Re: directional derivative,--
- From: toni.lassila@xxxxxxxxx
- Date: 25 Apr 2007 07:33:58 -0700
On Apr 24, 4:53 pm, JEMebius <jemeb...@xxxxxxxxx> wrote:
David C. Ullrich wrote:
On Tue, 24 Apr 2007 01:43:40 +0100, JEMebius <jemeb...@xxxxxxxxx>
wrote:
What about the well-known rectangular circular half-cone,
the graph of (x, y) -> z = sqrt(x^2 + y^2)?
That has no directional derivative in _any_ (non-zero) direction
(at the origin, which is presumably the point you're talking about.)
At least not according to what I've always thought was the
standard definition, as at
http://en.wikipedia.org/wiki/Directional_derivative
That is absolutely correct, if it is indeed standard to consider only entire lines through
the point in question. Is it standard in university and college math curricula? - I don't
know.
When defining mathematical concepts I want to stay as closely as possible to everyday
physical reality. So I identified "direction" with "half-line" rather than with "line".
The problem
"Prove that there is no real-valued function f such that f'(c;u) > 0
for a fixed point c in R^n and every nonzero vector u in R^n"
is very much a nonphysical one, so no such adherence to physical
reality is needed. Of course you can talk about one-sided directional
derivatives, but the theorem is not true for them.
.
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