Recent content by MarcusAgrippa

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.