Intuition Definition and 273 Threads

Let ##p: X-->Y## be a cover map. Then the induced homomorphism will inject the fundamental group of X into the fundamental group of Y; furthermore, the image of the fundamental group of X under p will be a subgroup of the fundamental group of Y I read the proofs, but I'd like to have an...

I may be doing Algebra I tutoring for high school students soon. What are some good resources for exercises and intuitive/novel explanations for topics some students find sticky, etc.? One resource I'm sure I'll be using is the Schaum's Outline of Elementary Algebra, 3ed. What I'd also like is...

Homework Statement So my problem is mainly intuitive one, in that this *feels* wrong, and am mostly looking for insight. If we have uniform 1D motion of a particle along ##x## with constant velocity ##v##, what is the rate of change (first derivative with respect to time) of the variable...

This isn't for math homework. I'm in self study and came across something in my book that I'm seeking clarification for. The equation: $$0.3\left(50-x\right)=6$$ The solution: $$3\left(50-x\right)=60$$ $$150-3x=60$$ $$-3x=-90$$ $$x=30$$ Simple enough. My question is in regards to this: The...

If $$\phi(t,x)$$ is a solution to the one dimensional wave equation and if the initial conditions $$\phi(0,x) , \phi_t(0,x)$$ are given, D'Alembert's Formula gives $$\phi(t,x)= \frac 12[ \phi(0,x-ct)+ \phi(0,x+ct) ]+ \frac1{2c} \int_{x-ct}^{x+ct} \phi_t(0,y)dy . \tag{1}$$ which is...

I am having conceptual issues about what the CMB actually is. I read that it is the remnant radiation at the moment of recombination roughly 380,000 years after the Big Bang. But what about this statement implies that we would be able to observe this radiation today? To put it naively, when I...

As I understand it, in the context of cosmological perturbation theory, one expands the metric tensor around a background metric (in this case Minkowski spacetime) as $$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}$$ where ##h_{\mu\nu}## is a metric tensor and ##\kappa <<1##. My question is, how...

Hey, I am struggling with developing an intuition behind 'Affine Subspaces'. So far I have read the theories concerning Affine Subspaces delivered by the course book and visited several websites, however none have made it 100% clear. I feel like I have some sort of intuition for it, but I fail...

For finding the inverse of a matrix A, we convert the expression A = I A (where I is identity matrix), such that we get I = B A ( here B is inverse of matrix A) by employing elementary row or column operations. But why do these operations work? Why does changing elements of a complete row by...

When a particle and its corresponding anti-particle interact do they always annihilate or are there other possible interactions that can occur, such as them scattering off of one another? If the former is true, why do they always annihilate? If the latter is true, is it the case that the most...

What is the intuitive reasoning for requiring that a Lagrangian describing a free-field contains terms that are at most quadratic in the field? Is it simply because this ensures that the EOM for the field are linear and hence the solutions satisfy the superposition principle implying (at least...

In field theory a typical Lagrangian (density) for a "free (scalar) field" ##\phi(x)## is of the form $$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi -\frac{1}{2}m^{2}\phi^{2}$$ where ##m## is a parameter that we identify with the mass of the field ##\phi(x)##. My question is...

As far as I understand it, the Lorentz factor ##\gamma(\mathbf{v})## is constant when one transforms between two inertial reference frames, since the relative velocity ##\mathbf{v}## between them is constant. However, I'm slightly confused when one considers four acceleration. What is the...

Basically a weight moves from starting point on a ramp down to the bottom of the ramp. The ramp is on wheels so the ramp will also move. The formula is based on the center of mass of the combined object. Center of mass does not move. XA0 = position of small mass before it slides down ramp XB0 =...

All, I am looking for a logical step that I must be missing in order to understand how heat energy is produced in an exothermic reaction. All of the standard explanations--more heat is given off than is taken in--don't seem satisfying. If heat is the result of the kinetic energy of molecules...

I've been reading through a derivation of the LSZ reduction formula and I'm slightly confused about the arguments made about the assumptions: $$\langle\Omega\vert\phi(x)\vert\Omega\rangle =0\\ \langle\mathbf{k}\vert\phi(x)\vert\Omega\rangle =e^{ik\cdot x}$$ For both assumptions the author first...

The mantra in theoretical physics is that global gauge transformations are physical symmetries of a theory, whereas local gauge transformations are simply redundancies (representing redundant degrees of freedom (dof)) of a theory. My question is, what distinguishes them (other than being...

Could somebody write me the intuition behind the concept of "Integral Over"? Please do not write me its formal definition, I can easily get it from textbook. What I am also looking for is its motivation behind it. Please give me also examples. For your convenience, the formal definition...

I have a few conceptual issues following a standard thought experiment to argue why light bends in a gravitational field and I'm hoping I can clear them up here. Consider an observer in a lift in free-fall in a uniform gravitational field and an observer at rest in the uniform gravitational...

I am trying to better understand likelihood ratio test and have found a few helpful resources that explicitly solve problems, but was just curious if you have any more to recommend. Links that perhaps work out full problems and also nicely explain the theory. Similar links you have found...

I thought I understood the concept of a correlation function, but I having some doubts. What exactly does a correlation function quantify and furthermore, what is a correlation length. As far as I understand, a correlation between two variables ##X## and ##Y## quantifies how much the two...

As I understand it, a curve ##x^{\mu}(\lambda)## (parametrised by some parameter ##\lambda##) connecting two spacetime events is a geodesic if it is locally the shortest path between the two events. It can be found by minimising the spacetime distance between these two events...

I read an intuitive approach on this website. You should read it, it's worth it: https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ I read that an imaginary exponent continuously rotates us perpendicularly, therefore, a circle is traced and we end up on -1 after...

I've been reading about canonical transformations in Hamiltonian mechanics and I'm a bit confused about the following: The author considers a canonical transformation $$q\quad\rightarrow\quad Q\quad ,\quad p\quad\rightarrow\quad P$$ generated by some function ##G##. He then considers the case...

As I understand it, the proper length, ##L## of an object is equal to the length of the space-like interval between the two space-time points labelling its endpoints, i.e. (in terms of the corresponding differentials) $$dL=\sqrt{ds^{2}}$$ (using the "mostly plus" signature). Furthermore, this is...

In particular how does matter "clump" together to form stars and planets, and how do Galaxy/star systems form? For the latter question is the answer simply that near massive enough bodies, the spacetime curvature is significant enough that the geodesics within its vicinity are closed curves...

As I understand it, the need for quantum field theory (QFT) arises due to the incompatibility between special relativity (SR) and "ordinary" quantum mechanics (QM). By this, I mean that "ordinary" QM has no mechanism to handle systems of varying number of particles, however, special relativity...

I was asked by a friend to explain why the frequency, ##f## and period, ##T## of a wave. The initial explanation I gave to them was as follows: Heuristically, the period of a wave is defined as ##T=\frac{\text{number of units time}}{\text{cycle}}##, and its frequency as ##f=\frac{\text{number...

Consider the case in which an incoming particle collides with stationary target particle producing new particles through the interaction. For example, $$e^{-}+e^{+}\rightarrow X+\bar{X}$$ My question is, why in general do the particles produced in such an interaction propagate outwards are...

I have been reading through "Complex Analysis for Mathematics & Engineering" by J. Matthews and R.Howell, and I'm a bit confused about the way in which they have parametrised the opposite orientation of a contour ##\mathcal{C}##. Using their notation, consider a contour ##\mathcal{C}## with...

In some elementary introductions to integration I have seen the Riemann integral defined in terms of the limit of the following sum $$\int_{a}^{b}f(x)dx:=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x^{\ast}_{i})\Delta x$$ where the interval ##[a,b]## has been partitioned such that...

Is there some way to - from an intuition standpoint - justify the fact that there should be a factor of ##a^{6}##, (where ##a ## is the particle diameter) in the Rayleigh Scattering formula? I've seen a few sources hint that there should be. I can follow the derivation from e.g a Lorentz atom...

In literature I have read it is said that the Hamiltonian ##H## is the generator of time translations. Why is this the case? Where does this statement derive from? Does it follow from the observation that, for a given function ##F(q,p)##, $$\frac{dF}{dt}=\lbrace F,H\rbrace +\frac{\partial...

