worksheet5.20100202.4b6831d3d9a423.63813979

# worksheet5.20100202.4b6831d3d9a423.63813979 - Worksheet on...

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Unformatted text preview: Worksheet on Tangent Vectors Describing curves and their equivalence Cartesian coordinates A curve is a one-parameter family of configurations. For example, for a single particle constrained to a plane, a curve is a one- parameter family of points in the plane, i.e., exactly what we normally mean by a physical curve. Using a chart we can describe a curve by its image in R 2 . The curve below has the shape of a circle segment. The configuration corresponding to s = 0 has cartesian coordinates x = cos H L = 1 and y = 5- sin H L = 5. The graph includes discrete points along the curve separated by D s = 0.1. In[104]:= c1 = ParametricPlot @8 Cos @ s D , 5- Sin @ s D< , 8 s,- .5, .5 <D ; p1 = ListPlot @ 8 Cos @ s D , 5- Sin @ s D< . Table @8 s fi x < , 8 x,- 0.5, 0.5, .1 <D , PlotStyle fi PointSize @ .02 DD ; Show @8 c1, p1 < , PlotRange fi 88 0.8, 1.1 < , 8 4.5, 5.5 << , AspectRatio fi 0.9 D Out[106]= 0.80 0.85 0.90 0.95 1.00 1.05 1.10 4.8 5.0 5.2 5.4 Two curves may agree at a point. In[107]:= c2 = ParametricPlot @8 2- Cos @ 2 s D , 5- Sin @ 2 s D< , 8 s,- .5, .5 <D ; p2 = ListPlot @ 8 2- Cos @ 2 s D , 5- Sin @ 2 s D< . Table @8 s fi x < , 8 x,- .5, .5, .1 <D , PlotStyle fi PointSize @ .02 DD ; Show @8 c1, p1, c2, p2 < , PlotRange fi 88 0.8, 1.1 < , 8 4.5, 5.5 << , AspectRatio fi 0.9 D Out[109]= 0.80 0.85 0.90 0.95 1.00 1.05 1.10 4.8 5.0 5.2 5.4 In this case, the rate of motion is different between the two curves. Although they coincide when s = 0 motion along the right curve is twice as fast as that along the left curve. Equivalence We say that two curves that coincide at some configuration, say with s = 0 along both curves, are equivalent provided that the rates of change with respect to s of the coordinate functions at that configuration are equal.of the coordinate functions at that configuration are equal....
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worksheet5.20100202.4b6831d3d9a423.63813979 - Worksheet on...

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