Chapter13Sec3ShowCode

Chapter13Sec3ShowCode - Initialization Cells (Code Needed...

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Unformatted text preview: 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, 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 Section 13.3: Arc Length and Curvature Arc Length 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 . 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 . 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 @ 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...
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Chapter13Sec3ShowCode - Initialization Cells (Code Needed...

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