Proof Definition and 999 Threads

Prove the following I literally have no idea where to start or what to do.

Homework Statement Carnot theorem states that no engine working between two temperatures T1 of source and T2 of sink can have a greater efficiency than that of the Carnot engine. Second law of thermodynamics:it is impossible for a self acting machine to transfer heat from a body at a higher...

Hello. Is there a quick proof for showing that the next prime is within twice the current prime? Edit: Never mind. Erdős had given a proof of this (of Bertrand's postulate to be precise) at a fairly young age. http://www3.nd.edu/~dgalvin1/pdf/bertrand.pdf

If we define Si=(1/2)× (reduced Planck's const)×sigma Then what will be (sigma dot vect{A})multiplied by (Sigma dot vect{B}) Here (sigma)i is Pauli matrix. Next one is, what will we get from simplifying <Alpha|vect{S}|Alpha> where vect{S} is spin vector & |Apha>is equal to " exp[{i×(vect{S} dot...

Can anyone help with a proper proof for the following relation, please? u(x) \frac{\partial u(x)}{\partial x} = \frac{1}{2} \frac{\partial u(x)^2}{\partial x} From simple calculations I agree that it's true, but it's been annoying me for a while that I can't find a proper mathematical proof...

Hi, I'm currently going through Griffith's Particle Physics gamma matrices proofs. There's one that puzzles me, it's very simple but I'm obviously missing something (I'm fairly new to tensor algebra). 1. Homework Statement Prove that ##\text{Tr}(\gamma^\mu \gamma^\nu) = 4g^{\mu\nu}##...

Hi, I need opinion about this problem. ================================================== question :Prove: If(a,b)= l and if ( "(a,b)=1" mean greatest common divisor of integers and b is 1 ) c|a (c divides a) and d|b (d divides b ) then (c,d)= 1. ( "(c,d)=1" mean...

Homework Statement potential energy function of : $$ U(x) = 4x^2 + 3 $$ And have to i) Work out the equation of motion ii) Prove explicitly that the total energy is conservedHomework Equations$$ F = \frac{dU}{dt} $$ The Attempt at a Solution I'm not too sure how to go about this...

So I was reading this book, "Euclidean and non Euclidean geometries" by Greenberg I solved the first problems of the first chapter, and I would like to verify my solutions 1. Homework Statement Homework Equations [/B] Um, none that I can think of? The Attempt at a Solution (1) Correct...

I want to prove that ##E = -g_{\mu \nu}u^\mu p^\nu## is the energy measured by an observer with velocity ##u^\mu## of an object with momentum ##p^\mu##. My reasoning is that in special relativity we know that ##\gamma m = E##. We can transform to coordinates where ##u'^\mu = (1,\vec{0})##. Since...

An absolute value property is $$\lvert a \rvert \geq b \iff a\leq-b \quad \text{ or } \quad a\geq b,$$ for ##b>0##. Is this true for the case ##a=0##? I mean if ##a=0, \lvert a \rvert =0## so ##0 \geq b##. But ##b## is supposed to be ##b>0##, so we have a contradiction. How can this property...

How would one prove the Stokes' theorem for general cases? Namely that $$ \int_{\partial M} W = \int_M \partial W$$ where ##M## is the manifold.

Homework Statement Show that: d<A(q,p)>/dt=<{A,H}>, where {A,H} is a Poisson Bracket Homework Equations Liouville theorem The Attempt at a Solution <A>=Tr(Aρ)⇒d<A>/dt=Tr(Adρ/dt)=Tr(A{H,ρ}) So, in order to get the correct result, Tr(A{H,ρ}) must be equal to Tr({A,H}ρ), but I don't think I can...

Homework Statement The sequence of positive numbers ##u_1,u_2,u_3...## is such that ##u_1<4## and ##u_{n+1}= \frac{5u_n+4}{u_n+2} ## i. By considering ##4-u_{n+1} ##, prove by induction that ##u_1<4## for ##n\geq 1## Mod note: The above is incorrect. In a later post the OP revised this to The...

Homework Statement "Prove: ##∀n∈ℕ##, ##3^n>n^2## Homework EquationsThe Attempt at a Solution (1) We will prove that ##3^n>n^2## at ##n=1## ##3=3^1>1=1^2## (2) Now assume that ##3^k>k^2## for some ##k>1## (3) We will prove that ##3^{k+1}>(k+1)^2## or ##3⋅3^k>k^2+2k+1## Note that...

I don't get the first part. why did he make the angle of sin equal to n pi.

