CLP-4 Vector Calculus (Feldman, Rechnitzer, and Yeager)

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Reparametrization

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. It often pays to tailor the parametrization 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.

Curvature

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 — 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 “curvy curve” rather than by a straight line. Probably the simplest “curvy curve” is a circle and that’s what we’ll use.

Sliding on a Curve

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:

• First, to set things up we’ll look at a bead sliding on a stiff wire.
• Then, we will imagine that we are skiing straight downhill and ask “Where on the hill can we become airborne?”.
• Then we will imagine that we are skateboarding in a culvert (a large pipe) and ask “When is it safe?”.

The Skier

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 .