On any smooth surface one can choose isothermal coordinates in a neighborhood of any point. What is the physical interpretation of this fact? Do the isothermals describe an equilibrium distribution of temperature? How would that be true say on a sphere?
Re: Isothermals If I understand your question, its basically just a mathematical approximation since the temperature distribution function is continuous. Basically the temperature can't jump, so if it is T at one point, it won't be too different from T at a point nearby.
Re: Isothermals No they do not describe equilibrium since as long as there is more than one isothermal heat must be flowing somewhere in the system. It flows from one isothermal to another, but does not flow along an isothermal. You cannot have part of a system in equilibrium.
Re: Isothermals equilibrium distributions of temperature that are not constant certainly exist. For an open planar region,they are solutions to the Dirichlet problem. Take a metal plate with some shape and keep its boundary at a constant temperature. eventually the temperature in the entire plate will be constant. this does not mean that heat does not flow. it means that the temperature doesn't change.
Re: Isothermals I guess another way to ask this question is , is there a physics proof for the existence of isothermal coordinates?
Re: Isothermals It would be helpful if you were a little clearer on what you are talking about. Are you talking about co-ordinates P, V, and T for a thermodynamic system? Are you talking about a surface in this P-V-T space? What is the substance that we are dealing with? A gas? Ideal gas? AM
Re: Isothermals I have been searching outside of this site and here is the answer. On a surface that is thermally isolated isothermals are lines of constant temperature and are equal to constant coordinate lines in isothermal coordinates. The physics proof might be that a surface that becomes thermally isolated achieves a constant temperature distribution. The isothermals determine isothermal coordinates and hence the complex structure. But I am guessing here.
Re: Isothermals I don't think AM meant this, I think he was referring to something I also wondered about - the common use of 3D PVT diagrams (by chemical engineers in particular). If we can follow a process along lines of constant temperature, pressure or volume to get from point A to point B in the diagram we have definite formula to perform the calculations on. General paths across surfaces in PVT space are not, in general calculable.
Re: Isothermals I don't understand what you mean by isothermal coordinates, plural. In any heat flow situation, the heat flux vector at any point has a definite direction (which may vary with time of course), therefore on a surface the isothermal line through that point is at right angles to the heat flux. In 3 dimension the isothermal surface through the point is normal to the heat flux. Those statements are true whether or not the heat flow is varying with time. For thermal problems, "equlibrium" often means "the heat flux is not time dependent". That is not the same as "the temperature is constant everywhere". In 2D, you can define a system of conformal coordinates where one set of coordinates are the isothermals, and the other set are aligned with the heat flux vectors. Is that what you meant by "isothermal coordinates"?
Re: Isothermals I am sorry to have been vague. By temperature constant I meant constant in time. I thought that isothermal coordinates was standard terminology. I guess not although the term was invented by a physicist. So the physics fact may be that a thermally isolated surface will reach an equilibrium where the temperature at each point is not changing in time. This reminds me of the Dirichlet problem. It this were true - and this is why I asked this question in a physics thread rather than a mathematics thread - then it would prove the existence of isothermal coordinates on any surface. I guess this also leads to the general problem of the long term asymptotic behavior of heat flow.
Re: Isothermals OK, Google has told me what isothermal coordinates are: http://en.wikipedia.org/wiki/Isothermal_coordinates So it is standard terminology, but not particularly about heat transfer.