worksheet5.20100202.4b6831d3d9a423.63813979

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online