Engineering Calculus Notes 313

Engineering Calculus Notes 313 - 3.5. SURFACES AND THEIR...

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: 3.5. SURFACES AND THEIR TANGENT PLANES 301 which are independent provided R = 0 and φ is not a multiple of π ; the latter is required because → ∂− p → → (nπ, θ ) = −(R sin(nπ ) sin θ )− − (R sin(nπ ) cos θ )− ı ∂θ → − = 0. Regular parametrizations of surfaces share a pleasant property with regular parametrizations of curves: → Proposition 3.5.7. A regular function − : R2 → R3 is locally p one-to-one—that is, for every point (s0 , t0 ) in the domain there exists → δ > 0 such that the restriction of − (s, t) to parameter values with p |s − s0 | < δ | t − t0 | < δ is one-to-one: (s1 , t1 ) = (s2 , t2 ) guarantees that → − (s , t ) = − (s , t ) . → p 11 p 22 Note as before that the condition (s1 , t1 ) = (s2 , t2 ) allows one pair of coordinates to be equal, provided the other pair is not; similarly, → − (s , t ) = − (s , t ) requires only that they differ in at least one → p 11 p 22 coordinate. Before proving Proposition 3.5.7, we establish a technical lemma. → → Lemma 3.5.8. Suppose − and − are linearly independent vectors. Then v w → → there exists a number K (v, w) > 0, depending continuously on − and − , v w such that for any θ → → →→ (cos θ )− + (sin θ )− ≥ K (− , − ). v w vw → → The significance of this particular combination of − and − is that the v w coefficients, regarded as a vector (cos θ, sin θ ), form a unit vector. Any → → other combination of − and − is a scalar multiple of one of this type. v w ...
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