Lecture 5 MA 261
Lecture 5 MA 261 MA 26100
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This page Class Notes was uploaded by Viktoryia Zhuleva on Sunday January 24, 2016. The Class Notes belongs to MA 26100 at Purdue University taught by Patricia E Bauman in Fall 2015. Since its upload, it has received 117 views. For similar materials see Multivariable Calculus in Applied Mathematics at Purdue University.
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Date Created: 01/24/16
Lecture 5 Vector Functions and Space Curves Definition A vector function is a function whose domain is a set of real numbers and whose range is a set of vectors m lt ft go he gt not gar hook Example Let rt lt t3 V 5 t lnt gt Find the domain of this function In order to find the domain of the entire function we have to find the domain of separate functions and then nd their common domain It means that the chosen values of t for each function in the domain must be true f t t3is defined for all t E 1R gt 2 V5 t is defined for 5 2 1 Mt lnt is defined for t gt 0 So if we combine all of the domains into one we will get 1 O lt t S 5 Definition the limit of TD lt f 1 gt ht gt as t gt a is defined as Iimft lt lim ft lim gt lim ht gt t a t a t a t a provided every limit on the right exists Example Compute lt Vt 3 11 sint gt ltirr11t x1 VZ2 t 1 t 1 1 1 11m2 11m 11m t 1 t 1 t 1t 1t1 t 1 t1 2 ltirr115int sin1 gt l So rt 2 2 s1n1 Definition A vector function rt lt f 1 gt ht gt is continuous at a if lim rt ra a In other words lim rt exists ra is defined and ra 2t lim rt a a Definition If f t gt and ht are continuous functions on an interval I the set of all points x y 2 when x f t y 2 gt z ht as 1 values in I is called a space curve or curve We take the direction of the space curve to be the direction as 1 increases Also x f t y 2 gt z 2 Mt are called parametric equations m Example Sketch the curve x cost y sint z t for V 39 O S t S 7t V First look at the relation between x y 2 Notice that x2 y2 r 1 since sin12 cost2 1 So ftgt moves counterclockwise as 1 increases hf 21 Also 2 increases as 1 increases Next step is to find individual vectors by plugging in values for parameter t 713 lt cost sint t gt T0 lt 100 gt m lt 01 gt 19372 Tn lt 10n gt 7137 lt 0 132 gt m lt 10 2n gt Example Sketch T39t lt xt t gt Domain is t 2 0 Since I is a nonnegative number xt2 t Which means x2 y r0 lt 00 gt r1 lt 11 gt r2 lt xZ 2 gt etc Two curves 7111 and 1750 collide at a point P x0 yo ZO if they arrive at P at the same time ie EOE 7750 lt x0y0zo gt for some t Two curves intersect if EOE 2 7753 lt x0 yo Z0 gt for some t and some S Example 7111 lt t 1 61 4 t 4 gt and 1750 lt t 3 t2 2t gt Find out if these two paths collide or intersect First let s see if they collide 7111 7750 t 1t 3 6t4t2 t 4 2t All of those three equations must be true in order for both paths to collide However in this case since the very first equation I 1 t 3 fails to be true paths do not collide Although curves don t collide they still might intersect at some point P Solve for 7111 2 1753 t 1S3 6t4s2 t 4 Zt Solve for S from the first equation and plug it into second S t 2 6t 4 t 22 6t4t24t4 t2 2tO tt 2O t00rt2 320rs4 In this case we have two roots for the equation they both can be the solution for the entire set two of them can be solutions or both of them can just not fit In order to check if two curves intersect we have to plug in two sets of t and S and check whether the equations are true It turns out that t O S 2 is the set of numbers that makes all equations to be true statements you can check it yourself And to find the point of curves intersection simply plug in either I or S into curve equations respectively Answer curves intersect at P 1 4 4
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