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.
My attempt:
$$
\begin{vmatrix}
1-\lambda & b\\
b & a-\lambda
\end{vmatrix}
=0$$
$$(1-\lambda)(a-\lambda)-b^2=0$$
$$a-\lambda-a\lambda+\lambda^2-b^2=0$$
$$\lambda^2+(-1-a)\lambda +a-b^2=0$$
The value of ##\lambda## will be positive if D < 0, so
$$(-1-a)^2-4(a-b^2)<0$$
$$1+2a+a^2-4a+4b^2<0$$...
Consider a Lagrangian, #L#, which is a function of, as well as other fields #\psi_i#, a traceless and symmetric tensor denoted by #f^{uv}#, so that #L=L(f^{uv})#, the associated action is #\int L(f^{uv}, \psi_i)d^4x #.
To vary w.r.t #f^{uv}# , I write...
I am struggling with Hamiltonian formulation of classical mechanics. I think I have grasped the idea of canonical transformations, including the idea of angle-action variables and invariant tori in phase space. Still, few points seem to elude my understanding...
Let's talk about a particle...
Hi guys,
I can't seem to be able to get to
$$ (\rho + p) \frac {d\Phi} {dr} = - \frac {dp} {dr} $$
from
$$T^{\alpha\beta}_{\,\,\,\,;\beta} = 0$$
the only one of these 4 equations (in the case of a spherically symmetric static star) that does not identically vanish is that for ##\alpha=r##...
I learned that for a bilinear form/square form the following theorem holds:
matrices ## A , B ## are congruent if and only if ## A,B ## represent the same bilinear/quadratic form.
Now, suppose I have the following quadratic form ## q(x,y) = x^2 + 3xy + y^2 ##. Then, the matrix representing...
Hello! I am a bit confused about how to interpret symmetric error when doing error propagation. For example, if I have ##E = \frac{mv^2}{2}##, and I do error propagation I get ##\frac{dE}{E} = 2\frac{dv}{v}##. Which implies that if I have v being normally distributed, and hence having a...
Hello,
I am often designing math exams for students of engineering.
What I ask is the following:
Can I choose any real 3x3 symmetric matrix with positive eigenvalues as a realistic matrix of inertia?
Possibly, there are secret connections between the off-diagonal elements (if not zero)...
Let us define matrix ##\mathbf{B}_n=[b_{ij}]_{n\times n}## as follows $$[b_{ij}]_{n\times n}:=\begin{cases} b_{ij} = \alpha\,,\quad j=i\\ b_{ij}=\beta\,,\quad j=i\pm1\\ b_{ij}=1\,,\quad \text{else}\end{cases}\,,$$ where ##\alpha\,,\beta\in\mathbb{R}## and ##n\geq2##. ##\mathbf{B}_4##, for...
I am sure you are all familiar with the cross product in 3D space.
i cross into j gives k.
Cyclic
Negative, if reversed, etc.
I am sure you are all familiar with the definition as: norm of the first vector, norm of the second, sine of the angle, perpendicular (but direction using right hand...
Goldstein 3rd Ed, pg 339
"In large classes of problems, it happens that ##L_{2}## is a quadratic function of the generalized velocities and ##L_{1}## is a linear function of the same variables with the following specific functional dependencies:
##L\left(q_{i}, \dot{q}_{i}, t\right)=L_{0}(q...
Suppose I measure the circumference of a circular orbit round a massive object and find it to be c. Suppose I then move to a slightly higher orbit an extra radial distance δr as measured locally. If space was flat I would expect the new circumference to be c + 2πδr. Will the actual measurement...
My friend asked for help with this precalculus question. I could not help him. So, I decided to post here.
Find a and b when the graph of
y = ax^2 + bx^3 is symmetric with respect to (a) the y-axis and (b) the origin. (There are many correct answers.)
I don't even know where to begin.
Hello
Say I have a column of components
v = (x, y, z).
I can create a skew symmetric matrix:
M = [0, -z, y; z, 0; -x; -y, x, 0]
I can also go the other way and convert the skew symmetric matrix into a column of components.
Silly question now...
I have, in the past, referred to this as...
