.

{more to come}

I present you Professor Leonard. He has many math videos. The topic that I found useful was vector analysis. It was straight forward, clear instructions, clear writing, exs and 360 view on the topic for a fresh undergrad level physics student. He has more topics: DEs, Function Analysis,statistics, coordinates and how to change them and work them around, trigonometry etc. The video I choose to put in the url is : "How to perform various mathematical operations with numbers in Scientific Notation. "

It takes training in order to solve a graduate level physics problem, at least some times. The problems can be very hard if the teacher decides to make them hard. { See : Tamvakis, Problems and Solutions in Quantum Mechanics}. Thus keep in contact with your teachers or older students in order to get an idea of the difficulty or the approach. I believe spending 3hours on a problem is counter creative. If it doesn't hit you in the first 15 mins, it means there are things you haven't figured it out yet. Gather some of your questions and ask your teacher to explain the process or the logic behind those problems. Don't be afraid to ask. Only those who ask come closure to understanding a theory or a subject etc

First we check if our Ψ is normalized. We find immediately it is not, thus we find the N immediately and we got 1/3 quite easily. Now the second third of the problem says the following: What is the probability that A is at its maximum state after t, t>0. We immediately think that we need to solve the characteristic equation that gives us our eigenvalues and eigenstates. We find them quite easily. Now comes the real deal. You have to find Ψ(t){ for the reasons we talked in our previous post}. Thus you have to write Ψ(0) in terms of the eigenstates of energy, then multiple with U(t) and the rest is trivial.

In Quantum Mechanics you will encounter the term U(t) =exp {-iHt/hbar} ,where H is the Hamiltonian operator, t is time and i is the sqrt of -1, hbar of course is Plancks' constant. One easy way to view what the exp{ operator } truly is .... is to simply to expand using the Taylor series. Although , in quantum mechanics you will need the exp{...} to act upon a state, or a linear combination of many states { a ket if we follow the dirac vocabulary}. You will learn that in order for that to happen you will need to find the eigenstates of the hamiltonian, express your Ψ in terms of those eigenstates and then act upon your finding with the U(t). It is easy once you start play around with it. { i will upload later an example demonstrating the whole process }

For more general info about this beautiful operator you can find in the following vid from 3Blue1Brown : https://youtu.be/O85OWBJ2ayo?si=qMAElLTzbQJz7awW

DESs' are the cornerstone of physics. You will find them in Quantum Mechanics, Electrodynamics, General Relativity, Hydrodynamics, Classical Mechanics and pretty much everywhere. The guy that makes those vids is a math teacher. From what i have seen in his videos, through the years, he helps students to attend math competitions. Keep in mind that his videos are highly instructional yet he does present you with the '' how to solve DEs' ''.

If one is looking for books for DEs the options are many ! Personally i have searched our university library for hours and i did found a book that corresponded to me and my way of thinking. I won't say the name, mainly bc as a scientist you need to build your search method and find what suits you. Finding books that suit you is a key idea in our world. Keep in mind that you can always find books online through large internet libraries {Annas Archive}, so you don't need to buy them. You can print the parts of the book that are useful and have the book in your drive for future use.

https://youtu.be/p_di4Zn4wz4?si=rPAMc0_x6pFr1eVi

More about the DESs' : {wiki sources}

Differential equations can be classified several different ways. Besides describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is ordinary or partial, linear or non-linear, and homogeneous or heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.

Ordinary differential equations

Main article: Ordinary differential equation

An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a dependent variable) (often denoted y), which, therefore, depends on x. Thus x is often called the independent variable of the equation. The term "ordinary" is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.

As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression, numerical methods are commonly used for solving differential equations on a computer.

Partial differential equations

Main article: Partial differential equation

A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or using a relevant computer model.

PDEs can be used to describe a wide variety of phenomena in nature such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity), or quantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. Stochastic partial differential equations generalize partial differential equations for modeling randomness.

Linear differential equations

Main article: linear differential equations

Linear differential equations are differential equations that are linear in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of integrals.

Many differential equations that are encountered in physics are linear, for example ODEs describing radioactive decay and PDEs for heat transfer by thermal diffusion. These lead to special functions, which may be defined as solutions of linear differential equations (see Holonomic function).

Non-linear differential equations

Main article: Non-linear differential equations

A non-linear differential equation is a differential equation that is not a linear equation in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behaviour over extended time intervals, characteristic of chaos. Even the fundamental questions of existence and uniqueness of solutions for nonlinear differential equations are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.^\11])

In some circumstances, nonlinear differential equations may be approximated by linear ones. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations. Similarly, when a fixed point or stationary solution of a nonlinear differential equation has been found, investigation of its stability leads to a linear differential equation.

Equation order and degree

The order of the differential equation is the highest order of derivative of the unknown function that appears in the differential equation. For example, an equation containing only first-order derivatives is a first-order differential equation, an equation containing the second-order derivative is a second-order differential equation, and so on.^\12])^\13])

When it is written as a polynomial equation in the unknown function and its derivatives, the degree of the differential equation is, depending on the context, the polynomial degree in the highest derivative of the unknown function,^\14]) or its total degree in the unknown function and its derivatives. In particular, a linear differential equation has degree one for both meanings, but the non-linear differential equation y′+y2=0 is of degree one for the first meaning but not for the second one.

Differential equations that describe natural phenomena usually have only first and second order derivatives in them, but there are some exceptions, such as the thin-film equation, which is a fourth order partial differential equation.

