In thermodynamics exist equations with the form:

dx/x = c/y^2 dy

Where c, x and y are dimentionful quantities. By taking indefinite integrals one gets

log x = – c / y + constant

If log x is an element of an one dimensional affine space (as said above), what exactly (or exactly enough for a non mathematician ;-) ) means the equality? Where does “constant” belong? and – c / y ? There are dimensional inconsistencies? If x units are Kelvin, what is the log x unity?

Thanks.

]]>Excellent; thank you very much. I guess now I have a lot of food for thought and if anything worthwhile strikes my mind I will post back.

]]>(2) The identification between projective spaces in this context comes from the following observation: if V is a real vector space and R is a one-dimensional vector space, and one forms the tensor product (which is a vector space of the same dimension as V) then the projective spaces and are canonically isomorphic, by identifying with for any non-zero (one can check that the class of does not depend on r).

In the case of F=ma, acceleration takes values in the three-dimensional vector space , where is the three-dimensional space of displacement vectors, and is the one-dimensional vector space of dimensionality equal to time raised to the negative two power. Similarly force takes values in where is the one-dimensional vector space of dimensionality equal to that of mass. By the previous discussion, the projective space and are canonically isomorphic, thus allowing one to assign meaning to the statement that force and acceleration have the same direction.

(3) one can indeed work in coordinates if desired, though of course the co-ordinate independence of one’s constructions becomes less clear when doing so.

]]>1) how do I build or go from vector spaces to projective spaces?

2) what does it mean for projective spaces to be canonically (naturally?) identifiable (isomorphic?)??

3) is this related to the expression of any (dimensionful) vector as a sort of linear combination of dimensionless unit vectors () and then being able to naturally identify the corresponding unit vectors somehow? If that is the case, I do not here intuitively get what it means to say they are naturally isomorphic, as opposed, for instance, to the identification we make between a given vector space and the dual of its dual! Here it is easy to see the independence of the starting basis! ]]>