Symmetric Definition and 566 Threads

Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are intricately related, and hence are discussed together in this article.
Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music.This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art and music.
The opposite of symmetry is asymmetry, which refers to the absence or a violation of symmetry.

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  1. K

    MHB Derivatives of symmetric expressions

    So I was bored in math class and came up with this series of related questions, that I cannot answer: Is there a clean expression for $f'(x),$ where $$f(x)=\prod_{i=1}^{n}\dfrac{(x-i)}{(x+i)}?$$ What about for $f''(x)?$ Or for $$f(x)=\prod_{i=1}^{n}\dfrac{(x^2-i)}{(x^2+i)}?$$
  2. binbagsss

    Deriving Max Symmetric Space from Spherically Symmetric - Sean M. Carroll

    I'm looking at Lecture Notes on General Relativity, Sean M. Carroll, deriving the FRW metric, and I'm a little confused with the use of eq 8.4 . I thought instead it should be using the general form of a spherically symmetric metric, not a vacuum spherical symmetric - eq 7.13 has been derived...
  3. P

    Image of a Matrix and symmetric matrix

    Hi, Well I hope it's not a thread that already is in the storage here. I want to understand the image of a matrix. not only calculating it but also why I'm doing that. Here are my questions: 1) They say d = Lx has a solution if d ∈ ImL. I know that the image of a matrix is calculated by Lx = x...
  4. andyrk

    F(x) symmetric about the line x=2

    Why is the function: y = f(x) = a(x-1)(x-2)(x-3) symmetrical about the line x = 2? I mean how can we be sure that it is? Is there any method to check it?
  5. binbagsss

    Birkhoff's Theorem: Spherically Symmetric Vacuum Solution Static or Stationary?

    In the text I'm looking at, the Schwarzschild metric derivation, and it argues to the form ## ds^{2}= -e^{2\alpha(r)} dt^{2} + e^{\beta(r)}+r^{2}d\Omega^{2} ## [1]. Up to this point some of the ##R_{uv}=0## components have been used, not all. It then says we have proven any spherically symmetric...
  6. binbagsss

    Schwarzchild metric spherically symmetric space or s-t?

    This is probably a stupid question, but, is the Schwarzschild metric spherically symmetric just with respect to space or space-time? Looking at the derivation, my thoughts are that it is just wrt space because the derivation is use of 3 space-like Killing vectors , these describe 2-spheres, and...
  7. G

    Is Nature Symmetric? A Deeper Look

    I guess that nature do not care about "directions". If I'm performing an experiment on (for example) light the result is invariant on the direction of the laboratory. So why we observe directionality on the nature that we are observing around us? Why "things" are definitively not symmetric...
  8. U

    Anti-symmetric and Symmetric Helium

    Homework Statement (a) Find the spatial wavefunction (b)Show anti-symmetric wavefunctions have larger mean spacing (c) Discuss the importance of this (d)Determine the total orbital angular momentum (e)Hence find the ground state term for Z=15[/B] Homework EquationsThe Attempt at a Solution...
  9. O

    Symmetric Equation of Line in 3D converts to 2 Planes or 2 Lines?

    Homework Statement Hi, An equation of the form Ax + By + C = 0 is a standard equation of a line in 2D. An equation of the form Ax + By + Cz + D = 0 is an equation of a plane. Is it possible to: Describe a plane in space, written in standard form, such that one variable is missing from the...
  10. H

    Transitive subgroup of the symmetric group

    Hi, I need help in proving the following statement: An abelian,transitive subgroup of the symmetric group Sn is cyclic,generated by an n-cycle. Thank's in advance
  11. R

    How to turn these symmetric equations into the general form?

