Intuition Definition and 276 Threads

What exactly are the theoretical motivations for considering space and time as a four dimensional continuum? Is it a natural consequence of requiring that the speed of light is independent of the frame of reference that it is measured in, since this implies that time and time are not absolute...

I am trying to explain to someone the formal notion of a limit of a function, however it has made me realize that I might have some faults in my own understanding. I will write down how I understand the subject and would very much appreciate if someone(s) can point out any...

I understand that the derivative of a function ##f## at a point ##x=x_{0}## is defined as the limit $$f'(x_{0})=\lim_{\Delta x\rightarrow 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$$ where ##\Delta x## is a small change in the argument ##x## as we "move" from ##x=x_{0}## to a neighbouring...

Homework Statement I did poorly on my exam, which I thought was very fair, and am now trying to understand certain aspects of perturbation theory. There are a total of three, semi related problems which i have questions about. They are mainly qualitative in nature and involve an intuitive...

Is there an intuitive way to understand why nature selects the path that minimizes the action? I've seen it proven that the Euler-Lagrange equations are equivalent to Newton's laws (at least in Cartesian coordinates). So I can understand it mathematically. But on a more common-sense level...

There's this problem that was given in my physics class.. "Three identical, negatively charged particles are situated at rest in a straight line, as shown below. All three are released simultaneously, and are free to move. Find the maximum velocities attained by particles A, B, and C. They have...

What is the intuition behind divergences for the sunset diagram? I know that there is quadratic divergence by why no quartic divergence or higher?

Apologies if this is a really trivial question, but I've never been quite sure as to the usage of the terminology dual space. I get that given a vector space ##V## we can construct a set of linear functionals that map ##V## into its underlying field and that these linear functionals themselves...

Hi , while trying to prove gauss law for electric field from gauss theorem i came up with this problem . as Gauss' differential law expresses ∇⋅E = ρ/εο what i get from that is ... the divergence of electric field which is the flux density is related to the charge density ...but i can't get...

Hello, I am having trouble understanding the answer to the above question. The answer is 6!/3! and below is my working. I am having trouble arriving at the above answer and here is my approach. The question is to find the permutations of KEEPER minus all the duplicates. Therefore, to count...

[I asked this question over a year ago, but I thought I'd try again.] Let ##I\subseteq \mathbb R## be an interval and ##f:I\to\mathbb R## be a ##C^\infty## function. I have the following characterizations: 1) ##f'\geq 0## everywhere iff ##f## is increasing. 2) ##f''\geq 0## everywhere iff...

Does anyone perhaps have a good way for me to get a lasting 'intuition' about what inverse hyperbolics are? I look at, for example, the well known sin x; it is periodic. Then, it seems, sinh x is a reflection of sin x about the line y=x. (I found an example at 7. The Inverse Trigonometric...

The title sort of says it all, but I'll clarify a bit. Is there any intuition for what Lagrangians are and what action is. I'm asking in all generality, not just for classical mechanics.

I've been grappling with the idea in my head as to how I would explain to someone exactly what equality between two mathematical objects actually means. This maybe a very stupid question, so apologies in advance, but if I'm honest I struggle to come up with an answer that doesn't involve using...

Hello, I hope this is the right forum section. I'm having trouble understanding how calculating the cross product arrives at the final result. When I do something simpler like multiplying a vector by a scalar, I can easily visualize in my head how each component "shrinks" or "grows". With the...

I realize there have been multiple threads on this and believe me I tried my hardest to find these answers from them and other resources. Questions: 1) Is the van de graaff generator a particle collider? I am under the impression that it is meant to follow up on the Cockroft-Walton accelerator...

So, I know that there are a lot of questions about good books on quantum mechanics and I have read each one of them, and I go on and bought Griffiths' Introduction to Quantum Mechanics. But the fact is that it did not build me a very good intuition as it emphasized the computational part of it...

