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math&physics with intuition
Приєднався 27 тра 2011
Here I share my videos/lectures on scientific subjects such as mathematics, modern physics (Relativity, Quantum Mechanics, etc), but also engineering.
Lorentz and Poincare groups, representations, angular momentum, Lie algebras
Infinitesimal Poincare transformations of fields describe how physical quantities transform under translations and Lorentz transformations, forming the basis of relativistic quantum field theory. In this post, we analyze the transformation of a field with a Lorentz index, exploring its infinitesimal change, including contributions from spacetime dependence and index transformation.
Using the generators of the Poincare group for translations, boosts and rotations, the infinitesimal transformation can be expressed compactly and applied to both vector and spinor fields. These concepts are critical in constructing representations of the Poincare group in modern physics.
📽️ Watch the video to delve deeper into these transformations, including their derivation and applications. This video is also part of a course titled “Master Variational Calculus & Advanced Mathematical Methods”.
#Relativity #Physics #LorentzTransformations #FieldTheory #SpinorFields #quantumfieldtheory #qft #feynman #mathematics #mathematician #lovemath #mathlover #physics #physicslover #quantumphysics #theoreticalphysics #mathchallenge #physicschallenge #pathintegrals
Using the generators of the Poincare group for translations, boosts and rotations, the infinitesimal transformation can be expressed compactly and applied to both vector and spinor fields. These concepts are critical in constructing representations of the Poincare group in modern physics.
📽️ Watch the video to delve deeper into these transformations, including their derivation and applications. This video is also part of a course titled “Master Variational Calculus & Advanced Mathematical Methods”.
#Relativity #Physics #LorentzTransformations #FieldTheory #SpinorFields #quantumfieldtheory #qft #feynman #mathematics #mathematician #lovemath #mathlover #physics #physicslover #quantumphysics #theoreticalphysics #mathchallenge #physicschallenge #pathintegrals
Переглядів: 0
Відео
From the path integral to the effective action, renormalization group and running of couplings
Переглядів 18612 годин тому
In quantum field theory, understanding how interactions change with energy scales is key to predicting particle behavior across different regimes. One way to do this is by examining the “effective action” at a given energy cutoff scale, denoted by Λ. At this scale, the effective action incorporates all interactions for energies below Λ, allowing us to study how particle fields behave over speci...
Functional derivatives and generalization of Euler Lagrange equations in QFT
Переглядів 537День тому
Understanding Functional Derivatives in Quantum Path Integrals Functional derivatives are essential in the study of quantum path integrals, helping us calculate expectation values and make connections between classical and quantum mechanics. This example shows how setting up these derivatives can yield equations that resemble classical physics with quantum effects added. #QuantumMechanics #Path...
From Path Integrals to Perturbation Theory and then to Feynman Diagrams
Переглядів 45114 днів тому
This video seeks to show the fundamental relationship between path integrals, perturbation theory, and Feynman diagrams. These diagrams offer an insightful graphical approach to perturbation theory, emerging seamlessly from path integral formalism. In this analysis, we expand the path integral for a particle in a potential field. Through successive expansions, each term encapsulates specific in...
How convolutions in the path integral lead to the Schrodinger equation
Переглядів 447Місяць тому
This post delves into the path integral formulation of quantum mechanics. For a free particle, the process involves evaluating convolutions through time slices and applying the Fourier transform, leading to the derivation of the Schrödinger equation. This mathematical approach starts from the classical action to arrive at the quantum evolution. #QuantumMechanics #PathIntegrals #SchrodingerEquat...
Heuristic Derivation of Path Integrals from Classical Field Theory (including Schwinger Principle)
Переглядів 238Місяць тому
A heuristic derivation of the path integral, starting from the Schwinger dynamical principle. This approach bridges classical field theory and quantum mechanics by considering small variations in the action. The Schwinger principle is derived by relating changes in the action to quantum transitions. This forms the foundation of the path integral formalism, where the sum over all possible field ...