    I was solving this problem and I didn't want to do it the really long way by finding the equation of B(t) by first finding T(t) and N(t). So i took the cross product of r' and r'' so that they would be in the direction of B. Found the parametric equation of the plane but the book answer was in...
  12. Y

    MHB Symmetric and anti-symmetric matrices

    Hello all, I have 3 matrices, A - symmetric, B - anti symmetric, and P - any matrix All matrices are of order nXn and are not the 0 matrix I need to tell if the following matrices are symmetric or anti symmetric: 1) 5AB-5BA 2) 4B^3 3) A(P^t)(A^t) 4) (A+B)^2 5) BAB How would you approach...
  13. A

    Comp Sci Eigenvalues and eigenvectors of a real symmetric matrix in Fortran

    Homework Statement I try to run this program, but there are still some errors, please help me to solve this problems Homework EquationsThe Attempt at a Solution Program Main !==================================================================== ! eigenvalues and eigenvectors of a real...
  14. TheFerruccio

    Integrating until symmetric bilinear form

    Homework Statement I am looking for some quick methods to integrate while leaving each step in its vector form without drilling down into component-wise integration, and I am wondering whether it is possible here. Suppose I have a square domain over which I am integrating two functions w and...
  15. eseefreak

    Reflexive, Symmetric, Transitive - Prove related problem

    Homework Statement Let A=RxR=the set of all ordered pairs (x,y), where x and y are real numbers. Define relation P on A as follows: For all (x,y) and (z,w) in A, (x,y)P(z,w) iff x-y=z-w Homework Equations R is reflexive if, and only if, for all x ∈ A,x R x. R is symmetric if, and only if, for...
  16. evinda

    MHB Proving Symmetric Difference Properties of Sets

    Hello! (Wave) Given the following definition: If $A,B$ are sets, we define the set $A \triangle B =(A \setminus B) \cup (B \setminus A)$ and we call it symmetric difference of $A,B$. I have to prove the following sentences: $A \cap B= \varnothing$, then $A \triangle B=A \cup B$ $A...
  17. C

    Schwarzschild Spacetime: Ellipsoidal for Moving Observers?

    In special relativity a sphere in the rest frame for some observer looks like an ellipsoid for an observer with a relative velocity. Can we use the same reasoning for the Schwarzschild spacetime? Namely that a spherically symmetric spacetime produced by a spherical mass look ellipsoidal for an...
  18. K

    Relations- reflexive, symmetric, anit-symmetric, transitive

    Suppose that R1={(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)}, R2={(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)}, R3={(2,4),(4,2)} , R4={(1,2),(2,3),(3,4)}, R5={(1,1),(2,2),(3,3),(4,4)}, R6={(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)}, Determine which of these statements are correct. Check ALL correct answers...
  19. camilus

    Pfaffian and determinants of skew symmetric matrices

    Can anyone explain or point me to a good resource to understand these operators? I'm trying to the understand determinants for skew symmetric matrices, more specifically the Moore determinant and it's polarization of mixed determinants. Can hone shed some light? I'm confused as to how the...
  20. C

    Calculating Symmetric Components in a Three Phase 4 Wire System

    Hello fellow engineers! I am a student doing a simple course in Electrical Engineering. I've got an enquiry regarding this question " 1. In a three phase 4 wire system with phase to neutral voltage of 230V, a balanced set of resistive loads of 8 ohms are connected between each phase and...
  21. M

    Writing 3rd Order Tensor Symmetric Part in Tensor Form

    Can some one write for me the Symmetric part of a third order tensor (as a tensor form) Thanks .
  22. C

    Riemannian curvature of maximally symmetric spaces

    A maximally symmetric is a Riemannian n-dimensional manifold for which there is n/2 (n+1) linearly independent (as solutions) killing vectors. It is well known that in such a space $$R_{abcd} \propto (g_{ab}g_{cd} - g_{ac}g_{bd}) .$$ How is this formula derived for a general maximally...
  23. R

    Spherically symmetric charge density

    Homework Statement Imagine a spherically symmetric charge density p(r)=Cr for r<=a, p(r)=0 for r>a. a) Find the electric field E(r) and potential V(r). Are they continuous at r=a? b) Suppose additional charge is placed uniformly on the surface at r=a with surface density sigma. Find E(r) and...
  24. N

    Microfluidics: asymmetric and symmetric t-junctions

    I have been looking on the web but I can't seem to find a textbook answer that describes the difference between microfluidic t-junctions that are symmetric and those that are asymmetric. From some papers I've gathered that you can make t-junctions asymmetric by changing the hydraulic resistance...
  25. N

    Asymmetric and Symmetric T-Junctions

    I have been looking on the web but I can't seem to find a textbook answer that describes the difference between microfluidic t-junctions that are symmetric and those that are asymmetric. From some papers I've gathered that you can make t-junctions asymmetric by changing the hydraulic...
  26. N

    Puzzling precession of the torque-free symmetric top

    Hello everyone, as I know the regular precession of torque-free symmetric top is such a cliche, I'll try to keep derivations short. The goal is, as I pointed out, to inspect a behaviour of the torque-free symmetric top in terms of precession rate, rotation rate and nutation angle. One way is to...
  27. Greg Bernhardt

    What Are Symmetric Groups and Their Mathematical Significance?

