Geometry

Topology

Deep Dive: Pointwise and Uniform Convergence 2

         

A sequence of curves that approximate the unit circle is shown.

True or False? These curves converge pointwise to the unit circle.

Details.

There is an open circle at the point (2,0),(2,0), and a linear portion with slope 1n+1\frac{-1}{n+1} at Stage nn that continues until it meets the unit circle, and then follows the circle around counterclockwise to (1,0).(1,0).

Let fn(θ)f_n(\theta) be the point where a ray making an angle of θ\theta with the positive xx-axis intersects the Stage nn curve fn,f_n, and let f(θ)f(\theta) be the point where the same ray intersects the unit circle. A picture of f2(θ)f_2(\theta) for an angle θ\theta is shown below.

True or False? For each θ\theta in the interval [0,2π),[0, 2\pi),

limnfn(θ)=f(θ).\lim_{n \to \infty} f_n(\theta) = f(\theta).

A sequence of curves that approximate the unit circle is shown.

True or False? The derivatives of the curves converge to the derivative of the circle.

Details. There is an open circle at the point (2,0),(2,0), and a linear portion with slope 1n+1\frac{-1}{n+1} at Stage nn that continues until it meets the unit circle, and then follows the circle around counterclockwise to (1,0).(1,0).

Let dn(θ)d_n(\theta) be the derivative dydx\frac{dy}{dx} of the Stage nn curve at the point where it intersects a ray making an angle of θ\theta with the positive xx-axis, and let d(θ)d(\theta) be the derivative dydx\frac{dy}{dx} of the unit circle at the point where the same ray intersects the unit circle.

True or False? For each θ\theta in the interval (0,2π),(0, 2\pi),

limndn(θ)=d(θ).\lim_{n \to \infty} d_n(\theta) = d(\theta).

A sequence of curves that approximate the unit circle is shown (as the stages progress, the slope of the linear portion approaches 0.)

True or False? Let L(fn)L(f_n) represent the length of the Stage nn curve, and L(f)L(f) represent the length of the unit circle. Then

limnL(fn)=L(f).\lim_{n \to \infty} L(f_n) = L(f).

A sequence of curves that approximate the unit circle is shown (as the stages progress, the slope of the linear portion approaches 0.)

True or False? The derivatives of the curves converge uniformly to the derivative of the circle.

Details. Let dn(θ)d_n(\theta) be the derivative dydx\frac{dy}{dx} of the Stage nn curve at the point where it intersects a ray making an angle of θ\theta with the positive xx-axis, and let d(θ)d(\theta) be the derivative dydx\frac{dy}{dx} of the unit circle at the point where the same ray intersects the unit circle.

True or False? For each θ\theta in the interval (0,2π),(0, 2\pi), the sequence {dn(θ)}\{d_n(\theta)\} converges uniformly to d(θ).d(\theta).

Note. Uniform convergence in this example is equivalent to:

For every ϵ>0,\epsilon > 0, there is a natural number NϵN_{\epsilon} such that

n>Nϵdn(θ)d(θ)<ϵ for all θ in (0,2π).n>N_{\epsilon} \Rightarrow \left|d_n(\theta) - d(\theta) \right| < \epsilon \text{ for all } \theta \text{ in } (0, 2\pi).

Two sequences of curves are shown. In both cases, we have a sequence of curves that is converging pointwise to the unit circle, and whose derivatives are converging pointwise to the corresponding derivative of the unit circle.

In which case is the convergence uniform and the lengths of the curves converge to the length of the unit circle?

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