Given two Ck parametrized curves ?:(a,b)??Rn and?:(c,d)??Rn, we say ? is a repar

Question

Given two Ck parametrized curves ?:(a,b)??Rn and?:(c,d)??Rn,

we say ? is a reparametrization of ? if there exists a Ck function ? : (a,b) ?? (c,d) with a Ck inverse ??1 : (c,d) ?? (a,b) such that ?=???(i.e. ?(t)=?(?(t))foranyt).

Now let ? : (a,b) ? R2 be a Ck (k ? 1) parametrized curve given by ?(t) = (x(t), y(t)). Suppose x?(t0) ?= 0 for some t0 ? (a, b). Prove that there exists a ? > 0 such that

? : ( t 0 ? ? , t 0 + ? ) ?? R 2
can be reparametrized as a graph of some Ck function y = y(x), sitting

over an open interval (x(t0) ? ?, x(t0) + ?) in the x-axis.

Explanation / Answer

We say a vector function fW.a; b/ ! R

3

is C

k

(k D 0; 1; 2; : : :) if f and its ?rst k derivatives, f

0

, f

00

, . . . ,

f

.k/, are all continuous. We say f is smooth if f is C

k

for every positive integer k. A parametrized curve is a

C

3

(or smooth) map ?W I ! R

3

for some interval I D .a; b/ or

Related Questions

Navigate

• Browse (All)
• Browse G
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.