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Unformatted text preview: MA1104 Multivariable Calculus Lecture 4 Dr KU Cheng Yeaw Thursday Jan 22, 2008 Recap Last lecture ... • we studied vectorvalued functions and their calculus • we defined arc length and curvature of a curve. We derived formulae for curvature for smooth curves. Overview In this lecture, • we conclude our study of vectorvalued functions by looking at the TNB frame. • we begin the study of (scalar) functions of several variables. First, we learn to visualize them by their (3D) graphs. • before we study their calculus (derivative and stuff), we first extend the notion of limit and continuity to functions of several variables. TNB Frame So far we have only used one single frame of reference, that is we write all vectors in terms of the standard unit vectors i , j and k . This frame can be inconvenient, and it is static. We shall define another frame of reference which is called the TNB frame. Consider an object moving along a smooth curve traced out by the vectorvalued function r ( t ) = h f ( t ) ,g ( t ) ,h ( t ) i . We shall define a reference frame that moves with the object. To do this, we require three mutually orthogonal unit vectors. One of these unit vector is the unit tangent vector T ( t ) = r ( t )  r ( t )  . Recall that T ( t ) must be orthogonal to T ( t ) for all t , this gives us a second unit vector, which is called the principal unit normal vector defined by N ( t ) = T ( t )  T ( t )  . In what direction does N ( t ) points? By the definition, N ( t ) points in the direction that T ( t ) points, which is orthogonal to the tangent vector. But we can say more. Notice from the chain rule T ( t ) = d T ( t ) dt = d T ( s ) ds ds dt . Therefore N ( t ) = T ( t )  T ( t )  = d T ds ds dt d T ds ds dt = d T ds d T ds since ds dt =  r ( t )  > for a smooth curve. Recall that κ = d T ds . If κ > , then we have N ( t ) = 1 κ d T ds , and so N ( t ) has the same direction as d T ds which is the instantaneous rate of change of the unit tangent vector with respect to arc length. d T ds points in the direction in which T turns as arc length increases. So N always points to the concave side of the curve. To get a third unit vector orthogonal to both T ( t ) and N ( t ) , we take their cross product. We define the binormal vector B ( t ) to be B ( t ) = T ( t ) × N ( t ) . It remains to check that B ( t ) is a unit vector. Since T ( t ) and N ( t ) are orthogonal, we have  B ( t )  =  T ( t )  N ( t )  sin π 2 =  T ( t )  N ( t )  = 1 , as desired. This triple of three unit vectors T ( t ) , N ( t ) and B ( t ) forms a frame of reference, called the TNB frame that moves along the curve traced out by r ( t ) . Normal and Osculating Plane For each point on a curve, the plane passing through that point and determined by N ( t ) and B ( t ) is called the normal plane ....
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 Spring '08
 Kuchengyeaw
 Arc Length, Multivariable Calculus, lim

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