Reynolds transport theorem: derivatives of time-dependent volume integrals
Переглядів 1112 місяці тому
The time derivative of a volume integral often appears in physics, particularly in the context of conservation laws and field theories. For example, in electromagnetism, the continuity equation relates the time derivative of the charge density to the divergence of the current density. The derivation shown in the video is an informal way of thinking about the transport theorem (which is related ...
Gradient in spherical coordinates (derivation using concepts of tensor calculus and linear algebra)
Переглядів 1232 місяці тому
Presented here is a detailed algorithm for computing the components of the gradient in spherical coordinates. Although this is a simple exercise usually seen in basic multivariable calculus courses, the pictures illustrate some concepts which might be inferred from tensor calculus. For further insights, refer to the accompanying video where these relations are rigorously proven and implemented ...
Derivation of the strain tensor in spherical coordinates
Переглядів 1162 місяці тому
Presented here is a detailed algorithm for computing the components of the strain tensor in spherical coordinates. The picture illustrates each step, starting with the definition of the displacement vector (in spherical coordinates) and transformation matrix (from cartesian to spherical), and culminating in the derivation of the strain tensor matrix. For further insights, refer to the accompany...
Fourier series expansion for thin plates
Переглядів 842 місяці тому
Application of Fourier series in two dimensions. For thin plates, there is a simple method for finding the displacement and stress when a plate is simply supported (displacements and bending moments are zero on the boundary). The idea is to use a double Fourier series. #FourierSeries #Mathematics #STEM #MathIsBeautiful #AdvancedMath #MathLover #fourier #mathematician #physics #physicslover #the...
Physics of thin plates derived from the variational principle (bending and twisting moments, etc)
Переглядів 522 місяці тому
Exploring the classical theory of thin plates, we delve into the relations between moments, shear forces, external forces, and derivatives of displacements. These fundamental equations describe the behavior of a thin plate under small deformations, with (possibly) external forces acting upon it. We derive the equations from an action principle. #ThinPlates #StructuralMechanics #EngineeringMathe...
Classical Theory of Thin Plates derived from the Action Principle
Переглядів 972 місяці тому
Exploring the classical theory of thin plates, we delve into the constitutive relations between stresses and strains, as well as their connection to displacement and deformation. These fundamental equations describe the behavior of a thin plate under small deformations, with no external forces acting upon it. Through a rigorous mathematical framework, we derive that the minimization of the ener...
Minimal surfaces in calculus of variations
Переглядів 982 місяці тому
In mathematics, a minimal surface is a surface that locally minimizes its area. Non-trivial solutions like the catenoid and the helicoid satisfy the minimal surface equation. #FourierSeries #Mathematics #MathIsBeautiful #ScienceAndMath#LearningMath #MathInspiration #MathematicalArt #AdvancedMath #MathLovers #mathematician #lovemath #physics #physicslover #theoreticalphysics #mathchallenge #math...
Markowitz Model for risk minimization in a financial portfolio
Переглядів 733 місяці тому
The goal of portfolio optimization is for an investor to achieve the best possible return while minimizing risk. In the Markowitz portfolio theory, risk is represented by the variance of the portfolio. Therefore, the problem is to minimize the portfolio variance while maintaining a given return. #finance # #FourierSeries #Mathematics #MathIsBeautiful #ScienceAndMath #MathGenius #LearningMath #M...
