Engineering Calculus Notes 313

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

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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 ...
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## 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.

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