In the book "Statistical physics for cosmic structures" at p. 171 a read a definition of scale invariance (leading to the so called scale invariant power spectrum) given as the requirement that ##\sigma^2_M(R=R_H(t)) = constant##, where ##R_H(t)## is the horizon, i.e. the maximal distance that...

Hi there, I'm trying to get a better intuitive handle on the concept of rest mass and rest energy - the energy term associated with rest mass. Introductory Physics textbooks often give statements along the lines of "mass is a form of energy" or "mass can be converted to energy" to explain...

I have not read any other QM books,i have little knowledge on that subject and want a books that uses mathematics in academic levels but is easy to get the grips on and also builds intuition and explains the phenomenons in a good manner.I do not want a book that emphasizes on mathematics or...

Im trying to get some intuition for convex neighbourhoods which is neighbourhoods ##U## such that for any two points ##p## and ##q## in U there exists a unique geodesic connecting ##p## and ##q## staying within ##U##. QUESTION 1: These kind of neighbourhoods can be shown to always exist for...

Hi, I am doing an introductory course on fluid mechanics, and I'd like some intuition (that's why I'm posting on the engineering forums and not on the math forums, even if I'm studying for a degree in math) about the concept of the total derivative and, particularly, its convective component...

Hi all, I've occasionly seen people multiply a matrix by its transpose, what is the use and intuition of the product? Any help appreciated.

Does anyone know of any mathematics books that are not textbooks? Something that has problems but is more focused on building mathematical intuition rather than just laying out formulas and what not. This is a pretty vague question/descriptor for a book, my apologies. If you need more...

The problem was present in a physics 1 exam, and I'm reasonably sure I know the answer, yet my friend contradicts me in my conclusion. Homework Statement Find magnitude of acceleration of system ABC. Masses of A, B and C are all equal and each has mass 2.00 kg. Let gravitational acceleration g...

Hey all, I've just started tensor analysis but do not understand why in contravarient uses 1 and covarient uses 2, could someone please explain these? Perhaps my understanding of the definitions is causing me to misunderstand why its written like this. Any help appreciated.

I am having trouble connecting some of the differential geometry in gr to what is actually measurable in the real world. As far as i understand we can measure physical quantities in terms of coordinates x^\mu on the tangent space of a 4-dim Pseudo-Riemannian manifold M. So let's say we are...

Background: I am an upper level undergraduate physics student who just completed a course in classical mechanics, concluding with Lagrangian Mechanics and Hamilton's Variational Principle. My professor gave a lecture on the material, and his explanation struck me as a truism. Essentially, he...

Hi there, I have a question about incompressible Stokes flow in a channel between solid walls (with no-slip boundary conditions at ##y = 0, L_y##). It is my intuition that, if the flow direction is ##x## (periodic), and the direction normal to the walls is ##y##, then there cannot be a net...

Hello brand new to the forums and I just have one question. So I decided to follow mechanical engineering as my degree of choice and I stand firm with my decision, I'm good at building things and finding unique solutions to problems, but the one thing I'm bad at is math... I'm a little above...

Homework Statement The assembly shown is used to control the tension T in a tape that passes around a frictionless spool at E. Collar C is welded to rods ABC and CDE. It can rotate about shaft FG but its motion along the shaft is prevented by a washer S. For the loading shown, determine (a) the...

Could someone explain to me the intuition behind RMS values? I understand how you calculate them; you take the average of the squares and square root them. I am just wondering why. For example when it comes to voltage, how did someone just think up of this magical way to compute an effective...

##dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy## I'm confused as to how the total derivative represents the total change in a function. My own interpretation, which I know is incorrect, is that ##\frac{\partial z}{\partial x} dx## represents change in the x...

The Friedmann equation states that $$(\frac{\dot a}{a}) = \frac{8\pi G}{3} \dot \rho + \frac{1}3 \Lambda - \frac{K}{a^2},$$ where ##a, \rho, \Lambda, K## respectively denotes the scale factor, matter density, cosmological constant and curvature. Now, I'm trying to get at an intuition on...

