On 2 Jun 2013 18:55:54 GMT, Bart Goddard <firstname.lastname@example.org> wrote:
>email@example.com wrote in >news:firstname.lastname@example.org: > >>>Sorry, I think I can answer my own question. The Jacobian (as defined >>>by Rudin, anyway) means the determinant. >> >> As far as I know the Jacobian always means the determinant. Or the >> absolute value of the determinant. Who defines it otherwise? >> (In what seems to me to be very standard terminology, the >> matrix in question is the _derivative_.) > >My prof, in 1980, stressed that the derivative was not the >matrix, but the linear transformation which the matrix represents.
Well of course, that's what the derivative _really_ is. My point was just that the matrix (or the linear transformation) is definitely not the Jacobian.
As you suggest below, saying the matrix is the derivative seems like a much better idea in low-level classes, where abstraction is something one wants to avoid. But among grownups yes, of course the derivative is that linear transformation.
>We were using Spivak's Calculus on Manifolds, which I don't have >near me at the moment, so I don't know if Spivak maintained the >same distinction. (That prof was Jim Munkres.) > >I've never found the distinction valuable, so I don't make it >when I teach vector calc stuff. > >B.