Engineering Calculus Notes 320

Engineering Calculus Notes 320 - any curve in S ( t ) = p (...

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

View Full Document Right Arrow Icon
308 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION the parameters constant will give a vector in this plane: holding s constant at s = s 0 , we can take r = r 0 + t to get −→ γ ( t ) = −→ p ( r 0 + t,s 0 ) whose velocity at t = t 0 is −→ v r (0) = −→ p ∂r and similarly, the velocity obtained by holding r = r 0 and letting s = s 0 + t will be −→ v s (0) = −→ p ∂s . Because P is a regular point, these are linearly independent and so form direction vectors for a parametrization of a plane T ( r 0 ,s 0 ) −→ p ( r 0 + r,s 0 + s ) = −→ p ( r 0 ,s 0 ) + r −→ p ∂r + s −→ p ∂s . By looking at the components of this vector equation, we easily see that each component of T ( r 0 ,s 0 ) −→ p ( r 0 + r,s 0 + s ) is the linearization of the corresponding component of −→ p ( r,s ), and so has Frst order contact with it at t = 0. It follows, from arguments that are by now familiar, that for
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: any curve in S ( t ) = p ( r ( t ) ,s ( t )) = ( x ( r ( t ) ,s ( t )) ,y ( r ( t ) ,s ( t )) ,z ( r ( t ) ,s ( t ))) the velocity vector v (0) = p r dr dt + p s ds dt lies in the plane parametrized by T p . It is also a straightforward argument to show that this parametrization of the tangent plane has frst order contact with p ( r,s ) at ( r,s ) = ( r ,s ), in the sense that v v v p ( r + r,s + s ) T ( r ,s ) p ( r + r,s + s ) v v v = o ( b ( r, s ) b ) as ( r, s ) ....
View Full Document

This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

Ask a homework question - tutors are online