Theorem Definition and 1000 Threads

Is this an abuse of Rolle's theorem? Rolle's theorem If a real-valued function f is continuous on a proper closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the open interval (a, b) such that f'(c) = 0. ##[x_1, x_1]##...

Hi All. I have a doubt concerning the limit: $$ \lim_{n \to \infty} \frac{\pi (n)}{Li(n)} = 1 $$. This mathematical statement does not imply that both functions converge to the same value. The main reason is that both tend to infinity as n tend to infinity. I would like to ask you if it is...

Hi, I have a question about identifying closed and open surfaces. Usually, when I see some exercises in the subject of the divergence theorem/flux integrals, I am not sure when the surface is open and needed to be closed or if it is already closed. I mean for example a cylinder that is...

Homework Statement Prove the following statement: Let f be a polynomial, which can be written in the form fix) = a(n)X^(n) + a(n-1)X^(n-1) + • • • + a0 and also in the form fix) = b(n)X^(n) + b(n-1)X^(n-1) + • • • + b0 Prove that a(i)=b(i) for all i=0,1,2,...,n-1,n Homework Equations 3. The...

Homework Statement https://projects.exeter.ac.uk/fluidflow/Courses/FluidDynamics3211-2/DimensionalAnalysis/dimensionalLecturese4.htmlaccording to this link , when we form the pi group , we need to put an exponent for the non-repeating variable ,( in this case , delta P is non-repeating variable...

Homework Statement [/B] Polynomial f(x) is divisible by ##x^2-1##. If f(x) is divided by ##x^3-x##, then the remainder is... A. ##(x^2-x)f(-1)## B. ##(x-x^2)f(-1)## C. ##(x^2-1)f(0)## D. ##(1-x^2)f(0)## E. ##(x^2+x)f(1)## Homework Equations Remainder theorem The Attempt at a Solution [/B]...

Homework Statement i was told by my lecturer that when we choose the repeating variables in pi buckingham theorem , we can choose based on 3 property , which is geometry property which consists of (length , width and area) , then followed by flow property ( velocity , acceleartion, discharge)...

Hi, Etherington't reciprocity theorem states that distances measured by angular separation and by luminosity differ. My question is which one (if any of them) is the actual distance. I can understand they might differ in an expanding universe, but there's still a physical distance in such one...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.3 The Tensor Algebra ... ... I need help in order to get an understanding of an aspect of Example 10.11 and Definition 10.7 in Section 10.3 ... The relevant text in...

Bloch's theorem states that the wave functions for electrons in a periodic potential have the form: ψn,k(r) = un(r)exp(ik⋅r) , where un has the same periodicity as the potential. Bloch's theorem is used to calculate energy bands, and my question is: Does the n in un label the different bands...

Homework Statement Hi everybody! I have a math problem to solve, I'd like to check if I understand well the Banach fixed-point theorem in the case of Euclidean norm and how to deal with maximum norm. Check if the following functions ƒ: ℝ2 → ℝ2 are strictly contractive in relation to the given...

Homework Statement Homework Equations one dimensional Reynold's transport theorem The Attempt at a Solution [/B] I started with this equation, and tried to expand it using the equation given in #2. This is the farthest I have gotten so far. I got stuck from here. I do not know how to...

Hi, Does anyone know why k has to be real in an infinite system for bloch's theorem. I understand that the wavefunction becomes unphysical in an infinite system as it diverges. Why does that mean k has to be real? f(x)=u(x)exp(ikx)

Homework Statement For the network of constant current shown in Figure 4 it is known that R1 = 50 Ω and , R = 10 Ω. When the switch P is in the 1-position , current I = 50 mA and Ip = 70 mA known i . When the switch P is in the 2-position , current I' = 40 mA and Ip' = 90 mA are known ...

Or basically anything that isn't a positive integer. So I can prove quite easily by induction that for any integer n>0, De Moivre's Theorem (below) holds. If ##\DeclareMathOperator\cis{cis} z = r\cis\theta, z^n= r^n\cis(n\theta)## My proof below: However I struggle to do this with...

Homework Statement Homework Equations The Attempt at a Solution I only know that they gave the parameterization of the circle: r(t) = <cost, sint, 2>. My problem is, did they already give the curl of F in the line integral? I don't understand why dx, dy, and dz are separated like that.

In de Broglie's original proof of the theorem of phase harmony, the frequency of the moving wave of energy mc^2 (not the internal periodic phenomenon wave) is multiplied by the following term ##freq * ( t - \frac{\beta * x}{c} ) ## Does anyone have an idea where the fraction comes from? All...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.3 The Tensor Algebra ... ... I need help in order to get a basic understanding of Theorem 10.8 which is a Theorem concerning the direct sum of a family of subspaces as a...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.3 The Tensor Algebra ... ... I need help in order to get a basic understanding of Theorem 10.8 which is a Theorem concerning the direct sum of a family of subspaces as a...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help in order to get a basic understanding of Theorem 10.2 regarding the basis of a tensor product ... ... My apologies if my...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help in order to get a basic understanding of Theorem 10.2 regarding the basis of a tensor product ... ... My apologies if my...

I uploaded a picture of what I am stuck on. I understand the equation of motion 3.4.5a for a damped oscillator but I don't understand how to use binomial theorem to get the expanded equation 3.4.5b. I am no where near clever enough to figure this one out. I know how to use binomial theorem to...

Hi everyone, I am currently working on a Coanda UAV and I am aware that there's no mathematical model to express the lifting effects of Coanda. It is more of a physical description of airflow movement. Correct me if I am wrong! Thus, I am using the generic expression of lift to describe the...