Derivation of the strain tensor in the theory of elasticity
Переглядів 1183 місяці тому
This summary covers the key points and equations on the strain tensor. #FourierSeries #Mathematics #MathIsBeautiful #ScienceAndMath #MathGenius #LearningMath #MathInspiration #MathematicalArt #AdvancedMath #MathLovers #mathematician #lovemath #physics #physicslover #theoreticalphysics #mathchallenge #math #maths #theoryofelasticity
Black Scholes equation: mathematical derivation
Переглядів 1053 місяці тому
Black Scholes equation: mathematical derivation
Interesting Fourier series in 2 dimensions
Переглядів 613 місяці тому
Interesting Fourier series in 2 dimensions
Maupertuis principle derived from the more general action principle
Переглядів 3254 місяці тому
Maupertuis principle derived from the more general action principle
Poisson brackets and analogies with Quantum Mechanics
Переглядів 2084 місяці тому
Poisson brackets and analogies with Quantum Mechanics
Calculation of Inverse Laplace Transform using residues
Переглядів 994 місяці тому
Calculation of Inverse Laplace Transform using residues
Lorentz transformations and the Invariant Element in Special Relativity (Mathematical Derivation)
Переглядів 1214 місяці тому
Lorentz transformations and the Invariant Element in Special Relativity (Mathematical Derivation)
Intro to the Laplace Transform from the Fourier Transform
Переглядів 784 місяці тому
Intro to the Laplace Transform from the Fourier Transform
differential equation with Dirac delta and boundary conditions
Переглядів 1784 місяці тому
differential equation with Dirac delta and boundary conditions
Problem on multiple convolutions, Fourier Transforms, Distributions (edited)
Переглядів 2285 місяців тому
Problem on multiple convolutions, Fourier Transforms, Distributions (edited)
Faddeev Popov Ghost and Path Integrals
Переглядів 1845 місяців тому
Faddeev Popov Ghost and Path Integrals
A Nice Operator Identity with Nested Commutators
Переглядів 695 місяців тому
A Nice Operator Identity with Nested Commutators
Representations of Bessel function of order zero
Переглядів 355 місяців тому
Representations of Bessel function of order zero
Synthesis of Non Gaussian Signals with a Prescribed Fatigue Damage Spectrum (Research on Vibrations)
Переглядів 505 місяців тому
Synthesis of Non Gaussian Signals with a Prescribed Fatigue Damage Spectrum (Research on Vibrations)
Engineering Graphical User Interface for analyzing random vibration signals
Переглядів 615 місяців тому
Engineering Graphical User Interface for analyzing random vibration signals
thanks for connecting lots of dots and making it a holistic explanation
how 23:36?
What that equation is telling you is that the number of particles between 𝑥 and 𝑥+𝑑𝑥 at a later time 𝑡+𝜏 depends on the displacement of neighboring particles from nearby positions at the earlier time 𝑡. Intuitively, as the displacement Δ becomes larger, the probability 𝜑(Δ) decreases, meaning that particles far from 𝑥 contribute less. Thus, the main contributions to the particle count at 𝑥 come from particles that were close to 𝑥 at time 𝑡.
Is there any book which I can refer too?
The equation seems general; what are the assumptions needed for the full expansion to converge?
convergence is a "tricky" concept when we think of path integrals. The "awesome" thing about the physics behind them is that, even if these expressions technically diverge, we can extract lots of useful information
Do you know of the Dyson-Schwinger equation?
yes, I will make a video about it soon.
Just a minor nitpick, it can get confusing if the integration variable and the integration limit use the same symbol (like "x"). I get that it works for the simplicity of writing. Love the video though.
Thanks it helped me a lot❤
wonderful explanation ! Where can i find the proof of hilbert transform for periodic function being and integral of co-tangent ?
Sir when you were able to understand all this during you academic journey... i was unaware of probability calculus till now, I am already close to finishing up my bachelors ?
“einstein derived lorentz” wait what???😮😮😮
Can I do it without complex analysis?
Some questions: 1. At 8:21, shouldn't there be a factor of 1/2? since the inner product is expanded into the sum of 2 terms 2. At 9:01, can the operator δ be treated as a constant with respect to the basis |ψ>? (since it escapes the bra-ket) 3. At 11:11, it seems like the LHS of the two equations stay the same but the RHS of the second equation got exponentiated? and I might be mistaken here, but shouldn't an integration remove all of the "δ" in the equation, including |ψ+δψ>?