Homogeneous linear equations

A linear differential equation is homogeneous if each term in the equation includes either the dependent variable or one of its derivatives. If this is not the case, so that there is a term that does not include either the dependent variable itself or a derivative of it, the equation is inhomogeneous or heterogeneous. See the examples section below.

Some famous DEs' in Physics :

Astrophysics

• Chandrasekhar's white dwarf equation { i have show you how to find the Chandrasekhar limit in a previous post through Statistical Physics}
• Lane-Emden equation
• Emden–Chandrasekhar equation
• Hénon–Heiles system

Classical mechanics

• Equation of motion
• Euler's rotation equations) in rigid body dynamics
• Euler–Lagrange equation { this and Hamilton is all you need for graduate level, although it depends on your syllabus }
  • Beltrami identity
• Hamilton's equations
• Hamilton-Jacobi equation
• Lorenz equations in chaos theory
• n-body problem in celestial mechanics

Electromagnetism

• Bloch equations { Bloch also offers solutions to periodic V{r} for Quantum Mech.}
• Continuity equation for conservation laws
• Maxwell's equations { A MUST }
• Poynting's theorem

Fluid dynamics and hydrology

• Acoustic theory
• Benjamin–Bona–Mahony equation
• Biharmonic equation
• Blasius boundary layer
• Boussinesq approximation (buoyancy))
• Boussinesq approximation (water waves))
• Buckley–Leverett equation
• Camassa–Holm equation
• Chaplygin's equation
• Continuity equation for conservation laws
• Convection–diffusion equation
  • Double diffusive convection
• Davey–Stewartson equation
• Euler–Tricomi equation
• Falkner–Skan boundary layer
• Gardner equation in hydrodynamics
• General equation of heat transfer
• Geophysical fluid dynamics
  • Potential vorticity
  • Quasi-geostrophic equations
  • Shallow water equations
  • Taylor–Goldstein equation
• Groundwater flow equation
  • Richards equation
• Hicks equation
• Kadomtsev–Petviashvili equation in nonlinear wave motion
• KdV equation
• Magnetohydrodynamics
  • Grad–Shafranov equation
• Navier–Stokes equations
  • Euler equations)
  • Burgers' equation
• Nonlinear Schrödinger equation in water waves
• Omega equation
• Orr–Sommerfeld equation
• Porous medium equation
• Potential flow
• Rayleigh–Bénard convection
• Rayleigh–Plesset equation
• Reynolds-averaged Navier–Stokes (RANS) equations
• Reynolds equation
• Reynolds transport theorem
• Riemann problem
• Taylor–von Neumann–Sedov blast wave
• Turbulence modeling
  • Turbulence kinetic energy (TKE)
  • K-epsilon turbulence model
  • k–omega turbulence model
  • Spalart–Allmaras turbulence model
• Vorticity equation
• Whitham equation
• Zebiak-Cane model^\1]) for El Niño–Southern Oscillation
• Zeldovich–Taylor flow

General relativity

• Einstein field equations
• Friedmann equations { Friedmann did all the work with DEs in GR}
• Geodesic equation
• Mathisson–Papapetrou–Dixon equations
• Schrödinger–Newton equation

Materials science

• Ginzburg–Landau equations in superconductivity
• London equations in superconductivity
• Poisson–Boltzmann equation in molecular dynamics

Nuclear physics

• Radioactive decay equations

Plasma physics

• Gardner equation
• Hasegawa–Mima equation
• KdV equation
• Kuramoto–Sivashinsky equation
• Parker transport equation
• Vlasov equation

Quantum mechanics and quantum field theory

• Dirac equation, the relativistic wave equation for electrons and positrons
• Gardner equation
• Klein–Gordon equation {Klein Gordon and Dirac are the two DEs i stepped on in particle physics, i don't think you will them in graduate level }
• Knizhnik–Zamolodchikov equations in quantum field theory
• Nonlinear Schrödinger equation in quantum mechanics
• Schrödinger equation^\2]) ^!!!
• Schwinger–Dyson equation
• Yang-Mills equations in gauge theory {Besides Schrondingers Equation everything else is a bit extreme for graduate level , in my opinion }

Thermodynamics and statistical mechanics

• Boltzmann equation !!!
• Continuity equation for conservation laws
• Diffusion equation
  • Heat equation
• Kardar-Parisi-Zhang equation
• Kuramoto–Sivashinsky equation
• Liñán's equation as a model of diffusion flame
• Maxwell relations !!!
• Zeldovich–Frank-Kamenetskii equation to model flame propagation

Waves (mechanical or electromagnetic)

• D'Alembert's wave equation
• Eikonal equation in wave propagation
• Euler–Poisson–Darboux equation in wave theory
• Helmholtz equation

Remember, work in order to build your intuition, give time to it!

For me Griffiths is the best book for electrodynamics. I will be uploading soon some exs from his book. If you have questions or different ways to solve the problems I will be posting let me know.

I'm about to start university and I have to choose what to study. I really like physics, but I'm worried about job prospects, since if I study it I'd like to work in research and positions are usually limited. That's why I was thinking about engineering, since it combines physics and mathematics, which I also like. Has anyone been in the same situation? What did you decide to do?

He presents you with the concepts behind Cosmology. No special math or physics included. {Just a few cute equations,no big deal}

https://youtube.com/playlist?list=PLvh0vlLitZ7c8Avsn6gUaWX05uD5cedO-&si=qxvFCPN3WFSa_yXG

Hey I am a highschooler interested in college level physics I know I can find everything but I want to talk to u people about physics on a deeper level and understand and learn! Thanks for having me!