This preview shows page 1. Sign up to view the full content.
Unformatted text preview: and x ! U , there exists B (x, r) ) U . Fix h < r, then the
segment joining the point x + hei with x is contained in B (x, r). Deﬁne the
curve . : [a, b + 1] * RN as follows
&
 (t)
if t ! [a, b] ,
. (t) :=
x + (t ' b) hei if t ! [b, b + 1] .
Using (ii), we have that
A
A
f (x + hei ) =
g = f (x) +
' = f (x) + b A b+1 A j =1 gj (x + (t ' b) hei ) h)ij dt gi (x + (t ' b) hei ) h dt = b = f (x) + N
b+1 + h gi (x + sei ) ds, 0 where in the last equality we have used the change of variable s = (t ' b) h. It
follows by the mean value theorem that
A
1h
f (x + hei ) ' f (x)
=
gi (x + sei ) ds = gi (x + sh ei ) ,
h
h0
where sh is between 0 and h. As h * 0, we have that sh * 0 and so x+sh ei * x.
Using the continuity of gi , we have that there exists
lim h$0 f (x + hei ) ' f (x)
= lim gi (x + sh ei ) = gi (x) ,
h$0
h which proves the claim.
The equivalence between (ii) and (iii) is left as an exercise.
Next we give a simple necessary condition for a ﬁeld g to be conservative.
61...
View
Full
Document
 Spring '14

Click to edit the document details