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Chapter2vectorvaluedfunction 150105020944-conversion-gate02
- 1. VECTOR-VALUED FUNCTION rahimahj@ump.edu.my Prepared by :MISS RAHIMAH JUSOH @ AWANG
- 2. ectorposition v aasexpressedbecanD-3inequationcurveorlineA kjir zyx :equationparametricin the )(tfx )(tgy )(thz kjir )()()((t) thtgtf )(),(),((t) thtgtfr
- 3. DOMAIN Example 1 Determine the domain of the fo11owing function cos ,1n 4 , 1 So1ution: The first component is defined for a11 's. The second component is on1y defined for 4. The third component is on1y defined for t t t t t t r 1. Putting a11 of these together gives the fo11owing domain. 1,4 This is the 1argest possib1e interva1 for which a11 three components are defined. t
- 4. rahimahj@ump.edu.my kjir )4(3)(ofgraphSketch the(b) line.the sketchThen(2,3,-1)?and(1,2,2)pointsthepasses thatlinestraightaofequationlinetheisWhat(a) 2 ttt Example 2
- 9. rahimahj@ump.edu.my functionfollowingtheofeachofgraphSketch the 1,)( tt r(a) (b) ttt sin3,cos6)( r Solution : The first thing that we need to do is plug in a few values of t and get some position vectors. Here are a few, (a) Example 3
- 10. rahimahj@ump.edu.my The sketch of the curve is given as follows (red line).
- 11. rahimahj@ump.edu.my Solution : (b) As in Question (a), we plug in some values of t.
- 12. rahimahj@ump.edu.my The sketch of the curve is given as follows (red line).
- 13. functionfollowingtheofeachofgraphSketch the kjir cttatat sincos)( Example 4 CIRCULAR HELIX
- 23. rahimahj@ump.edu.my INTEGRATION OF VECTOR FUNCTIONS ))(())(())(()( thenb],[a,onofand,, functionsintegrablesomefrom)()()()(If ise.componentwdonealsoisfunctionsvectorofnIntegratio b a kjiF kjiF b a b a b a dtthdttgdttfdtt thgf thtgtft
- 24. rahimahj@ump.edu.my Example 7 Solution : kji kji kjiF kjiF F 802-42 ])5()2[( 4)52()43()( 4t)52()43()( if)(Find 3 1 4223 3 1 3 3 1 3 1 3 1 2 32 3 1 ttttt dttdttdtttdtt tttt dtt
- 25. rahimahj@ump.edu.my b a dt dy dt dx L 22 b a dt dz dt dy dt dx L 222 In 2-space In 3-space In general, b a b a tL dt d L )('or r r
- 26. Notes: Smooth Curve The graph of the vector function defined by r(t) is smooth on any interval of t where is continuous and . The graph is piecewise smooth on an interval that can be subdivided into a finite number of subintervals on which r is smooth. r 0t r
- 27. rahimahj@ump.edu.my Find the arc length of the parametric curve 4 3 0;2,sin,cos)( 33 tztytxa 10;2,,)( ttzeyexb tt Find the arc length of the graph of r(t) 42;6 2 1 )()( 23 ttttta kjir 20;2sin3cos3)()( tttttb kjir 4 3 : LAns 1 : eeLAns 58: LAns 132: LAns Example 8 Example 9
- 29. If r(t) is a vector function that defines a smooth graph, then at each point a unit tangent vector is t t t r T r UNIT TANGENT VECTOR 3 a) Find the derivative of 1 sin 2 b) Find the unit tangent vector at the point where 0. t t t te t t r i j k Example 10
- 30. curve.thetont vectorunit tangetheiswhere asbydenoted),(curvethevector to normalunitprinciplethedefinewe,0If T Nr T t dtd rahimahj@ump.edu.my )(' )(' T T t t dtd dtd T T N UNIT NORMAL VECTOR
- 31. ).(curvethetoly,respectivevector unitprincipaltheandnt vectorunit tangetheareandwhere asdefinediscurveaofvectorbinormalThe tr NT B rahimahj@ump.edu.my BINORMAL VECTOR NTB
- 32. Find the unit normal and binormal vectors for the circular helix cos sint t t t r i j k Example 11
- 33. curve.thetont vectorunit tangetheiswhere asdefinedis)(curvesmoothaofcurvatureThe T r t rahimahj@ump.edu.my )(' )(' t t dtd dtd r T r T CURVATURE 3 r r r
- 34. Curvature is the measure of how “sharply” a curve r(t) in 2-space or 3-space “bends”.
- 35. Find the curvature of the helix traced out by 2sin ,2cos ,4t t t tr Example 12
- 36. Radius of Curvature asdefinediscurvatureofradiusitsthen ),(curvesmooththeofcurvaturetheisIf tr 1
- 37. thentime,iswhere),r(ectorposition v bygivencurvethealongmovesparticleaIf tt rahimahj@ump.edu.my dt d t r v )(velocity 2 2 )(onaccelerati dt d dt d t rv a dt ds t )(speed v
- 38. rahimahj@ump.edu.my Example 13 .2when particletheofonacceleratiandspeedvelocity,theFind sincos)( bygivenisafter timeparticleaofectorposition vThe 3 t tttt t jir Solution : kji kjiv kji r v 1242.09.0 )2(3)2(cos)2(sin 2when )3()(cos)(sin velocityobtain thewe,w.r.tatingDifferenti 2 2 t ttt dt d t
- 41. rahimahj@ump.edu.my C Ce cc C t te c t ctce tdtdttdtet dtd t t t i kjir kji kji kji kjir rv 2 )0(2sin )0( 3 1 )0( cCwhere 2 2sin 3 1 ) 2 2sin () 3 1 ()( 2cos)( havewe,Since 30 321 3 32 3 1 2 Solution :
- 43. Find the position vector R(t), given the velocity V(t) and the initial position R(0) for 2 2 ; 0 4t t t e t V i j k R i j k Example 15
- 44. rahimahj@ump.edu.my rahimahj@ump.edu.my “All our dreams can come true, if we have the courage to pursue them”