Hey guys! I need help proving why this proof is wrong. I know it's wrong, but I can't figure out why. Anyway: i = sqrt -1 i^4 = 1 1^4 = 1 Substution: i^4 =1^4 i = 1 1 = sqrt -1 1^2 = -1 1 = 1^2 1= -1 If you have any questions, feel free to ask.

are spacelike and timelike orthogonal?what is the mathematical proof

Okay, these are my last questions and then I'll get out of your hair for a while. For 1, I have already done a proof by contradiction, but I'm supposed to also do a direct proof. Seems like it should be simple? For 2, this seems obvious because it's the definition of an integral. My delta is...

In Yang-Mills theory, the gauge transformations $$\psi \to (1 \pm i\theta^{a}T^{a}_{\bf R})\psi$$ and $$A^{a}_{\mu} \to A_{\mu}^{a} \pm \partial_{\mu}\theta^{a} \pm f^{abc}A_{\mu}^{b}\theta^{c}$$ induce the gauge transformation$$F_{\mu\nu}^{a} \to F_{\mu\nu}^{a} -...

I have no idea how to incorporate the limit into the basic definitions for a Riemann integral? All we have learned so far is how to define a Riemann integral and the properties of Riemann integrals. What should I be using for this?

Note: Please only give hints please! No answers because I want the satisfaction of solving it. 1. Homework Statement A mass M at height h above flat round and falling vertically with velocity v breaks up explosively into 2 parts. The kinetic energy given to the system in the explosion is E...

Homework Statement Suppose R1 and R2 are relations on A and R1 ⊆ R2. Let S1 and S2 be the transitive closures of R1 and R2 respectively. Prove that S1 ⊆ S2. Please check my proof and please explain my mistakes. thank you for taking the time to help. Homework Equations N/A The Attempt at a...

{\displaystyle \sum_{n=1}^{\infty}a_{n}} is converage, For N\in \mathbb{N}\sum_{n=N+1}^{\infty}an is also converage proof that \lim_{N\rightarrow\infty}(\sum_{n=N+1}^{\infty}an)=0 {\displaystyle \sum_{n=1}^{\infty}a_{n}} is converage, For N\in \mathbb{N} \sum_{n=N+1}^{\infty}an is...

Homework Statement I am trying to understand the proof that ##\lim S## is a closed set in the metric space ##M##, where ##\lim S = \{ p \in M ~|~ p \mbox{ is a limit point of } S\}##. Here is the definition of a limit point: ##p## is a limit point of ##S## if and only if there exists a...

Let $0 \le a,b,c \le 1.$ Prove the inequality:$\sqrt{a(1-b)(1-c)}+ \sqrt{b(1-a)(1-c)}+\sqrt{c(1-a)(1-b)} \le 1 + \sqrt{abc}$

Let $F:P(A)->P(A$) be monotone and $C$ be the union of sets whose image is invariant under F. Prove $F(C)=C$ https://i.stack.imgur.com/3Wjdg.png

Homework Statement I need to prove by induction that ##(n!)^{2} \le (2n)!##. I'm pretty sure about my preliminary work, but I just need some suggestions for the end. Homework Equations It is well known from a theorem that if ##a \le b## and ##c \ge 0##, then ##ca \le cb##. The Attempt at a...

I am very open minded and I would fully trust in Cantor's diagonal proof yet this question is the one that keeps holding me back. My question is the following: In any given infinite set, there exist a certain cardinality within that set, this cardinality can be holded as a list. When you change...

This is a simple exercise from Spivak and I would like to make sure that my proof is sufficient as the proof given by Spivak is much longer and more elaborate. Homework Statement Prove that \lim_{x\to a} f(x) = \lim_{h\to 0} f(a + h) Homework EquationsThe Attempt at a Solution By the...

Quantum theory, although hard to understand with intuition has a lot of experimental proof. Do the more modern theories e.g. String theory, or black hole theories have any experimental proof, or are they theories that the mathematics have led to? Without proof, do they deserve so much credit...

Hi. I have been looking at the proof that the parity operator is hermitian in 3-D in the QM book by Zettili and I am confused by the following step ∫ d3r φ*(r) ψ(-r) = ∫ d3r φ*(-r) ψ(r) I realize that the variable has been changed from r to -r. In 3-D x,y,z this is achieved by taking the...

Homework Statement Let A,B,C,D be commuting n-square matrices. Consider the 2n-square block matrix ##M= \begin{bmatrix} A & B \\ C & D \\ \end{bmatrix}##. Prove that ##\left | M \right |=\left | A \right |\left | D \right |-\left | B \right |\left | C \right |##. Show that the result may not be...