In the symmetric eigenvalue problem, Kv=w^2*v where K~=M−1/2KM−1/2, where K and M are the stiffness and mass matrices respectively. The vectors v are the eigenvectors of the matrix K~ which are calculated as in the example below. How do you find the directions of the eigenvectors? The negatives...
The equation $$\frac{\hbar^2}{2m}\frac{d^2u}{dr^2}-\frac{Ze^2}{r}u=Eu$$ gives the schrodinger equation for the spherically symmetric functions ##u=r\psi## for a hydrogen-like atom.
In this equation, substitute an assumed solution of the form ##u(r)=(Ar+Br^2)e^{-br}## and hence find the values...
Hello, I was reading few papers discussing modified gravity theories and their use in understanding galaxies with no dark matter by checking for anomalous velocity dispersion. Now, the author was using 4 gravity theories MOND, Weyl, MOG and Emergent gravity. The thing is he had provided the...
Summary:: Let ##A \in \Bbb R^{n \times n}## be a symmetric matrix and let ##\lambda \in \Bbb R## be an eigenvalue of ##A##. Prove that the geometric multiplicity ##g(\lambda)## of ##A## equals its algebraic multiplicity ##a(\lambda)##.
Let ##A \in \Bbb R^{n \times n}## be a symmetric matrix...
https://en.wikipedia.org/wiki/CPT_symmetry says "CPT theorem says that CPT symmetry holds for all physical phenomena" - e.g. we could imagine decomposition of given phenomena into Feynman diagrams and apply CPT symmetry to all of them.
However, for some o processes such reversibility seems...
I'm trying to prove the associative law of symmetric difference (AΔ(BΔc) = (AΔB)ΔC ) with other relations of sets.
A naive way is to compare the truth table of two sides. However, I think the symmetric difference is not a basic one, it is constructed form other relations, that is AΔB =...
Can be Einstein equation rewrited into some simpler form, when suppose only spherically symmetric (but not necessarily stationary) distribution of mass-energy ?
If yes, is there some source to learn more about it ?
Thank you.
edit: by simpler form I mean something with rather expressed...
Spatial slices of the Robertson-Walker metrics are maximally symmetric so they must have a constant curvature. Is it true that in three Riemannian dimensions that a constant curvature scalar determines whether the volume is finite or infinite? Carroll seems to have given a counter-example for...
I was reading zee's group theory in a nutshell.
I understand that we can decompose a 2 index tensor for rotation group into an antisymmetric vector(3), symmetric traceless tensor(5) and a scalar(trace of the tensor). Because "trace is invariant" it put a condition on the transformation of...
Hi,
The textbook asks me to use subgroup axioms to prove why a set of permutations that interchange two specific symbols in S4 is or isn't a subgroup of the symmetric group, and the same for a set of permutations that fix two elements.
My guess is that the set of permutations that interchange...
I looked in the instructor solutions, which are given by:
But I don't quite understand the solution, so I hope you can help me understand it.
First. Why do we even know we are working with wavefunctions with the quantum numbers n,l,m? Don't we only get these quantum numbers if the particles...
I am trying to understand but without a succes why symmetric magnetic field around ##Z## axis make that ##\hat \phi## magnetic field is zero
I can't understand why it physically happens and also how can I derive it mathematically?
What does the word symmetric means when talking about magnetic...
Why does the constraint:
$$R_{ijkl}=K(g_{ik} g_{jl} - g_{il}g_{jk})$$
Imply that the resulting space is maximally symmetric? The GR book I'm using takes this relation more or less as a definition, what is the idea behind here?
What is exactly the definition of symmetric limit?
It's the first place in the book that I see this notation, and they don't even define what it means.
How does it a differ from a simple limit or asymptotic limit?
I found a few hits in google, but it doesn't seem to help...
Now, with the given set of natural numbers, we can deduce the relation ##R## to be as following
$$ R = \big \{ (1,6), (2,7), (3,8) \big \} $$
Now, obviously this is not a reflexive and symmetric. And I can also see that this is transitive relation. We never have ##(a,b) \in R## and ##(b,c) \in...
Hi,
if I have a equation like (just a random eq.) p_dot = S(omega)*p. where p = [x, y, z] is the original states, omega = [p, q, r] and S - skew symmetric.
