Engineering Calculus Notes 190

Engineering Calculus Notes 190 - 178 CHAPTER 2. CURVES...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 178 CHAPTER 2. CURVES Definition 2.4.1. A vector-valued function p ( t ) defined on an interval I is regular on I if it is C 1 and has nonvanishing velocity for every parameter value in I . (The velocity vector at an endpoint, if included in I , is the one-sided derivative, from the inside.) A vector-valued function p ( t ) on an interval I is a regular parametrization of the curve C if the image of I under p equals C (i.e., p maps I onto 15 C ). When p ( t ) is a regular parametrization of the graph of a function f ( x ), the tangent line agrees with the usual one, namely the graph of the linearization of f ( x ). This suggests that the tangent line is a geometric object, attached to a curve independent of the vector-valued function used to parametrize it. To study this question, we first study the different ways a graph can be parametrized. Lemma 2.4.2. If p ( t ) and q ( s ) defined on the interval I (resp. J ) are both regular parametrizations of a graph...
View Full Document

Ask a homework question - tutors are online