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1. Yes, that is the limiting process that I described. I assume you know that you need to cancel the epsilon from the first three terms in the numerator of your limit. 2. Yes, you are correct in saying that Killing's equation is a necessary consequence of the isometry condition. Your final...

It is called the directional derivative, or the covariant derivative.

As I recall, a nice elementary text on turbulence is Tennekes and Lumley. They give some nice intuitive explanations of the meaning of the quantities of interest. The bible of turbulence is the Yaglom and Monin two volume work.

Yes, qualitatively. The mean value of the products u'v' is not related to vorticity (as far as I know) but measures the correlation of the components of the vector fluctuation. Look up a statistics book on the the meaning of correlation coefficients.

The variance is the mean square deviation from the mean. The reason one does not use the standard deviation is because it involves a nasty square root. But they measure essentially the same thing: how much the distribution is spread out around the mean. I am not sure what you are denoting by v...

Turbulence is a statistical theory. The velocity field is a random variable. That is, at each point in space at a given time, the value of the velocity v(x,t) is a random variable. Technically, one should imagine that one has set up in a laboratory ten trillion identical copies of the fluid...

The Killing equation is certainly a necessary condition, as you note. The converse of the theorem is more difficult to prove, and a theorem may not admit an exact converse but only a partial converse. In fact, most theorems that involve differentiation usually only admit partial converses...

The displacement y of the string from its equilibrium position (y=0) is a function of both x and t - it depends on where along the string you look, and on the time at which you look at it. Figure 16.19 shows the position of the string at a given time t. So when you find the slope of the tangent...

BSc majoring in physics and astrophysics. They begin astrophysics in the first year. We use Bennett in the first semester and are struggling to find a text pitched at the correct level for the second semester.

My apologies. I assumed this was a freshman problem. I guess I should call myself "Dr Stupid", right?

Don't forget the negative sign.

Use the conservation of energy to work this out. To reach the moon, the velocity of the bullet must never reverse. To reverse, it must pass through the value zero. Take the potential energy of the Earth and of the Moon into account. In the limiting case, the velocity will be zero at the first...

The curl of the E-field is zero only if the E-field is time independent, i.e. electrostatics. In the dynamic case, the curl of the E-field is the negative rate of change of the B-field, which is not zero in general. This makes the dynamic E-field non-conservative, with the path integral around a...

Since the chamber into which the gas expands is a vacuum, the gas pushes against nothing at all. So, if the outer walls are rigid, the gas does no work. Not so? The expansion is rapid, much faster than the thermal adjustment time, so the expansion process involves zero heat intake. The internal...

Do you have any other recommendations? I am looking for a book to use with an undergraduate class - freshman level - that is not as difficult as Carroll and Ostlie but more challenging than Bennett, from which I currently teach. The good books I know all appear to be out of print.

There is no hard and fast definition of "mathematical physics", or of its allied term "theoretical physics". The best that I can do is tell you how I use these terms. For me, a "mathematical physicist" is someone who is interested principally in the mathematics that underlies and is exploited...

You would find 2 books to be useful as a beginner: 1. Bennett, The cosmic Perspective - this is an introductory text that with a minimum of physics and mathematics examines the principal astrophysical systems in the universe, provides an understanding of their properties and has a wealth of...

Relativity is about how space and time is measured in different inertial frames of reference. Zero subscripted quantities generally refer to the value of that quantity measured in an inertial frame in which the object is at rest. Unsubscripted quantities refer to the value measure in an inertial...

The time dependent factor f(t) is not part of the delta function. It therefore can be taken out of the integration, since the integral does not involve time. This leaves you wth an integral over all space of the delta function, which equals 1.

Write the Lagrangian density in terms of the derivatives of the vector potential. Then use the fact that ## \frac{\partial (\partial_\mu A_\nu)}{\partial (\partial_\sigma A_\tau)} = \delta^\mu_\sigma \delta^\nu_\tau ##

Always use ## \sum^{\infty}_{m=0}a_mx^{m+k}##

Ultimately, you are looking for polynomial solutions. A polynomial need not begin at x^0. So you allow it to begin at x^k. You could begin by substituting into the equation the series ##y(x)=\sum^{\infty}_{m=0}a_mx^{m}## But then, if the solution polynomial does begin with the x^k term, your...

Open means that the set includes all points in its interior, and does not include any of its boundary points. Convex means that, if you join any two points in the set with a line segment, the line segment lies completely in the interior of the set. There are more precise definitions of both...

I suppose we always want to find reasons for why things are as they are. If you have only ever seen one Lagrangian for a dynamical system, and you are unaware that there may be many others that lead to the same equations of motion, you may be tempted to try to "understand" why the Lagrangian...