1) I see what you mean. The way to interpret the action operator inside that inner product is this: the operator acts like a differential operator (on the right state, but also on the left state). Let me remind that this is a "heuristic" derivation, so the notation is not always rigorous here. Any books on QFT that deal with the Schwinger formalism should cover this more rigorously. 2) "delta" represents the variation of the state due to a small change of the field. In general, the state changes if we perturb the field, so I would not consider your interpretation correct. "delta" can act both on the "bra" and on the "ket". 3) yes, at 11:11 we have a simple exponentiation. The variable of integration on the left is the entire inner product between the "bra" and the "ket" you see in the expression. The result should be the logarithm as written in the video (the delta in the inner product does not disappear)
Another way to think about your first question. Consider the evolution of the change in the action between two states A and B. We can write the inner product: < B | δS | A >. Based upon the considerations in the video, δS should depend only on the final and initial states, so it could be written as: δS= S_B - S_A (where S_B is an operator that acts only on state B, and S_A acts only on state A). Let's assume C is another state (that you can think of as being "in between" A and B). You can write: δS = S_B - S_A = ( S_B - S_C ) + ( S_C-S_A ) Therefore: < B | δS | A > = < B | ( S_B - S_C ) | A > + < B | ( S_C - S_A ) | A > Now, those differences in the action are going to create some small perturbations. In particular, S_B - S_C will act on < B |, whereas ( S_C - S_A ) will act on | A >. This serves as another "justification" of the formula you were confused about.
I have sometimes wondered if you can write every law of physics (thermodynamics, fluid dynamics...) with differential forms.
Hey! Thank you so much for making this video. What I attempted was, I used MoC to solve for tau's PDE, and (while I don't 100% understand it yet), but realized that tau must be a function of C, and then I had a(C) just like in the paper. But then he says a = phi(v), and I'm "in general" lost about what's happening here. Secondly, I got stuck when I tried to substitute x' = x - vt back into xi, and (where gamma = lorentz factor), got a * 1/gamma^2 * (x - vt) and I got confused about what to do with the extra gamma. Then saw your video and you absorbed it into a(v), and here too I'm clueless. Can you please help /explain this? This is the only part in the derivation I'm not 100% clear on (perhaps my total lack of theoretical exploration in real analysis is the problem here), I barely managed to learn the characteristics method from youtube. Thank you so much :)
sir please recommended a book where i find this topic related examples
Holy crap. I understood your thumbnail! My flavor preferences are chocolate, vanilla, peppermint, asparagus, charm and pistachio. Up has a strange aftertaste but strange just tastes like marshmallows.
Thanks for the video. Can you please let us know a practical example how to use Reynolds transport theorem ?
In hydrodynamics you have a Eulerian picture and lagragian picture and the transport theorem is exactly the transformation between the operators between 2 pictures
More specifically if think f as density rho the theorem gives Lagrangian derivative of mass is in fact the conservation law of mass in eular picture which is the right hand side of the equation
Elegant
Thanks for this video!!!