Hi! I try to get some intuitive understanding on the equipartition theorem stating that in thermal equilibrium, energy is evenly distributed among all degrees of freedom of a physical system. This is indeed intuitive for a system consisting of composite particles with translational and...

Throughout my life, most of the things I've learned have come naturally, and seem to commit to memory without the need for much effort. However now while I'm in the process of self-studying math and physics, i find that i constantly seem to forget everything i learn and can only make progress...

Homework Statement A block starts sliding on an inclined plane inclined at an angle θ with the horizontal. The first half of the plane is frictionless and the second half has a co-efficient of friction μ. When the block reaches the bottom of the slope, it has velocity zero. Calculate the...

I know not all shapes satisfy this relationship, but what is your intuition behind this formula that finds area (base*height=area). I think it would be best if we focus just on rectangles and squares since they seem to be the most elementary case. I can think of a proof dealing with integer...

Homework Statement I'm curious about how the trapezoidal rule is derived for approximating definite integrals. Homework Equations According to my calculus book the equation is $$h[(1/2)y_{0} + y_{1} + y_{2} + ... +y_{n-1} + (1/2)y_{n}]$$ The Attempt at a Solution I'm curious as to why the...

Hi there, I'm having trouble understanding the Fourier transform of a function where the result in the frequency domain has imaginary components. For example, if you take the Fourier transform of Sin[t] , the result is I Sqrt[\[Pi]/2] DiracDelta[-1 + \[Omega]] - I Sqrt[\[Pi]/2]...

We can get a lot of good intuition for how first and second derivatives work by interpreting a sign restriction. Let ##I\subseteq \mathbb R## be an interval and ##f:I\to\mathbb R##. 1) If ##f## is differentiable, then ##f## is monotone iff ##f'\geq 0## everywhere. 2) If ##f## is twice...

Dear PF, although I've gone through many particle phyics lectures and textbooks, I still have problems with wrap my mind around the whole scattering theory and cross section topics. 1. Is there a deep reason why cross sections for charged, point-like particles decrease with the center-of-mass...

Could anybody explain what divisors and the Riemann-Roch theorem are intuitively, motivating them, without any jargon or vagueries (i.e. using actual math), and preferably offering a nice example necessitating this stuff? I'm sure there is a nice way to explain it in an absolutely natural...

The intuitive picture I have of giving a set a topology, is that of giving it a shape in the sense of connecting the points and determining what points lie next to each other (continuity), the numbers of holes of the shape, and what parts of it are connected to what. However, the most...

Hello! How can I justify that the infinite series 1 - 1 + 1 - 1 + 1 - 1... is divergent? If I were to look at this, I see every two terms canceling out and thus, and assume that it is convergent since the sum doesn't blow up. That's what my intuition would tell me. I know I can use...

Hello ! Could you please give me some kind of intuition of the physical meaning of the second spatial derivative ? I see it all the time, but I have difficulty comprehending it to the same level I have done with the second time derivative. Thanks !

hey pf! i have a few question about the physical intuition for divergence, gradient, and curl. before asking, i'll define these as i have seen them (an intuitive definition). \text{Divergence} \:\: \nabla \cdot \vec{v} \equiv \lim_{V \to 0} \frac{1}{V} \oint_A \hat{n} \cdot \vec{v} da...

hello pf! i am wondering if anyone here knows of a geometric, intuitive explanation for the laplace transform? if so, please direct me to the source of if you could, explain to me your understanding? thanks!

For the following circuit: How would you deduce the number of currents there are? I am having trouble predicting how many are there (i am convinced there are 3). Also, in general, what are some tips for finding how many currents there are?

I am reading up on principal bundles and currently I'm trying to get to grips with the definition of a connection on such a space. The definition is as follows: A connection on P is a unique separation of the tangent space ##T_uP## into the vertical subspace ##V_u P## and the horizontal...