My feeling is this - "feeling" because I am not sure that my justification is correct - if the form of the Lagrangian for a given system is not unique, does it make sense to look for an interpretation for the particular form of the Lagrangian that you have chosen to use? Why interpret one and...

Good question. In spite of the various rationalizations that you report and that I have seen, I personally don't know whether the question is even sensible. The Lagrangian is not uniquely defined. Many Lagrangians give rise to the same equations of motion. In the end, it is the equations of...

If you are doing indefinite integrals, my guess is that the answer you want is u(x,y).

Elementary vector mechanics. Try a book like Serway, Physics for scientists and engineers, or any freshman Physics text.

Are you sure? That answer does not look right to me. u should be a function of two variables.

Thanks. Very nice. I seem to have a lot of unnatural desires.

Try reading Marsden and Tromba, Vector Calculus. Or any other freshman math book, like Thomas and Finney. You have not been very explicit in what you ask. It is possible to write down a general symbolic expression for this integral. But you probably have something more particular in mind...

In colloquial speech, "crest" might refer loosely to the curved section of the wave above the equilibrium position. In physics, we usually mean the point along the wave where the wave reaches its maximum displacement.

The tension will be the same along the entire string, assuming that there is zero friction at the bridge. Friction at the bridge might cause a difference in the tension on the two sides. The rest is a simple vector problem. You need to measure, or estimate, the angles made by the string on the...

Fantastic! Well done!

Use proper time. This requires neither a Lorentz transformation, nor any calculation involving length contraction or time dilation. Proper time is the time read by an on-board clock - the number of ticks of the clock, if you like. We all have an on-board clock, a ticker that one day will stop...

Looks like your units may be wrong. The density you give is atoms/m^3. Scanning the units in the above formula, I think Mu (your u) should be in kg, and the density in the Jeans formula should be kg/m^3. That might give you the additional powers of 10 that you need to get a reasonable answer...

You are deducing the results for a system in thermal interaction with an heat bath by considering the heat bath + the system as an isolated system and using the max-entropy principle for the combined system. See H B Callen's book on thermodynamics.

Looks like the transformation law for the skew symmetric matrix. To understand it, you will need to understand the notation first. It may help you to know that the statement reads: the skew symmetric matrix of the transformed angular velocity vector is equal to the transform of the skew...

Brilliant. Amazing how simple solutions often escape one's attention! Something about trees and the wood ... Of course, this also works with any surface that can be "rolled out" onto a plain with our distorting the surface, such as a cylinder, or any ruled surface like a single sheeted...

I'm not sure what kind of answer you want. The Fourier transform expresses a function as a linear combination of the complex exponentials of the form e^{ikx} or e^{i \omega t}. These functions are linearly independent functions that span an infinite dimensional linear (or, vector) space. The...

Do you know the theory of maxima and minima in the differential calculus? You have considered a path consisting of two segments. There are other such paths. Instead of walking to the corner, walk on one face to some intermediate point up one edge (say x from the base), and then across a second...

First, you have not written an equation, but a cubic expression. Probably an oversight on your part. Your expression contains - \lambda^3 . So it must arise from an expression of the form - (\lambda -a)(\lambda -b)(\lambda -c) . You can get rid of the negative in the characteristic equation...

Get a formula for the angle between the fields in the new frame. It will involve v and the given angle in the old frame. Set the new angle to zero and solve for v. The most obvious way to proceed is by adapting a coordinate system to your data - we can choose whatever coordinates we please, so...

You need to let your course administrator for Calculus III what your situation is. With a doctor's certificate and a copy of your father's death certificate, you ought to be given an exemption for the quizzes that you missed. That may put you into a position where you might be able to score...

Looks correct.

Convert the frequency \omega_1 of the wave in the lab frame to \omega_1' in the frame of the mirror. The, as you correctly state, the reflected wave has \omega_2' = \omega_1' . Convert the frequency \omega_2' of the wave in the mirror frame back to the lab frame. The conversion is done by...

The reason for setting problems like this is to familiarise you with the math and to help you to develop skill in computation. To use software at this stage of the game is counterproductive. In your problem, you should have recognised that you need to find the series for sin x and the series...

Then, apart from the fact that you have ignored the second branch of the square root function, your answer is correct. It may be better to quote the solution function u(x,t) in implicit form rather than to solve it explicitly for u. That way you include both branches of the square root function.

I am not sure on what aspect of the above you are asking for a comment. I don't like your persistent use of the word "model". If you are attempting to describe the real world by mathematics, the word "model" is appropriate. In that situation, one is constructing a mathematical model of some...

I think that they may want a slightly more general answer. Have you quoted the question precisely as it was stated?