Intuition Definition and 276 Threads

There are two theorems from multivariable calculus that is very important for manifold theory. The first is the inverse function theorem and the second is the "constant rank theorem". The latter states that (Constant rank theorem). If ##f : U\subset \mathbb{R}^n \to \mathbb{R}^m## has...

Homework Statement A particle is moving along the graph of y=sqrt(x). At what point on the curve are the x-coordinate and the y-coordinate of the particle changing at the same rate? Homework Equations y = sqrt(x) y' = 1/(2sqrt(x)) The Attempt at a Solution The solution to the...

I'm trying to understand why the voltage across parallel circuit elements is the same. I keep finding the answer "because they are both connected to the same voltage source." Why is it the case, in a physical sense, that voltage MUST be the same across, say, capacitors that are connected in...

So far I have learned about Coulomb's law, the electric field, gauss's law, the electric potential and now capacitance. I feel that although I "know of" these topics, I don't actually "flow with them". Ignoring the math for a second; I want to form an understanding. And I think calculating the...

I'm trying to build up some intuition with Gauss's law, calculating the flux through surfaces given certain charge configurations etc. For example, for any number of different charges enclosed in a box or sphere, if I move far away can I consider the object as a point source having the sum of...

The explanation below illustrates why I think the method of successive approximations is merely a sneaky way of working with power series when you're not formally allowed to use a Taylor series expansion for a function (i.e. when it doesn't exist, as in proving the existence theorem on ode's for...

I have been pondering about the concept of the warp drive and my intuition can't seem to understand the idea fully. I will try to exlain what i have found out so far. To keep everything simple and stripped down to only the bare essentials, let's discuss about it in one-dimentional universe...

I'm taking an undergraduate level GR course, and from my text (Lambourne), the author describes a geodesic as a curve that "always goes in the same direction", and says that the tangent vector to the curve at some point u+du (where u is the parameter variable from which all the vector components...

Just wanted to know if I'm approaching this problem correctly. 1. The Problem A. If the terms of a sequence of all positive terms go to zero, then the sequence must converge? True or false. Provide an example. B.If the terms of a series of all positive terms go to zero, then the series...

So I been working with quaternions as you all know. I get them basically, but to really understand their rotation properties i decided to better understand matrices and how they relate to real valued functions. a matrix is a transformation of a vector from one vector space to another through...

I feel like I don't pick up on things as quickly as my peers. I really need to slowly conceptually picture every concept or else I don't understand what we're talking about---and during lectures there's really no time to stop and visualize formulae as we go step by step through derivations. When...

I am studying scattering from these notes. There I came across Green's function in one dimension which is computed as \langle x|G_o|x'\rangle = -\frac{iM}{\hbar ^2k}\exp(ik|x-x'|) I understand Green's function as a sort of propagator from x' to x. There are two observations that can be made...

Suppose a particle is moving along the $x$ axis with velocity $v$. It starts at the point $-5$ and has constant negative acceleration $a$. You need to find what time the particle reaches the origin. My teacher solved this question, and used the kinematics equation Δx = v_0t+1/2 at^2...

Hi! I'm trying to get some intuition for the notion of the topology of a set. The definition of a topology ##\tau## on a set ##X## is that ##\tau## satisfies the following: - ##X## and ##Ø## are both elements of ##\tau##. - Any union of sets in ##\tau## are also in ##\tau##. - Any finite...

If we have two sequences and the ratio of their limit is greater than zero, why does this mean that they either both converge or diverge? I don't understand why the test works. Also, what about lim[(1/x)/(1/x^2)] = lim x = ∞? The series of 1/x^2 converges but series of 1/x diverges...

Hello! I'm finally starting to get a grip around quantum field theory. The last hang up is the following: I've been told that since we are quantizing a field, the field strength is the observable. Now analogous to QM we then define a field of hermitian operators, ##\phi(x)##, which give a...

