Feedback about the text<\/p>\n
The CLP-4 Vector Calculus text is still undergoing testing and changes. Because of this we request that if you find a problem or error in the text then:<\/p>\n
1. Please check the errata list that can be found at the text webpage.<\/p>\n
2. Is the problem in the online version or the PDF version or both?<\/p>\n
3. Note the URL of the online version and the page number in the PDF<\/p>\n
4. Send an email to clp@ugrad.math.ubc.ca<\/a> . Please be sure to include<\/p>\n There are invariably many ways to parametrize a given curve. Kind of trivially, one can always replace by, for example, But there are also more substantial ways to reparametrize curves<\/a>. It often pays to tailor the parametrization<\/a> used to the application of interest. For example, we shall see in the next couple of sections that many curve formulae simplify a lot when arc length is used as the parameter.<\/p>\n So far, when we have wanted to approximate a complicated curve by a simple curve near some point, we drew the tangent line to the curve at the point. That’s pretty crude. In particular tangent lines are straight \u2014 they don’t curve. We will get a much better idea of what the complicated curve looks like if we approximate it, locally, by a very simple \u201ccurvy curve\u201d rather than by a straight line. Probably the simplest \u201ccurvy curve\u201d is a circle and that’s what we’ll use.<\/p>\n We are going to investigate the motion of a particle of mass sliding on a frictionless , smooth curve that lies in a vertical plane. We will consider three scenarios:<\/p>\n The difference between the bead on the wire and the skier on the hill is that while the hill is capable of applying an upward normal force (i.e. it can push you upward to keep you from falling to the centre of the Earth), it is not capable of applying a downward normal force. That is the hill cannot pull down on you to keep you on the hill. Only gravity can keep you grounded. There are two main possibilities .<\/p>\n","protected":false},"excerpt":{"rendered":"\n
Reparametrization<\/h3>\n
Curvature<\/h3>\n
Sliding on a Curve<\/h3>\n
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The Skier<\/h3>\n