Forms Definition and 482 Threads

I am very confused about that in some literature the Maurer Cartan forms for a matrix group is written as ##{\omega _g} = {g^{ - 1}}dg## what is ##dg## here? can anyone give an example explicitly? My best guess is ## dg = \left( {\begin{array}{*{20}{c}} {d{x^{11}}}& \ldots &{d{x^{1m}}}\\...

Homework Statement Are the Gibbs and Boltzmann entropies always equivalent? Homework Equations $$ S=k_{B}ln\Omega $$ [Boltzmann entropy, where ##\Omega## is the number of available microstates $$ S=-k_{B}\sum_{i}p_{i} ln(p_{i}) $$ [Gibbs entropy, where ##p_{i}## is the probability of a...

Hello, I would like to know if anybody here has used the book "Differential Forms and the Geometry of General Relativity" by Tevian Dray and how they found it. Thanks!

I have a certain set of problems (i.e. https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/limcondirectory/LimitConstant.html), where many problems are in an indeterminate form ($\frac{0}{0}$) but if we apply L'Hopital's rule it yields an incorrect answer. Instead, I have to simplify the...

If there is enough gravity and the universe starts to collapse in on itself, would life forms that live during this phase of the universe get younger as time passes and they would need to expend energy to stay old?

so the solar wind is the result of resonance of orbiting planets, and solar cycles too, after this paper [Link to crackpot paper deleted]

No question this time. Just a simple THANK YOU For almost two years years now, I have been struggling to learn: differential forms, exterior algebra, calculus on manifolds, Lie Algebra, Lie Groups. My math background was very deficient: I am a 55 year old retired (a good life) professor of...

Hello, As you might discern from previous posts, I have been teaching myself: Calculus on manifolds Differential forms Lie Algebra, Group Push forward, pull back. I am an engineer approaching this late in life and with a deficient background in math. It is all coming together and I almost...

Homework Statement Taking derivatives of generating functions is another useful operation. This is done termwise, that is, if F(x) = f0 + f1x + f2x2 + f3x3 + · · · , then F' (x) ::= f1 + 2f2x + 3f3x2 + · · · . For example, ##\frac{1}{(1-x)^2} = (\frac{1}{(1-x)})'= 1 + 2x + 3x^2 +· · · ##...

Hi; I've been doing some research for my master's on gauge theory in the language of fiber bundles, and something occurred to me. Both GR and the Standard Model (SM) can be described in terms of connections (potentials) and curvatures (field strengths), but there's this generalization of GR...

Hi. I'm trying to self-study differential geometry and have come across interior products of vectors and differential forms. I will use brackets to show the interior product and I would just like to check I am understanding something correctly. Do I need to manipulate the differential form to...

Homework Statement Write down an orthonormal basis of 1 forms for the rotating C-metric [/B] Use the result to find the corresponding dual basis of vectorsSee attached file for metric and appropriate equations The two equations on the left are for our vectors. the equations on the right...

I don't understand why we can write the elements of the lorentransformation in the form ## {\Lambda}^{\mu}\:_{\nu} = [exp(-\frac{i}{2}{\omega}^{\rho\sigma}M_{\rho\sigma})]^{\mu}\:_{\nu} ## I know that we can write it in the form ## {\Lambda} = exp(t\Theta) ## where ## \Theta ## are elements...

I know that energy is recognized through motion. Even in the mass-energy equivalence a velocity is present even though it is a rest-energy (Not really sure if this would count as a potential energy since there is no 'field' of acceleration that the mass is in) So does kinetic and potential...

Homework Statement Find the minimum value of ## x_1^2+x_2^2+x_3^2## subject to the constraint: ## q(x_1,x_2,x_3)=7x_1^2+3x_2^2+7x_3^2+2x_1x_2+4x_2x_3=1 ## Homework EquationsThe Attempt at a Solution I am not really sure how to think about it. I have seen the opposite way but have not seen this...

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... I am currently focussed on Chapter 2: Linear Algebra Essentials ... and in particular I am studying Section 2.8 The Dual of A Vector Space, Forms and Pullbacks ... I need help with...

What do Physics Forums members regard as the best first introduction to differential forms ...?

Can anyone suggest a good text or a good online set of notes from which to make a first approach to the topic of differential forms ... ? Similarly a first approach to to tensors ... ? The thought is to use these notions in order to gain an understanding of differential geometry and ... later...