How does the equation appear if i only want a system to have the state z? do I get z_dot = -q*x + p*y. Or is the symmetric not valid so I...
The given definition of a linear transformation ##F## being symmetric on an inner product space ##V## is
##\langle F(\textbf{u}), \textbf{v} \rangle = \langle \textbf{u}, F(\textbf{v}) \rangle## where ##\textbf{u},\textbf{v}\in V##.
In the attached image, second equation, how is the...
Setup: Let ##\hat{\mathbf{e}}_1,\hat{\mathbf{e}}_2,\hat{\mathbf{e}}_3## be the basis of the fixed frame and ##\hat{\mathbf{e}}'_1,\hat{\mathbf{e}}'_2,\hat{\mathbf{e}}'_3## be the basis of the body frame. Furthermore, let ##\phi## be the angle of rotation about the ##\hat{\mathbf{e}}_3## axis...
Hi, in the lecture notes my professor gave us, it is stated that, due to Kramers theorem, the energy in a band must satisfy this condition:
$$E(-k) = E(k)$$
But, judging from actual pictures of band structures I don't find this condition to be true. Here's a (random) picture
I guess it looks...
Hi,
I stumbled upon an identity when studying tensor perturbations in cosmology. The formula states that
$$
\int d^2\hat{p} f(\hat{p}.\hat{q})\hat{p}_i \hat{p}_k e_{jk}(\hat{q}) = e_{ij}(\hat{q})/2 \int d^2\hat{p} f(\hat{p}.\hat{q})(1-(\hat{p}.\hat{q})^2),
$$ where ##e_{ij}## is a symmetric...
Again in pg. 166 eq 7.2
https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.astro.caltech.edu/~george/ay21/readings/carroll-gr-textbook.pdf&ved=2ahUKEwi1gdbj3ODgAhXRWisKHXW_D-sQFjACegQIBhAB&usg=AOvVaw1YY2mM7uccdbX4nTxFgQO5
Here ##u^1=\theta,u^2=\phi## and v=r.
The tangent vector on...
Let ##Q_ik## be a symetric tensor, so that ##Q_ik= \frac{m}{2} \dot x_i \dot x_j + \frac{k}{2} x_i x_j## (here k is also a sub, couldn't do it better with LaTeX).
How do we derive such a tensor, with respect to time? And what could such a tensor mean in a physical sense? It really looks like the...
Homework Statement
A spherically symmetric charge distribution produces the electric field E=(200/r)r(hat)N/C, where r is in meters.
a) what is the electric field strength at 10cm?
b)what is the electric flux through a 20cm diameter spherical surface that is concentric with the charge...
Homework Statement
Consider matrices A = [1 2;2 4] and P = [1 3;3 6]. Using B = P^-1*A*P, verify that similar matrices have the same eigenvalues. Find the eigenvectors y for B and show that x = P*y are eigenvectors of A.
Homework Equations
B = P^-1*A*P,
x = P*y
The Attempt at a Solution
I...
Homework Statement
Let ##n## be a natural number and let ##\sigma## be an element of the symmetric group ##S_n##. Show that if ##\sigma## is a product of disjoint cycles of orders ##m_1 , \dots , m_k##, then ##|\sigma|## is the least common multiple of ##m_1 , \dots , m_k##.
Homework...
Hey! :o
Let $n\in \mathbb{N}$ and $M=\{1, 2, \ldots , n\}\subset \mathbb{N}$. Let $d:M\times M\rightarrow \mathbb{R}$ a map with the property $$\forall x, y\in M : d(x,y)=0\iff x=y$$
Let \begin{equation*}G=\{f: M\rightarrow M \mid \forall x,y\in M : d(x,y)=d\left (f(x), f(y)\right...
Hi! I have the following problem I don't really know where to start from:
A bowl with axial symmetry is built in flat Euclidean space ##R^3##, and has a radial profile giveb by ##z(r)##, where ##z## is the axis of symmetry and ##r## is the radial distance from the axis. What radial profile...
Compute the order of each of the elements in the symmetric group ##S_4##.
Is the best way to do this just to write out each element's cycle decomposition, or is there a more efficient way?