    Definition/Summary The symmetric group S(n) or Sym(n) is the group of all possible permutations of n symbols. It has order n!. It has an index-2 subgroup, the alternating group A(n) or Alt(n), the group of all possible even permutations of n symbols. That group has order n!/2. For n >= 5...
  28. R

    Radially symmetric "Breathing" resonance of a sealed water filled tube

    Please could someone explain to me what is meant by the radially symmetric "breathing resonance" of a sealed water filled tube or cell? That is with the use of transducers this can be achieved, but what does it mean? is it talking about generating a standing wave in the fluid? It relates...
  29. A

    Is stress tensor symmetric in Navier-Stokes Equation?

    Hello, In CFD computation of the Navier-Stokes Equation, is stress tensor assumed to be symmetric? We know that in NS equation only linear momentum is considered, and the general form of NS equation does not assume that stress tensor is symmetric. Physically, if the tensor is asymmetric then...
  30. J

    Representation symmetric, antisymmetric or mixed

    Hi, While studying Lie groups and in particular the representations I have some trouble with determining whether a representation is a symmetric, antisymmetric or mixed tensor product of the fundamental representations. I'm working with SO(5) to get an understanding about the actual...
  31. M

    What are the practical applications of symmetric equations?

    It seems that the only applicable use I've seen is in finding intercepts on various axes. Are there any other instances where this form would used? What else can this be used for?
  32. K

    Resolving Symmetric Movement Paradox w/ Respect to Stationary Observer

    I am trying to understand this apparent "paradoxes" but probably i am missing something important. Imagine that accourding to stationary observer on Earth two twin cats are moving in the opposite directions with speed -v and v. When the two cats meet the stationary observer at the beginning O...
  33. kq6up

    General Solution for Eigenvalues for a 2x2 Symmetric Matrix

    Homework Statement From Mary Boas' "Mathematical Methods in the Physical Science" 3rd Edition Chapter 3 Sec 11 Problem 33 ( 3.11.33 ). Find the eigenvalues and the eigenvectors of the real symmetric matrix. $$M=\begin{pmatrix} A & H \\ H & B \end{pmatrix}$$ Show the eigenvalues are real and...
  34. B

    Cylindrically symmetric line element canonical form

    Hello, What is the most general cylindrically symmetric line element in the canonical form? Best regards.
  35. C

    How Do You Derive Euler's Equation of Motion for a Rigid Body?

    Homework Statement Derive Euler's equation of motion for a rigid body: $$\dot{\vec{L}} + \vec{\omega} \times \vec{L} = \vec{G},$$ where ##\vec{L}## is the angular momentum in the body frame, ##\omega## is the instantaneous velocity of the body's rotation and ##\vec{G}## is the external torque...
  36. kq6up

    Understanding Skew Symmetric Matrices for Physics - A Helpful Guide

    I am a bit dense when it comes to linear algebra for some reason. I am reviewing math to prepare for a physics grad program, and I am using Mary Boas "Mathematical Methods in the Physical Sciences". She presents the idea of a skew symmetric matrix in the problem set rather than in the text. I...
  37. C

    Obtaining representations of the symmetric group

    Homework Statement Consider the following permutation representations of three elements in ##S_3##: $$\Gamma((1,2)) = \begin{pmatrix} 0&1&0\\1&0&0\\0&0&1 \end{pmatrix}\,\,\,\,;\Gamma((1,3)) = \begin{pmatrix} 0&0&1\\0&1&0\\1&0&0 \end{pmatrix}\,\,\,\,\,; \Gamma((1,3,2)) = \begin{pmatrix}...
  38. B

