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09.26 - Section 10.7 Vector functions and space curves...

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Section 10.7: Vector functions and space curves Show “The Unit Tangent Vector” (Curves 1 and 3) Group work problem: Consider a vector function v ( t ) = f ( t ), g ( t ), h ( t ) . Is it necessarily true that | v ( t )| = | v ( t )| for all t ? Why or why not? (Try to give an answer that avoids messy calculations, since they’re unenlightening and error-prone.) Section 10.8: Arc length and curvature Main ideas? … ..?.. The arc length formula: o L = a b | r ( t )| dt The curvature formulas: o κ = | d T / ds | (the definition) o κ ( t ) = | T ( t )|/| r ( t )| o κ ( t ) = | r ( t ) × r ′′ ( t )| / | r ( t )| 3 The independence of arc-length and parametrization Parametrization via arc-length The geometric definition of curvature (the osculating circle); show “Osculating Circle” (Curves 1–3)
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Problem: What is the length of the curve traced out by r ( t ) = sqrt(3) e t i + cos e t j + sin e t k as t goes from 0 to 1? ..?.. r ( t ) = … ..?.. sqrt(3) e t i e t sin e t j + e t cos e t k | r ( t )| 2 = … ..?..
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