Is it true that every limit that takes on an indeterminate form can be evaluated? Is it proper to say that a limit problem has a solution if the limit does not exist?

Homework Statement Given A = [2, 9, 8], B = [6, 4, −2] and C = [7, 15, 7], show that AB and AC are perpendicular, then find D so that ABCD forms a rectangle. Homework Equations Dot Product The Attempt at a Solution The vector AB = B - A = [4,-5,-10] The vector AC = C - A = [5,6,-1] AB⋅AC...

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I need help with Exercise 1.2.9 (a) ... Exercise 1.2.9 (a) reads as follows:https://www.physicsforums.com/attachments/5101I am somewhat overwhelmed by this exercise ... can someone...

Homework Statement The angular velocity is ω, R is the radius of the vessel. at rest the water has depth H. The face of the water form a paraboloid y=Ax2. find R for which the maximum height h of the water above the bottom doesn't depend on ω. Homework Equations Centripetal force: ##F=m...

I was doing some course work earlier today and noticed that I've seen two different equations for a line in 3D space. Usually the equation I use is: \vec r(t)=<x_{0}, y_{0}, z_{0}> + t<x, y, z> You plug in the various points with what the problem provides. However, a few times I have seen a...

Hi all, I have a non-economic background and I am currently interested in utility functions and how they are modeled in economics. Some internet research and I found out that there are obviously different ways of model utility. In my own studies of related topics I encountered the following two...

Hi I am trying to program excel to take the DFT of a signal, then bring it back to the time domain after a low pass filter. I have a code that can handle simple data for example t = [ 0, 1, 2, 3] y = [2, 3, -1, 4] So I think everything is great and so I plug in my real signal and things go off...

ok energy is supposed to change its nature as its used, say electricity. Yet there is rules in electrical work that call this degradation. I.E. the 3 foot drop rule for figuring out size of wire to allow current flow. Now if this is true and you can't recycle energy how then can we say that it...

What is the difference between 1) ΔG°' and ΔG° and 2) ΔG°' and ΔG ? ΔG° I think it is the gibbs free energy at standard state means at room temperature 1 atm pressure.

this starts as a calculus question, but springs into where i can get help with david bachman's A GEOMETRIC APPROACH TO DIFFERENTIAL FORMS second edition. looking at paul's notes cheat sheets http://tutorial.math.lamar.edu/cheat_table.aspx we have## \int \frac{1}{\sqrt{a^{2}-x^{2}}} =...

I heard that proportionality of kinetic energy with square of velocity, ##E_k\propto v^2##, can be derived with help of quadratic forms. It goes like: we guess that ##E_k\propto v^2## and we assume that momentum ##p\propto v##, then equation is valid in another inertial system. And so on. The...

The usual form for tension as a result of the symmetric Cauchy stress tensor is, $$\mathbf{t} = P \mathbf{\hat{n}}$$ or better $$t_i = {P_i}^j n_j$$ Buoyancy would be $$T = \int_{\partial V}{P_i}^j n_j da$$ integrated over a closed surface. I've assumed that the stress tensor ##P##, is, in...

I'd like to understand if non-atomic life forms are possible. For example, purely electrical ones. I'm not talking about those electron eating bacterias, they are still made out of atoms. Is there a possibility to form a stable structure out of electrons, or photons? Not necessarily in a vacuum...

What I understood so far is that whole universe was born from a single point at which there was no time (or all the time was at single point - no present, no future) and then a big bang happened (may be), creating time, energy, mass etc. I want to learn physics at much deeper levels. So, I...

Hi, I'm trying this summer to finish my mathematical methods book. I'm investigating right now the chapter of complex numbers, the end of the chapter has some applications in electricity and how can complex numbers make the work easier. The problem is that I didn't found it easier nor...

I want to make sure I understand the conditions required for L'Hospital's Rule to work. $$\lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}$$ If ##\lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}## exists. Should ##f## and ##g## be differentiable at ##a##? Or just...

Here's a list of all the indeterminate forms I'm familiar with: ##\frac{0}{0}##, ##\frac{\infty}{\infty}##, ##0⋅\infty##, ##\infty - \infty##, ##0^0##, ##1^{\infty}##, ##\infty^0## Suppose we want to evaluate the limit: $$\lim_{x→0} x^2 \cos{\frac{1}{x}}$$ We can find the value of this limit by...