Having read several introductory notes on Gauss's law, I have found it very frustrating that when the author comes to discussing the standard examples, in which one considers symmetric charge distributions, they do not explicitly discuss the symmetries of the situation, simply stating that, "by...

Maxwell's equations reveal an interdependency between electric and magnetic fields, inasmuch as a time varying magnetic field generates a rotating electric field and vice versa. Furthermore, the equations predict that even in the absence of any sources one can have self propagating electric and...

What is the intuition for why the frequency of light does not change as it passes from a less dense medium to a denser one (or vice versa)? Classically, if we treat light in terms of waves, then intuitively, is the reason why the frequency does not change because it is determined by the...

Hi, I'm looking at the following graph, but there are a few things I don't get. For instance: curl should always be zero in circles where the field lines are totally straight (right-most figure) curl should always be non-zero in circles where the field lines are rotating (center figure in 2nd...

Consider the 2-point correlator of a real scalar field ##\hat{\phi}(t,\mathbf{x})##, $$\langle\hat{\phi}(t,\mathbf{x})\hat{\phi}(t,\mathbf{y})\rangle$$ How does one interpret this quantity physically? Is it quantifying the probability amplitude for a particle to be created at space-time point...

In the canonical quantisation of a free scalar field ##\phi## one typical constructs a mode expansion of the corresponding field operator ##\hat{\phi}## as a solution to the Klein-Gordon equation...

I've been asked by someone with minimal background in physics to explain what vector and scalar quantities are and give examples. Here are my thoughts: A scalar is a quantity that has a magnitude only, it is completely specified by a single number. Importantly, it has no directional dependence...

This question really pertains to motivating why vectors have components whereas scalar functions do not, and why the components of a given vector transform under a coordinate transformation/ change of basis, while scalar functions transform trivially (i.e. ##\phi'(x')=\phi(x)##). In my more...

Apologies for a possibly very basic question. I was recently asked by someone to explain to them the answer to the following classical mechanics (uniform circular motion) problem: Consider two discs of different radii, attached to one another such that they are concentric, and each containing a...

I had someone ask me how rockets are able to accelerate in space and my initial answer was that the rocket fuel combusts and is heated into an energetic gas, the gas is accelerated out of the back of the rocket (i.e. the rocket exerts a force on the gas), then according to Newton's 3rd law, the...

While I can often come up with ideas pretty quickly, I have trouble keeping myself focussed at tasks that just 'have to be done'. I'm talking for example about doing calculations for which you already know what the result must be or for which you don't expect there will be anything 'cool' about...

In classical mechanics we use a 6n-dimensional phase space, itself a vector space, to describe the state of a given system at anyone point in time, with the evolution of the state of a system being described in terms of a trajectory through the corresponding phase space. However, in quantum...

I understand that if we have a quantum mechanical system, then its state at some given time ##t## is fully described by a state vector ##\lvert\psi(t)\rangle## in a corresponding Hilbert space. This state vector containing all possible information about the distributions (of all possible values)...

Hey, So I've taken a course learning theory to mosfets, Jfets, transistors etc. Recently we did an amplifier project with a summing component as a final project. I chose which component pretty randomly. I'm a little misled as to which component is better in certain circumstances. BJT's are...

In elementary calculus (and often in courses beyond) we are taught that a differential of a function, ##df## quantifies an infinitesimal change in that function. However, the notion of an infinitesimal is not well-defined and is nonsensical (I mean, one cannot define it in terms of a limit, and...

Consider a two-point function $$f(\mathbf{r}_{1},\mathbf{r}_{2})$$ If one requires homogeneity, then this implies that for a constant vector ##\mathbf{a}## we must have $$f(\mathbf{r}_{1},\mathbf{r}_{2})=f(\mathbf{r}_{1}+\mathbf{a},\mathbf{r}_{2}+\mathbf{a})$$ How does one show that if this is...

I know how to implement Lagrangian mechanics at a mathematical level and also know that it follows the approach of calculus of variations (i.e. optimisation of functionals, finding their stationary values etc.), however, I'm unsure whether I've grasped the physical intuition behind the...