    Showing Two Functions Are Symmetric About A Line

    Hello everyone, I have the functions y_1 = \frac{c}{b} + d e^{-bx} andy_2 = \frac{c}{b} - d e^{-bx} , where c \in \mathbb{R}, and b,d \in \mathbb{R}^+. What I would like to know is how to show that these two functions are symmetric about the line y = \frac{c}{d}. What I thought was that if y_1...
  39. T

    Finding the gyromagnetic ratio of an axially symmetric body

    Homework Statement So, we can presumably write that m = g L, where L is the angular momentum, g the ratio wanted, and m magnetic dipole moment of an axially symmetric body. Total mass is M, total charge Q, mass density \rho_m(r)=\frac{M}{Q}\rho_e(r), where \rho_e(r) is charge density...
  40. D

    Why is the stress-energy tensor symmetric?

    If we use the "flux of 4-momentum" definition of the stress-energy tensor, it's not clear to me why it should be symmetric. Ie, why should ##T^{01}## (the flux of energy in the x-direction) be equal to ##T^{10}## (the flux of the x-component of momentum in the time direction)?
  41. T

    Find the unique symmetric matrix A such that Y'AY=Y'GY

    I asked this question here, however the title of the thread (and the thread itself) was sloppy and unclear.I could not find a way to delete or edit. This is for a regression analysis course, and I've only taken one introductory course on linear algebra, so when I Google'd "finding a symmetric...
  42. K

    MHB Positive, definite matrix symmetric

    Under which conditions is a real positive definite matrix symmetric? I have crossposted here: http://math.stackexchange.com/questions/661102/positive-definite-matrix-symmetric
  43. alyafey22

    MHB Solutions of symmetric three variable equations.

    Inspired by this http://mathhelpboards.com/challenge-questions-puzzles-28/solving-system-equations-8521.htmlit is interesting to look at the solutions of xz+y=a xy+z=a zy+x=a where we look at the case a>0 Assume the following a=bc where a\neq 0 Then we have If we assume that c is a...
  44. M

    Finding 8 Relations on a Set of 3 Elements with the Same Symmetric Closure

    Homework Statement Show that if a set has 3 elements, then we can find 8 relations on A that all have the same symmetric closure. Homework Equations Symmetric closure ##R^* = R \cup R^{-1} ## The Attempt at a Solution If the symmetric closures of n relations are the same then...
  45. bhanesh

    What is the minimum rank of a skew symmetric matrix?

    What is minimum possible rank of skew symmetric matrix ?
  46. caffeinemachine

    MHB Zero-Trace Symmetric Matrix is Orthogonally Similar to A Zero-Diagonal Matrix.

    Hello MHB. During my Mechanics of Solids course in my Mechanical Engineering curriculum I came across a certain fact about $3\times 3$ matrices. It said that any symmetric $3\times 3$ matrix $A$ (with real entries) whose trace is zero is orthogonally similar to a matrix $B$ which has only...
  47. C

    Positive-definite symmetric matrix satisfying a certain property

    Homework Statement We have a finite group ##G## and a homomorphism ##\rho: G \rightarrow \mathbb{GL}_n(\mathbb{R})## where ##n## is a positve integer. I need to show that there's an ##n\times n## positive definite symmetric matrix that satisfies ##\rho(g)^tA\rho(g)=A## for all ##g \in G##...
  48. Sudharaka

    MHB The Existence of Symmetric Matrices in Subspaces

    Hi everyone, :) Here's a question I am stuck on. Hope you can provide some hints. :) Problem: Let \(U\) be a 4-dimensional subspace in the space of \(3\times 3\) matrices. Show that \(U\) contains a symmetric matrix.
  49. H

    Locally Maximally Symmetric Spacetimes

    Can one say that every general curved spacetime, locally is maximally symmetric? I know that one can say that every general curved spacetime is locally flat (and therefore maximally symmetric with R=0), but I'm talking about a very high curvature spacetime, where still we can consider nonzero...
  50. D

    Time Symmetric Quantum Mechanics

    I've been seeing more and more papers that seem to suggest Time Symmetric Quantum Mechanics (TSQM) is becoming the more parsimonious explanation to some newer experiments. For those unfamiliar with this formulation, it's a two-state-vector formulation, with one of the state vectors...
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