The sum of the $k$ th power of n variables $\sum_{i=1}^{i=n} x_i^k$ is a symmetric polynomial, so it can be written as a sum of the elementary symmetric polynomials. I do know about the Newton's identities, but just with the algorithm of proving the symmetric function theorem, what should we do...

Homework Statement Hi I am looking at the proof attached for the theorem attached that: If ##s \in R##, then ##\sum'_{w\in\Omega} |w|^-s ## converges iff ##s > 2## where ##\Omega \in C## is a lattice with basis ##{w_1,w_2}##. For any integer ##r \geq 0 ## : ##\Omega_r := {mw_1+nw_2|m,n \in...

So the definition of a bounded sequence is this: A sequence ##(x_{n})## of real numbers is bounded if there exists a real number ##M>0## such that ##|x_{n}|\le M## for each ##n## My question is pretty simple. How does one choose the M, based on the sequence in order to arrive at the...

I agree that this could have been done more simply(i'm not looking for an alternative proof), but I don't understand how it is wrong, any insight? Since Dn is an dihedral group, we know its elements are symmetries, Dn = (R1,R2,R3...Ri) and since R is a symmetry, we know it's a permutation, so...

I'm in the 6th week of a well-known MOOC course created by Kevin Devlin, "Introduction to Mathematical Thinking." I enjoy the course & did well in the first weeks with conditionals and truth tables, etc.; however now that we are entering into proofs, I'm running into trouble with algebra...

Repeatedly apply $\binom{n}{r}= \binom{n-1}{r}+\binom{n-1}{r-1}$ to show: $$\binom{n}{r}=\sum_{i=1}^{r+1}\binom{n-i}{r-i+1}$$ The closest i got was showing you could show different iterations with the binomial coefficients (Pascal's Triangle).

From Courant's Differential and Integral Calculus p.13, In an ordinary system of rectangular co-ordinates, the points for which both co-ordinates are integers are called lattice points. Prove that a triangle whose vertices are lattice points cannot be equilateral. Proof: Let ##A=(0,0)...

Homework Statement they say 1. ##e^{ln x}= x ## and 2.##e^-{ln(x+1)}= \frac 1 {x+1}## how can we prove this ##e^{ln x}= x ## and also ##e^-{ln(x+1)}= \frac 1 {x+1}##? Homework EquationsThe Attempt at a Solution let ## ln x = a## then ##e^a= x, ## a ln e= x,## →a= x, where ## ln x= x

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with the proof of Proposition 6.1.7 ... Proposition 6.1.7 and its proof read as follows: In the above text from Bland, in the proof of (1), we read the...

I'm stuck on this proof question: (¬(Q⇒¬P) ∧ ¬((Q∧¬R)⇒¬P )) ⇔ ¬(R ∨ (P ⇒¬Q)) I've tried to get rid of the negation and implications but I keep going in circles and I'm getting nowhere near to the equivalence required. I would appreciative if anyone can help me solve this because it's really...

Homework Statement Prove the formula for inertia of a ring (2D circle) about its central axis. Homework Equations I = MR^2 Where: M: total mass of the ring R: radius of the ring The Attempt at a Solution - So I need to prove the formula above. - First, I divide the ring into 4...

Homework Statement Does anyone know of a simple proof for this: https://s30.postimg.org/tw9cjym9t/expect.png E(X) = E(X|S)P(S) + E(X|S_c)P(S_c) X is a random variable, S is an a scenario that affects the likelihood of X. So P(S) is the probability of the scenario occurring and and P(S_c) is...

Whittaker (1st Edition, 1902) P.132, gives two proofs of Fourier's theorem, assuming Dirichlet's conditions. One proof is Dirichlet's proof, which involves directly summing the partial sums, is found in many books. The other proof is an absolutely stunning proof of Fourier's theorem in terms of...

Homework Statement Prove that a set S of vectors is linearly independent if and only if each finite subset of S is linearly independent. Homework EquationsThe Attempt at a Solution I think that that it would be easier to prove the logically equivalent statement: Prove that a set S of vectors...

Homework Statement [/B] Theorem attached. I know the theorem holds for a discrete subgroup of ##C## more generally, ##C## the complex plane, and that the set of periods of a non-constant meromorphic function are a discrete subset. I have a question on part of the proof (showing the second...

Homework Statement Homework Equations The Rotation Matrix The Attempt at a Solution I am sorry but I do not know how to even begin

Homework Statement Find all integer solutions to x2 + y2 + z2 = 51. Use "without loss of generality." Homework Equations The Attempt at a Solution My informal proof attempt: Let x, y, z be some integers such that x, y, z = (0 or 1 or 2 or 3) mod 4 Then x2, y2, y2 = (0 or 1) mod 4 So x2 +...