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Chapter13Sec3ShowCode

Chapter13Sec3ShowCode - Initialization Cells(Code Needed...

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Initialization Cells (Code Needed for Most Plot3D Statements) In[62]:= ticks @ min_, max_ D : = Join @ Table @8 i,If @ i 0, ,Style @ i,12 DD , 8 0.01,0 << , 8 i,Ceiling @ min D ,Floor @ max D , 2 <D , H * Numbered ticks * L Table @8 j, , 8 0.01,0 << , 8 j,Round @ min D ,Round @ max - 1 D ,1 <DD H * Un - numbered ticks * L H * 2D Options * L SetOptions @ Plot, ImageSize fi 350, AxesOrigin fi 8 0,0 < , AspectRatio fi 1, AxesStyle fi Directive @ Medium, Bold D , Ticks fi ticks D ; H * 3D Options * L SetOptions @ Plot3D, ImageSize fi 600, BoxRatios fi 1,Boxed fi False, AxesOrigin fi 8 0,0,0 < , AxesStyle fi Directive @ Medium, Bold D , Ticks fi ticks, Mesh fi None, ViewPoint fi 8 2,0.9,1 < , ViewVertical fi 8 0,0,1 < D ; SetOptions @ ContourPlot3D, ImageSize fi 600, BoxRatios fi 1,Boxed fi False, AxesOrigin fi 8 0,0,0 < , AxesStyle fi Directive @ Medium, Bold D , Ticks fi ticks, Mesh fi None, ViewPoint fi 8 2,0.9,1 < , ViewVertical fi 8 0,0,1 < D ; SetOptions @ ParametricPlot3D, ImageSize fi 600, BoxRatios fi 1,Boxed fi False, AxesOrigin fi 8 0,0,0 < , AxesStyle fi Directive @ Medium, Bold D , Ticks fi ticks, Mesh fi None,
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ViewPoint fi 8 2,0.9,1 < , ViewVertical fi 8 0,0,1 < D ; SetOptions @ RegionPlot3D, ImageSize fi 600, BoxRatios fi 1,Boxed fi False, AxesOrigin fi 8 0,0,0 < , AxesStyle fi Directive @ Medium, Bold D , Ticks fi ticks, Mesh fi None, ViewPoint fi 8 2,0.9,1 < , ViewVertical fi 8 0,0,1 < D ; Chapter 13: Vector Functions 2 Chapter13Sec3.nb
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Section 13.3: Arc Length and Curvature Arc Length ª Definition Consider the vector equation r H t L = X f H t L , g H t L , h H t L\ over the interval a £ t £ b . Suppose the functions f , g , h , f ', g ', h ' are all continuous on the interval and the curve is traversed exactly once as t increases from a to b . Then, the length traversed is L = a b @ f ' H t LD 2 + @ g ' H t LD 2 + @ h ' H t LD 2 t = a b I dx dt M 2 + J dy dt N 2 + I dz dt M 2 t = a b r ' H t L t . The arc length function is s H t L = a t r ' H u L u and by the Fundamental Theorem of Calculus, Part 1, we have ds dt = r ' H t L . ª Example Find the arc length function if the vector equation is r H t L = X sin H t L , cos H t L , t \ over 0 £ t £ Π . Reparameterize the function r (t) in terms of the arc length function, that is, if t = t H s L , then find r H s L = r H t H s LL to parameterize the curve with respect to arc length. In[68]:= Manipulate @ Show @ Plot3D @ 0, 8 x, - 1.5,1.5 < , 8 y, - 1.5,1.5 < , PlotRange fi 8 - 0.5,4 < , PlotStyle fi Directive @ Opacity @ 0 DD , Mesh fi None,BoundaryStyle fi None D , ParametricPlot3D @8 Sin @ t D , Cos @ t D , t < , 8 t, 0, b < , PlotStyle fi 8 Thick, Blue <D , Graphics3D @ Text @ Style @ x, Large D , 8 1.2, - 0.2,0 <DD , H * Label the x - axis * L Graphics3D @ Text @ Style @ y, Large D , 8 0.2,1.4,0 <DD , H * Label the y - axis * L Graphics3D @ Text @ Style @ z, Large D , 8 0,0.2,4.2 <DDH * Label the z - axis * L D , 88 b,Pi < , 0.1, Pi < D Chapter13Sec3.nb 3
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Out[68]= b x y z - 1 1 - 1 1 2 4 4 Chapter13Sec3.nb
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