The Cayley-Hamilton Theorem can be used to express the third invariant of the characteristic polynomial obtained from the non-trivial solution of the Eigenvector/Eigenvalue problem. I follow the proof (in Chaves – Notes on Continuum Mechanics) down to the following equation, then get stuck at...

Homework Statement why we can't form the pi group by using repeating variable of (μ, ρ , v) or (D, v , μ ) ? http://www-mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/fprops/dimension/node9.html http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Section5/dimensional_analysis.htm...

Hey all, the schema theorem shows that in all probability a genetic algorithm will converge to a solution. much like the second law of thermodynamics for optimization. Although, it is taught with the genes being $$ \in (0,1, *), * \in (0,1) $$ is there a proof for non binary genes? example...

Very simple question. So I am on a homework problem, and I want to make sure that I am using this theorem correctly. My book states that the Work-Kinetic Energy Theorem is: W=ΔK Now the solution to this problem involved multiple forces and thus each force is doing work. So my question is, is...

Homework Statement Determine the voltage across ##R_3## in the following figure assuming an input voltage ##v_0## of 10V is applied across the open terminals. Let ##R_1=3\Omega##, ##R_2=15\Omega##, ##R_3=10\Omega##, ##R_4=5\Omega## and ##R_5=2\Omega## and Homework Equations The Attempt at a...

While studying energy conservation on Morin I found this explanation about the work-energy theorem for a system. Using Koenig theorem $$\Delta K_\textrm{system}=\Delta K +\Delta K_\textrm{internal}$$ so we have I've got two main question on that: Why are only external forces considered for...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.2 Properties of Tensor Products ... ... I need help with an aspect of the proof of Theorem 10.3 ... ... basically I do not know what is going on in the second part of the proof...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.2 Properties of Tensor Products ... ... I need help with an aspect of the proof of Theorem 10.3 ... ... basically I do not know what is going on in the second part of the...

Homework Statement Sorry- I've figured it out, but I am afraid I don't know how to delete the thread. Thank you though :) Homework Equations Below The Attempt at a Solution Photo below- I promise its coming! I've started by using cylindrical coordinates, but I wasn't sure if spherical...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.2 Properties of Tensor Products ... ... I need help with an aspect of the proof of Theorem 10.3 regarding a property of tensor products ... ...The relevant part of Theorem 10.3...

Hi, I am reading "mathematical analysis" by Apostol right now for a course in analysis. Since I am trying to understand the author's proof of the above theorem(3.25 in the book), but I have something that I can't understand. He assumes that each of the nested sets contains infinitely many...

Divergence theorem states that $\int \int\vec{E}\cdot\vec{ds}=\int\int\int div(\vec{E})dV$ And Gauss law states that $\int \int\vec{E}\cdot\vec{ds}=\int\int\int \rho(x,y,z)dV$ If $\vec{E}$ to be electric field vector then i could say that $div(\vec{E})=\rho(x,y,z)$ However i can't see any...

Homework Statement Homework Equations Work energy theorem The Attempt at a Solution .5mv2 - .5mv2 = (kx - μmg)d final velocity is 0 ½mv2 =(kx - μmg)d solve for v2 ((kx - μmg)d2)m When this was wrong I tried integrating Fx but it was still wrong

Homework Statement Can anybody give me hint how to find K if F(x)= 3x+2 The integral lower part is not the same, , how to deal with his? Homework EquationsThe Attempt at a Solution Please ,I need hint to start

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with an aspect of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as follows: I do not...

How do I show that $\mathbb{Z}^{\times} _{20} ≅ \mathbb{Z}_{2} \times \mathbb{Z}_{4}$? I read that the chinese remainder theorem is the way to go but there are many versions and I can't find the right one. Most versions that I have found are statements between multiplicative groups, not from...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ... The relevant part of...

I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ... I need some help with a further aspect of the proof of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ... The text of Theorem 6.10 reads as follows: The above proof mentions Figure...

I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ... I need some help with an aspect of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ... The text of Theorem 6.10 reads as follows: The above proof mentions Figure 6.1 which is...

Torricelli's theorem relates the speed of a fluid exiting an opening in a reservoir to the height of the opening relative to the top of the reservoir... V=√2gh https://en.wikipedia.org/wiki/Torricelli%27s_law As seen in the wiki-link provided, the equation is essentially a Bernoulli's...

Hi, I was studying about the statistical ensemble theory and facing some problem to understand these concepts , I have understood that the ensemble is a collection of systems which are macroscopically identical but microscopically different . In some books they are called as systems with...

A waiter believes the distribution of his tips has a model that is slightly skewed to the left, with a mean of $\$8.20$ and a standard deviation of ​$\$5.60$. He usually waits on about 60 parties over a weekend of work. ​a) Estimate the probability that he will earn at least ​$\$600$. ​b) How...

What is the difference between quantum mechanics and realism ? quantum mechanics states on statistics while the hardy assumption of EPR is that hidden variables may describe exactly the outcomes of each individual test. Bell refutes the last idea. But, he didn't need to refute the case where...

Homework Statement Take a standard deck of cards, and deal them out into 13 piles of 4 cards each. Then, using the marriage theorem, we can show that it is always possible to select exactly 1 card from each pile, such that the 13 selected cards contain exactly one card of each rank (Ace, 2, 3...

The dark matter crisis: falsification of the current standard model of cosmology Pavel Kroupa (AIfA, Bonn) (Submitted on 11 Apr 2012 (v1), last revised 20 Jun 2012 (this version, v2)) The current standard model of cosmology (SMoC) requires The Dual Dwarf Galaxy Theorem to be true according to...

Homework Statement In the figure, a block of mass m is moving along the horizontal frictionless surface with a speed of 5.70 m/s. If the slope is 11.0° and the coefficient of kinetic friction between the block and the incline is 0.260, how far does the block travel up the incline? Homework...