Upon further reflection, I do not believe that oblique coordinates enter the problem at all. Rather, the local unit vectors (\hat{r}_0,\hat{\theta}_0) used to span the vector d\vec{r} (as defined in your next-to-last reply) are not principal axes, but may be rotated by the appropriate...
Ok, I get your point. BTW, do you see an alternative way to express dr (in last Wednesday's reply) that avoids differentiation of a(r,θ) and b(r,θ) if u itself were a first-order quantity, i.e., u ⋅ u << |u| ?
Now it's gotten more interesting! I follow your argument and math 'til the three bullet points; there you lose me: I would argue the second bullet point to be the correct answer regardless of whether vectors u and v (in your notation) are or are not orthonormal (as it happens, the functions...
Ok, all this is well-known to me, but we may now be able to get to the gist of my question with more ease: Given the presence of u(x) in the delta function, the central question is how should your second equation above be modified, or, to put it crudely, how should the components of u be...
JAmbaugh - many thanks for the detailed reply! I can see where my notation can be confusing: I use r and θ as unit vectors and (r,θ) to denote an explicit dependence on spherical coordinates.
I am familiar with the Jacobian transformation you describe. Unfortunately it leads to a...
The following integral arises in the calculation of the new density of a non-uniform elastic medium under stress:
∫dx ρ(r,θ)δ(x+u(x)-x')
where ρ is a known mass density and u = ru_r+θu_θ a known vector function of spherical coordinates (r,θ) (no azimuthal dependence). How should the Dirac...