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You might as well just say: start answering the question. Your answer is the most general answer in fluid dynamics! :rofl:

A closed vessel full of water is rotating with constant angular velocity \Omega about a horizontal axis. Show that the surfaces of equal pressure are circular cylinders whose common axis is at a height g/{\Omega}^2 above the axis of rotation. Any ideas? I do not know how to start.

If the velocity in a two-dimensional flow is given as \vec u = \left\langle {u(y),v(y),0} \right\rangle. Why must v be constant? I am not sure where to start. Can anyone help?

An infinitely long cylindrical bucket with radius a is full of water and rotates with constant angular velocity \Omega about its horizontal axis. The gravity is in the vertical direction. The velocity of the flow in cylindrical coordinates (whose z axis is the horizontal axis of the bucket) is...

How to show that these sets are nonempty (\mid means "divides")? Here N is an arbitrary large integer and q is some fixed integer. {R_{k,q}} = \{ k \in {\mathbb N}:(kN\mid k!) \wedge ((k - 1)N\mid k!) \wedge \cdots \wedge (N\mid k!) \wedge (k > Nq)\} {S_{k,q}} = \{ k \in {\mathbb...

Never mind. I do not think neither you nor I have a clue.

That was very helpful.

The reason why I think your approach is incorrect is because we are interested in energy density, but what are the units for spectral radiance? Is it watts per steradian per square meter per hertz?

I think the approach you are proposing is incorrect. There must be something easier because Maple cannot even calculate such a small quantity.

It is 10^6 I think. I forgot to add that part in my original post. In part a I applied the Stefan Boltzmann law. Is it correct?

So, for part b and c I need to find the frequency and then subtract the spectral radiances?

Are you sure this is the correct approach? How do I find the wavelength? Using Planck's Postulate or the de Broglie relations?

Homework Statement Calculate the Fermi energy for magnesium, assuming two free electrons per atom. Homework Equations {E_F} = \frac{{{\hbar ^2}}}{{2m}}{(3{\pi ^2}\rho )^{2/3}}, where \rho = q\frac{N}{V} and q is the number of free electrons. The Attempt at a Solution q = 2, so...

Homework Statement A blackbody is radiating at a temperature of 2.50 x 103 K. a) What is the total energy density of the radiation? b) What fraction of the energy is emitted in the interval between 1.00 and 1.05 eV? c) What fraction is emitted between 10.00 and 10.05 eV? Homework...

Homework Statement a) Should liquid water at room temperature be treated as indistinguishable particles? b) Should liquid helium at 4 K be treated as indistinguishable particles? The attempt at a solution Composite particles made up of spin 1/2 particles such as atoms made up of...

I am looking for a concrete example, which does not require a computer. Can anyone provide such an example?

Could you give an example with polynomials? Show how you would find the GB. Something that does not require a computational software program?

Thank you for your explanation, but could you give an example with polynomials? Something that does not require a computational software program?

Can anyone explain how it relates to the Gröbner bases?

I understand ideals and polynomial rings. I just want a brief summary.

Can anyone explain ideals and polynomial rings i.e. definitions, examples, the most important theorems, etc.?

Suppose I need a theorem or to know whether some result has been proven (or not), to prove something else. What are the best sources? Where would I find, for instance, if there is a proof that there exist (or does not exist) integers m and n such that {e^{m/n}} = \pi?

Is there a proof that there exist (or does not exist) integers m and n such that {e^{m/n}} = \pi? How would one prove such a statement?

It seems like if we let x = \sqrt 2 + \pi it does not work.

For example, let x = \sqrt 2 + \pi and assume x = p/q. Suppose we can show that xc, where c is an integer, must be an integer between two integers, namely a and b i.e. a < xc < b. If I prove that xc cannot be an integer, would it be reasonable to infer that we have a contradiction? I know it...