Hello I am a mechanical engineer who is teaching himself the math of exterior algebra and differential forms. It is not easy for me and I have had many SIMPLE stumbling blocks due to my not respecting algebra. May I ask for help on some simple aspects? (Please be patient with me.) My...

Please refer to p. 99 and 100 of Rovelli’s Quantum Gravity book (here). I wonder what is the signification of the “naturalness” of the definition of ##\theta_0=p_idq^i##? If I take ##\theta_0'=q^idp_i## inverting the roles of the canonical variables and have the symplectic 2-forms of the...

If we have,$$A=d[(\bar{\alpha}-\alpha)(dt+\lambda)]$$ where $$\alpha$$ is a complex function and $$\lambda$$ is a 1-form. t here represents the time coordinate. If we want to calculate $$d\star A=0$$ where $$\star$$ is hodge star, we get if I did my calculations correctly...

Homework Statement Two converging lenses having focal lengths of f1 = 12.5 cm and f2 = 19.5 cm are placed a distance d = 49.0 cm apart as shown in the figure below. The image due to light passing through both lenses is to be located between the lenses at the position x = 33.0 cm indicated. (a)...

Homework Statement Find the value of the 2-form dxdy+3dxdz on the oriented triangle with (0,0,0) (1,2,3) (1,4,0) in that order. Homework EquationsThe Attempt at a Solution I have tried various subtraction of these coordinates and applying them to the formula but the answer is in the back of...

Homework Statement Hi my question is whether an SOP = POS for a given function F(A, B, C, D). Or is the SOP a complement of POS?Tasked to convert derive a simplfied SOP from a 4 var Kmap with don't cares. My answer was, F = C'.D' + A'.B'.C which is correct. Then for part B they asked for the...

The flat-space source-free Maxwell equations can be written in terms of differential forms as $$d F = 0; \ \ d \star F = 0.$$ And in the theory of gauge fields, one can introduce a connection one-from A from which one can formulate general Maxwell equations (for Yang-Mills fields) by $$ dF + A...

I have been reading a lot about Differential Forms lately because its so sexy. I have a pretty good grasp of how wedge product, hodge star, and differential operator "d" work, and their application to physics (it took me some time to see how d*F=J). I want to continue reading about it because...

Homework Statement For a chemistry lab, we're trying to find the amount of CaCO3 in an antacid table by measuring the change in pressure due to the reaction of CaCO3 with HCl to produce CO2. This question in the lab report is confusing me to no end... Let’s say the % concentrations of CaCO3...

Homework Statement Why does Asin(kx)sin(wt) also represent a standing wave? Which two interfering waves may superpose to make it? Acos(kx+wt) and Acos(kx-wt) could if we were subtracting them, but we're adding so that doesn't make much sense? Also, is there something like a phase shift in the...

What is the right hand and left hand limit of (sinx)^tanx? I know this is an indeterminate form 0^0 but what are the RHL and LHL because though I intuitively know that 0^0 is indeterminate, I don't understand what the right hand and left hand limits are?

Hello every one . Can someone please prove me the TORSION form and the CURVATURE form ( Maurer-Cartan form) in details, and what is the canonical solder 1-form?!

Homework Statement Now consider a 2-torus ## S_1 × S_1## and a coordinate patch with coordinates ## (\alpha_1, \alpha_2)## such that ## 0 < \alpha_i < 2 \pi##. Let us introduce in this patch a 1-form of the type: $$\omega = (A + B\alpha_2 + C sin(\alpha_2 ) + D cos(2\alpha_1 + \alpha_2...

I have noticed the following 2 different forms for the Sturm-Liouville equation online, in different texts, and in lectures. [p(x) y']'+q(x)y+\lambda r(x) y = 0 -[p(x) y']'+q(x)y+\lambda r(x) y = 0 Does it make a difference? I am guessing not as the negative can just be absorbed into...

Hello I'm trying to work through a see-saw model derivation and I've become a bit stuck. I've tried lots of sources but the difference in conventions doesn't fill me with confidence when combining these sources. I need to get from ## \overline{ \nu_L^c } \nu_R^c + h.c ## to ## \overline{...