In Sean Carroll's general relativity book he gives a requirement that two (differentiable) manifolds be the same manifold that there exist a diffeomorphism ##\phi## between them; i.e. a one-to-one, invertible and ##C^{\infty}## map. Now I wanted to get some intuition why this is the best...

Hi, This is silly, but I'm confused as to how we use the unit circle as a representation of particle states. I've been given the formula (probability distribution)=α|0> + β|1> (or in radians sin∅|0> + cos∅|1> or something), where the probability of a particle being in a certain state is the...

Why is the probability of a particle being at a certain location proportional to the square of of "the" amplitude? What does "the" in that sentence represent more specifically? Why are amplitudes complex numbers? Any intuition or clarification would be very appreciated. Thanks!

Hi there, I just started to learn about gradients. I can calculate them with ease; but I don't think I really understand them conceptually. I understand the usual example of the temperature scalar field where the temperature in a room is a function of your position T(x, y, z). But when it comes...

Hello, on the topic of torque, my textbook says it's a vector quantity with direction either outward, or inward (perpendicular to the page). Can someone explain WHY it is outward/inwards? I saw this animation, but it doesn't really provide intuition.

Hey everyone, I've been reading up a bit on control systems theory, and needed to brush up a bit on my Laplace transforms. I know how to transform and invert the transform for pretty much every reasonable function, I don't have any technical issue with that. My only problem is that some...

Homework Statement This isn't really homework, but I've been reviewing calc & trig and realized that the area of one period of sin(x) = 4. Since sin(θ) can be understood as the y-value of points along a unit circle, I noticed that the area of a unit square that bounds the unit circle is...

So heat capacity is the change in the mean value of the internal energy when the temperature is changed: C = d<E>/dt But I would like a little more intuition than that. T^(-1) = dS/d<E>, so going back to the intuition that the inverse of the temperature is a measure of how disordered the system...

I am using a computational electromagnetics software that solves for the resultant electric and magnetic fields around materials represented as lattices of discrete dipoles. Some of the results are the extinction, absorption, and scattering efficiency factors of EM waves, represented as a graph...

I am trying to understand what is meant when people say space itself is expanding. When I hear that, I imagine a 3-d Cartesian coordinate system in which every axis gets scaled at a given rate (like in an animation). What I don't understand is, if everything gets scaled, then how can...

Homework Statement F(x,y,z) = xyzi+xyj+x^2yzk Surface is the top and four sides of cube with vertices at <+/-1,+/-1,+/-1> Homework Equations ∫∫curlF * ds = ∫F*dr The Attempt at a Solution At Z = 1, I broke up the surface into 4 lines, parameterized them and combined...

Homework Statement The numbers aren't important because I'm after getting a more intuitive sense of the problem. When an elevator is accelerating upwards with a mass in it on a scale. Why is the apparent weight the normal force? Homework Equations F=ma W=mg The Attempt at a...

Can someone explain to me the mathematical intuition that motivates the embedding of quantum operators between the conjugate wave function and the (non-conjugated) wave function? That is, we write: \Psi^{*}\hat{H}\Psi, that is: \Psi^{*}(\hat{H}\Psi), so that \hat{H} operates on \Psi (not...

Homework Statement A force F=bx3 acts in the x direction, where the value of b is 3.7N/m3. How much work is done by this force in moving an object from x=0.00m to x=2.6m?Homework Equations W=F*D ∫undu = (u(n+1)/(n+1))+C The Attempt at a Solution I know from previous problems that I'm going to...

How should I think about symmetry currents?... in particular, when there are no fields to "carry the charge", eg in a pure Maxwell theory or, maybe, in a CFT of free scalars? Perhaps it would help if someone elucidated the connection between the "charge" in Noether's theorem and the "charge" in...

