Re: Taylor's Theorem


Also, for multivariable calculus, does anyone have a reference to the
most accurate and most general form of Taylor's theorem?


I've seen two forms of the theorem with different hypotheses:

1) Requires k+1 order partial derivatives to be continuous on some
neighbourhood of the reference point. Then says for every x in that
neighbourhood, there exists a c in line segment joining ref. point and
x such that...

2) Requires k+1 order partial derivative to be continuous on line
segment joining reference point and x. Then says there exists a c in
this line segment such that....


The second form is more general than the first, but there must be a
reason the first form was stated that way in Apostol's Calculus volume
1 (among other places). Anyone know if the 2nd form is valid?

.