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Unformatted text preview: 14.1. Vector Functions and Space Curves Vector function and component functions. r ( t ) = h f ( t ) , g ( t ) , h ( t ) i = f ( t ) i + g ( t ) j + h ( t ) k , t [ a, b ] The domain is a set of numbers but the range is a set of vectors. Example 1. r ( t ) = h cos( t ) , sin( t ) , t i = cos( t ) i + sin( t ) j + t k , t [0 , 2 ] . Limit of a vector function. lim t a r ( t ) = h lim t a f ( t ) , lim t a g ( t ) , lim t a h ( t ) i . Continuous vector function. A vector function r ( t ) is continuous at a if lim t a r ( t ) = r ( a ). Space curve. If we consider r ( t ) as a position vector then when t varies, the position vector r ( t ) as a pointer is shaping a curve in the space, called a space curve . The points from this curve are with coordinates x = f ( t ) , y = g ( t ) , z = h ( t ) . Example 2. Sketch (describe) the curve given by the vector function in Example 1. This curve is called helix . Example 3. Describe the curve given by the vector equation r ( t ) = h 1 + t,- 1 + 2 t, 1- t i . Planar curve. r ( t ) = h t, t 2 i = t i + t 2 j . 1 Hence, a space curve permits one-parametric representation in terms of a vector function similar to the planar curves (the polar curves). Conversely, each vector function can be con- sidered geometrically as a space curve....
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This note was uploaded on 02/05/2010 for the course MATH 264 taught by Professor Dr.d.dryanov during the Fall '09 term at Concordia Canada.
- Fall '09