
Re: Question on Rudin's Principles of Mathematical Analysis 3rd edition  addressed to those who have access to book
Posted:
Jun 3, 2013 1:06 PM


On 2 Jun 2013 18:55:54 GMT, Bart Goddard <goddardbe@netscape.net> wrote:
>dullrich@sprynet.com wrote in >news:113nq812gfc5jgal5goutsdb190b6mg2r1@4ax.com: > >>>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 lowlevel 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.