Tricky equations 😅 but you did the job well
Wonder if the parameter nu in the constitutive relation can be linked to the conservation of mass
Yes, Poisson's ratio can indeed be linked to the concept of mass conservation. Mass conservation can be used to evaluate it. Mathematically, if a material is stretched along the x-axis, causing a strain 𝜖_𝑥, the resulting strains in the perpendicular y and z directions are given by: 𝜖_𝑦=𝜖_𝑧=−𝜈𝜖_𝑥 ν is Poisson's ratio, which is typically a positive number for most materials. Mass conservation implies that during the deformation of a material, the mass of the material remains constant. For a given volume element of the material, if we assume the material is incompressible, then its density remains constant, which implies that any increase in one dimension must be exactly compensated by a decrease in other dimensions to maintain the same volume. Consider a small cubic element of the material with an initial volume 𝑉0=𝐿𝑥 𝐿𝑦 𝐿𝑧, where 𝐿𝑥, 𝐿𝑦, and 𝐿𝑧 are the initial lengths along the 𝑥, 𝑦, and 𝑧,axes, respectively. After deformation, the new dimensions become 𝐿𝑥′=𝐿𝑥(1+𝜖_𝑥), 𝐿𝑦′=𝐿𝑦(1+𝜖_𝑦), and 𝐿𝑧′=𝐿𝑧(1+𝜖_𝑧) The new volume 𝑉′ after deformation is: 𝑉′=𝐿𝑥′𝐿𝑦′𝐿𝑧′=𝐿𝑥(1+𝜖_𝑥)⋅𝐿𝑦(1+𝜖_𝑦)⋅𝐿𝑧(1+𝜖_𝑧) For small strains, the volume can be approximated as: 𝑉′≈𝑉0(1+𝜖_𝑥+𝜖_𝑦+𝜖_𝑧) For mass conservation in the case of an incompressible material, the volume must remain constant, so 𝑉′=𝑉0 This implies: 𝜖_𝑥+𝜖_𝑦+𝜖_𝑧=0 Substituting the relationship between longitudinal and transverse strains (using Poisson's ratio): 𝜖_𝑥−𝜈𝜖_𝑥−𝜈𝜖_𝑥=0 This implies that for incompressible materials, Poisson's ratio 𝜈=1/2
@@math.physics thank you for the comprehensive explanation
Ah so the equation including W is for bending?
w(x,y) is the displacement of the generic point (x,y) of the plate in the direction z perpendicular to the plate. In the classical theory of thin plates, the derivatives of w are related to stresses, strains, bending moments, twisting moments, shear forces, etc.
🎉🎉 👏👏 🥇🏅
This equation needs alot of data 😅
Could you please explain to me why physicists before Einstein could not have arrived at the final formula?
Is it used for prediction model ?
There are several applications that demonstrate the versatility of the Black-Scholes model in the financial world, particularly in derivatives trading and risk management (I don't have much experience with that though)
nagyon jó
@@plranisch9509 köszi, azt hiszem, egy év alatt szépen fejlődtem (mármint ehhez a videóhoz képest) :)
Not following you. Where is the paper I see in the cover photo so I can just read it?
can u pls do a full review of the exam paper im in supirioe terza
Interesting problem
Bit tricky for me to understand but good explanation 🎉
How can we find length of a curve
Exuse me, I have one question. This is not really related to the topic of the video. Do you know how to plot magnetic field lines in the Schwarzschild metric in wolfram mathematica? Or do you know where i can read about this? You're my last hope :( i want to learn. Thank you in advance
🎉🎉 🏅🏅
Interesting problem
Thank you i'm watching all your videos.
The nitty grity is the most beautiful, yet doesn't appear so.
🎉🎉 🏅🏅
You are a genius, I mean it seriously, and thank you so much for your work. Really amazing and so cool to explain the Einstein theories!!!
very useful video, there's a lot of false information out there about this, falsely stating that kinetic energy in QM can be negative.
Ciao, fai anche ripetizioni? dove posso contattarti? Grazie
🎉🎉 🥇🥇🏆🏅 good content
Great proof. Please do more videos.
Really nice! The explanation is so clear and easy to understand.
I submit that what causes the two plates to come closer is Quantum space; specifically the gravity created by that space. Matter and quantum space repel each other. Around the earth the quantum space density is great and lessons by distance from the earth. The plates are pressed together by this dense quantum space. To prove my theory take the Casimir experiment to the ISS and do it where the quantum space, the pressure of it, gravity, is less. If I am correct the plates should not come together as readily since the pressure of the quantum space, gravity, is much less. Prove me wrong
Interesting problem 🎉🎉
how do you generalise to evaluate zeta(4) with integrate_0^1 1/(1-xyzw) dx dy dz dw --> Pi^4/90?
Interesting problem
I love your accent ❤❤❤
I like this proof. A smart little trick at the beginning and just raw dogging till the end. Euler would be proud.