Hi, I finished reading about the Schmidt decomposition from Preskill's notes today. I understand and follow his derivation but it still seems completely non intuitive to me. We have \mid\psi\rangle_{AB}=\sum_{i,u}a_{iu}\mid i\rangle_{A}\mid u\rangle_{B}=\sum_{i}\mid...

http://en.wikipedia.org/wiki/Pappus's_centroid_theorem (the second one) When we do this with a torus, wouldn't we gaps tiny gaps between the discs on the outer edge? The slices are closer together on the inner edge and this prevents them from getting any closer on the outer edge so shouldn't...

If at very large scales of matter our intuitive logic fails to explain it (Black Holes) and at very small scales of matter untuition again fails (Quantum effects) then is there a connection between those two groups of phenomena?

In the following two problems I am trying to get a deeper intuition of, the plane has 3 variables and is 3 dimensional and the hyperplane has four variables and is 3 dimensional as well. Can someone please show me why, practically, in the context of these problems? Question with hyperplane...

why was laplace transform developed i have googled it and found that it is something about shaping a family of exponential and vector projections etc i couldn't get it. some simply said that it was used to make a linear differential equation to algebraic equation but i couldn't understand how...

Would anyone be kind enough shed some light on the physics Intuition of a vector in a linear space over a complex field for me? Furthermore, what does z^z mean?

I just had a discussion with someone who said he thought about quotient rings of polynomials as simply adjoining an element that is a root of the polynomial defining the ideal. For example, consider a field, F, and a polynomial, x-a, in F[x]. If we let (x-a) denote the ideal generated by x-a...

I have used the attached photo to show that light takes the path described by snells law such that the time it takes from point A to B is minimized. But conceptually there is something wrong for me in this derivation. Because why would the light ray coming in, know that there is a different...

Does the line of a force applied need to go through the centre of mass to cause translational acceleration? I have follow up questions regarding the answer to this simple question which I can't find the answer for anywhere

So here's the equation y(x,t) = A cos (kx - wt) where A is the amplitude of vibration and k = 2pi/lambda i have already dealt with SHM or SHO and i completely understand the equations like x = Acos(wt + phi) but i didnt really dealt with waves and the equation of traveling wave is kinda...

A funny thing happened to me recently... I solved a complicated math problem (for me anyway), and it was almost as if I had no idea of what I was doing. I just started writing... and it kind of came out, unconsciously... I think I know why Poincare said that intuition creates while logic...

Let's say you have a function that is continuous in (a,b] but discontinuous at x=a and you integrate it from a to b. For example, \int^{1}_{0} \frac{1}{\sqrt{x}}dx I understand that the integral exists, and it can be easily computed by using the limit as x approaches 0 from the positive...

I took Diffy Q about a year ago and that was the first math class that felt like a cookbook class. Ever since that class I feel like I've gotten worse and thinking less intuitive about math. I think I got discouraged because so many of those methods for solving in Diffy Q just popped up out of...

I have some problems understanding the intuition behind the integration by parts technique. I don't quite see why you solve for \int u(x)v\prime (x), instead of one of the other parts, what makes it easier to solve for that particular term? And in general when working with integration...

I don't understand the intuition/proof of why arc length = arc angle * radius. This may be partially because I don't fully understand the concept of radians, but anyhow please help.

I am not a mathematician, so I always used to wonder about following. "I never heard Einstein conducting any experiments, But how he predicted the physics properties the way it is explained in Relativity Theorems. These are some of the breakthroughs of that time, But without the intuition...

If the force applied on ab object due to gravity is -mg why is applying a force of mg give upward raise the object if the two forces are equal and opposite in direction. I would think the object would remain still until you apply a force greater than mg.

I was working on a problem with an incline plane and two masses connected by a pulley. One mass on the plane and the other hanging down the side. We had to first calculate the acceleration ignoring friction and then we were give a coefficient of kinetic friction and asked to recalculate...

Can anyone help me develop some insight as to what positive and negative voltage means? I've heard of the analogy to water flowing where amps are viewed as units of water flowing through a certain point per unit time and volts are viewed as how high the water is flowing downwards to the...