{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Engineering Calculus Notes 190

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

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

View Full Document Right Arrow Icon
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 C = gr ( f ) , then there is a unique function t ( s ) defined on J and mapping onto I such that −→ q ( s ) = −→ p ( t ( s )) for all s J. The function t ( s ) is C 1 with nonzero derivative everywhere in
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern