Slide_03 - MA1104 Multivariable Calculus Lecture 3 Dr KU...

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Unformatted text preview: MA1104 Multivariable Calculus Lecture 3 Dr KU Cheng Yeaw Monday Jan 19, 2008 Recap Last lecture ... • we studied equation of lines and planes. We observed the simplicity and usefulness of vectors to describe lines and planes. • we extended our knowledge of 3-D surfaces by introducing quadric surfaces in addition to planes and spheres. We visualize them using traces. Overview In this lecture, • we will study vector-valued functions and their calculus (limit, continuity, derivative, integral). We observe the similarities between calculus of vector functions and single variable calculus. • we introduce the notion of arc length and curvature. Arc length measures the length of a curve. Curvature measures the sharpness of a curve. Vector-valued Functions We may use a curve in R 3 to describe the path traced out by a moving object in space. However, the curve alone does not tell us the position of the object at a particular time. It turns out that it is better and more convenient to describe its position at any given time by a position vector. This is the concept of a vector-valued function. A vector-valued function r ( t ) is a mapping from its domain D ⊆ R to its range R ⊆ V 3 , so that for each t ∈ D , r ( t ) = v for exactly one vector v ∈ V 3 . We write a vector-valued function as r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k or r ( t ) = h f ( t ) ,g ( t ) ,h ( t ) i for some scalar function f , g and h (called the component functions of r ) . Example 1. Sketch the curve traced out by the vector-valued function r ( t ) = sin t i- 3cos t j + 2 t k . Solution. There is a relationship between x and y here: x 2 + y 3 2 = sin 2 t + cos 2 t = 1 which is the equation of an ellipse in 2-D. In 3-D, since the equation does not involve z , it becomes the equation of an elliptic cylinder whose axis is the z-axis. The curve will wind its way up the cylinder anticlockwise as t increases. The starting point is (0 ,- 3 , 0) . We call this curve an elliptical helix . Arc Length in R 3 A natural question about a curve in space is ‘How long is it?’. This is the arc length of the curve. Suppose that a curve is traced out by the endpoint of r ( t ) = h f ( t ) ,g ( t ) ,h ( t ) i where f , f , g , g , h , h are all continuous for t ∈ [ a,b ] where the curve is traversed exactly once as t increases from a to b . We can approximate the arc length of this curve as follows: Step 1. Partition the interval [ a,b ] into n subintervals of equal size: a = t < t 1 < ··· < t n = b , where t i- t i- 1 = 4 t = b- a n for all i = 1 , 2 ,...,n . Step 2. Let s i denote the arc length of that portion of the curve traced out as t increases from t i- 1 to t i . We can approximate s i by the distance of the point ( f ( t i ) ,g ( t i ) ,h ( t i ) from ( f ( t i- 1 ) ,g ( t i- 1 ) ,h ( t i- 1 )) (since f , g and h are continuous). By the distance formula, s i ≈ p ( f ( t i )- f ( t i- 1 )) 2 + ( g ( t i )- g ( t i- 1 )) 2 + ( h ( t i )- h ( t i- 1 )) 2 . Applying the Mean Value Theorem (why can we do this), we get...
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This note was uploaded on 03/19/2012 for the course MATH 1104 taught by Professor Kuchengyeaw during the Spring '08 term at National University of Singapore.

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Slide_03 - MA1104 Multivariable Calculus Lecture 3 Dr KU...

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