Recent content by Rearden

Thanks for pointing out that distinction, it clears things up.

Afraid I still don't follow. Why is the material derivative involved at all? In the ordinary course of physics we never swap a partial derivative for a total derivative, and as I understand it, the material derivative gives: \begin{equation} \frac {df(\mathbf x (\mathbf X,t),t)} {dt}...

But isn't the material derivative given by the total derivative wrt time, \begin{equation} \frac {df} {dt} = \frac {\partial f} {\partial t} + (\mathbf v \cdot \nabla) f \end{equation} whereas the integrand over the control volume just contains a partial derivative?

Hi, I'm struggling to understand one step in the derivation of the Reynold's Transport Theorem. I get as far as: \begin{equation} \frac{d}{dt} \int_{V(t)} f(\mathbf x, t) \, dV = \frac {d} {dt} \int_{V_0} f(\mathbf X,t) J(\mathbf X,t) \, dV = \int_{V_0} J \frac {\partial f} {\partial t}...

Neat way of phrasing it! I'm also fairly sure it's a "yes" given the original context of the problem: Poisson's equation (i.e. Newtonian gravity) is invariant under the Euclidean transformations. Because the space we see around us is manifestly Euclidean, I don't think the equation has any...

Hi everyone, I was wondering: if a space is invariant under Poincare transformations, does that mean it has to be Minkowski space? Or could it have some further isometries? By the same token, if a space is invariant under the orthogonal transformations, does it have to be Euclidean? I...

Sorry, I wasn't thinking hard enough about integration...I can see why it's an honest scalar now. Thanks everyone!

Hi, I'm a little confused as to the nature of mass density. I've always seen it referred to as a scalar. Now by conservation of mass, when you integrate mass density over a volume, you get a scalar quantity. But volume transforms like a scalar density of weight 1, so shouldn't mass density...

Exactly what I needed, Thanks a lot!

I think I understand. So does a four-velocity vector expressed in flat spacetime with a Cartesian basis have exactly the same components as the corresponding tangent vector on a manifold, despite the nominally different basis vectors?

Hi, I'm having trouble understanding how people can make calculations using the partial derivatives as basis vectors on a manifold. Are you allowed to specify a scalar field on which they can operate? eg. in GR, can you define f(x,y,z,t) = x + y + z + t, in order to recover the Cartesian...

Sorry to bother again...I now understand the derivation of the Noether current, but which field do I apply it to if I want the standard stress-energy tensor? Is this where "spacetime as a perfect fluid" comes into play?

That's exactly what I need! Thanks again

Hi, I was wondering if the stress-energy tensor arose naturally in special relativity in the same way that plain energy and momentum do via Lagrangians. I understand Noether's theorem for particles, but Wikipedia describes the stress-energy tensor as a Noether current; can anyone explain what...

Thanks for the help :) makes things much clearer