Are you sure this would work? I think the only problem is part 2. You cannot just define a < xc < b. You have to show that is the case. Correct? If I suppose x = \sqrt 2 + \pi. Would the same argument work? We know that both \sqrt 2 and \pi are irrational. If I I show that part 2 must hold, etc...

Yes, I know. I just gave an example. Suppose x = \sqrt 2 + \pi. Would the same argument work? I know I left some detail out, but essentially I show that xc has to be an integer. Using the same reasoning would I arrive at a contradiction?

Suppose, beforehand we show that xc is an integer, and then show it is not. By the way, is it true that \sqrt 2 c + \sqrt 3 c is never an integer? I think it is true. Then we have arrived at a contradiction.

Can anyone explain what is wrong with my reasoning? Suppose x = \frac{p}{q} and let x = \sqrt 2 + \sqrt 3 . Also, let a,b,c \in {\Bbb Z} and assume a < xc < b. If I show that xc must be an integer, and I know there does not exist c such that \sqrt 2 c, or \sqrt 3 c is an integer. Then...

Yes, I know. I actually have the link to that section in my first post. I am looking for similar formulas for the coefficients of the trigonometric Fourier series.

I cannot find it. I could not find it anywhere. I am looking for the most general formulas. Like the one for the exponential Fourier series (formulas which can be applied to any interval of the form [a, a + T]). Usually, there are different formulas for different intervals. I am looking the the...

Are there general formulas for Fourier coefficients on an integral [a, a + T], where T is the period. There is a general formula for the coefficients of exponential Fourier series. Are there general formulas for the coefficients of the trigonometric Fourier series that would work on any interval...

Thanks.

So you just equate the coefficients and then solve for both b and E, right?

Can you help? How do I show that b = 1/{a_0}? What should E be? Also, how to derive the expression for the ground state energy?

Homework Statement By substituting the wave function \psi (x) = Ax{e^{ - bx}} into the Schoedinger equation for a 1-D atom, show that a solution can be obtained for b = 1/{a_0}, where {a_0} is the Bohr radius. Homework Equations - \frac{{{\hbar ^2}}}{{2m}}\frac{{{d^2}\psi...

How to prove (not by induction) 1^{2}+2^{2}+\ldots+n^{2}=\frac{n(n+1)(2n+1)}{6}? What is the general approach for similar series, say, 1^{1}+2^{2}+\ldots+n^{n}?

It should be -0.1, right?

Do you mean it is wrong according to your interpretation or it is generally wrong? Should it be 0.1?

In this case T=\pi, but he said that I should work on the positive x-axis only. That's what bothers me. You can't do that, right? f(x) would not be periodic and it would not work because a Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines.

I need to use the the half-range sine expansion. Correct? However, the problem does not state that the function is periodic, nor that it is defined on [0,2\pi]. After all, the Gaussian function is not periodic. My instructor said that I should only consider [0,\infty), but then this would not...

I need to use the the half-range sine expansion. Correct?

It says on the interval [0,2\pi]. Does this mean f(x+2\pi k)=f(x), k\in\mathbb{Z}? A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. A real-valued function f(x) of a real variable is called periodic of period T>0 if f(x+T) = f(x) for...

Compute the sine coefficients for f(x)=e^{-x^{2}} on the interval [0,2\pi]. Does this mean f(x+2\pi k)=f(x), k\in\mathbb{Z}? Can x\in[0,\infty)?

My interpretation: find probabilities at both tails of the distribution i.e. \int^{0}_{-\infty}\psi^{2}(x)\;dx and 1-\int^{0.1}_{-\infty}\psi^{2}(x)\;dx.

Apparently there is an inconsistency in the question and there are two interpretations of the question. What would be the other one?

Can anyone help?

Homework Statement Air is mostly composed of diatomic nitrogen, N2. Assume that we can model the gas as an oscillator with an effective spring constant of 2.3 x 103 N/m and and effective oscillating mass of half the atomic mass. For what temperatures should vibration contribute to the heat...

Am I supposed to find probabilities at both tails of the distribution?

I will try to read the question more carefully next time.