Unformatted text preview: t b. To show that is continuous at the number t we shall show that for every number 0 we
have
lim u .
lim u
u t Suppose that ut 0. Using the fact that the function is continuous at t, we choose a number 0 such that
a t t t b
and such that the inequality
d t , u
4
holds whenever t u t . Now we choose a partition
u 0 , u 1 , u 2 , , u n
of the interval a, b such that
n b d uu 1 , uj j1 4 . Since the inclusion of extra points in this partition makes the sum
n d uj 1 , uj j1 larger, we may assume, without loss of generality that t is a point in the partition and that if t u k then
both of the points u k 1 and u k1 lie in the interval t , t . t
uk t
uk 1 Now given any m 1, 2, , n we clearly have 403 t
u k 1 m d uj 1 , uj um j1 u m is the length of the restriction of the curve to the interval u m , b , we also have and, since b n d uj , uj 1 b um . jm1 Therefore if m 1, 2, , n then since
n b m d uu 1 , uj n um d uj j1 , uj 1 b um j1 d uj 1 , uj jm1 we conclude that
m d uj um 1 , uj j1 4 . We conclude that
k 1 u k 1 d uj 1 , uj d uj 1 , uj d uk j1 4 k1 , uk 1 d u k , u k 1 j1 uk 1 4 4 4 4 Thus
lim u u t u k 1 u k 1 3 lim u .
ut
4 6. Suppose that is a rectifiable curve with domain a, b in a metric space X. Prove that for every number
there exists a number 0 such that whenever P is a partition of a, b and P we have
L P, L  . 0 Hint: Examine the method of proof of Darboux’s theorem.
7. Suppose that 1 and 2 are curves in R k with domain a, b . The sum 1 2 and dot product 1 2 of 1
and 2 are defined by the equations
1 2 t 1 t 2 t
and
1 2 t 1 t 2 t
for each t
a, b . If f is a real valued function defined on the interval a, b then we define f 1 by the
equation
f 1 t f t 1 t
for every t
a, b . Prove that if, for a given number t
a, b the derivatives 1 t and 2 t and f t exist
then
1 2 t 1 t 2 t
1 2 t 1 t 2 t 1 t 2 t f 1 t f t 1 t f t 1 t
All of these facts follow directly from the definitions.
8. Prove that if is a differentiable curve in R k and if the function is constant then is the constant
function zero. Interpret this fact geometrically.
If we define f t t 2 then, for each t, 404 0 f t t t t t 2 t t .
A geometric interpretation of this fact is that if the path runs around in a sphere with center O
then the velocity , being perpendicular to the line that runs from O to , is tangential to the
sphere.
9. If a curve in R k is differentiable at a number t and if t
t is defined by the equation
t
Tt
t O then the unit tangent T t of at the number
. This is standard stuff and can be found in any elementary calculus text.
a. Prove that if t exists then the derivative T t of T exists at the number t and we have...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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