Question

Let \(I=[0,2 \pi]\) and define \(f, g, h: E^{1} \rightarrow E^{2}(C)\) by $$ \begin{aligned} f(x) &=(\sin x, \cos x) \\ g(x) &=(\sin 3 x, \cos 3 x) \\ h(x) &=\left(\sin \frac{1}{x}, \cos \frac{1}{x}\right) \text { with } h(0)=(0,1) . \end{aligned} $$ Show that \(f[I]=g[I]=h[I]\) (the unit circle; call it \(\mathrm{A}),\) yet \(\ell_{f} A=2 \pi\) \(\ell_{g} A=6 \pi,\) while \(V_{h}[I]=+\infty\). (Thus the result of Problem 7 fails for closed curves and nonsimple arcs.)

Short answer

Each function maps to the unit circle: \(\ell_f A = 2\pi\), \(\ell_g A = 6\pi\), \(V_h[I] = +\infty\).

Step-by-step solution

Understand the function outputs

Firstly, we recognize that each function maps from one-dimensional space into two-dimensional space specifically onto circles. The function \(f(x) = (\sin x, \cos x)\) describes a unit circle in the plane as \(x\) varies from 0 to \(2\pi\). Similarly, the function \(g(x) = (\sin 3x, \cos 3x)\) will also map onto the unit circle, as it represents three revolutions around the circle for \(x\) in \([0, 2\pi]\). The function \(h(x) = (\sin \frac{1}{x}, \cos \frac{1}{x})\) describes a curve that also maps onto the unit circle.

Determine the image under each function

By evaluating each function over the interval \([0, 2\pi]\), they all map the interval to a set of points on the unit circle. Specifically: \(f[I]\) traces the unit circle once, \(g[I]\) traces it three times, and \(h[I]\) densely fills the unit circle as \(x\) approaches 0. Thus, all image sets equal the unit circle \(A\).

Calculate the length \(\ell_f A\) for \(f(x)\)

The arc length of the unit circle traced by \(f(x)\) is the length of one full revolution: \(\ell_{f} A = 2\pi\), which is the circumference of the circle with radius 1.

Calculate the length \(\ell_g A\) for \(g(x)\)

Since \(g(x)\) represents \(3\) full revolutions around the unit circle, the total length of \(A\) traced by \(g(x)\) is \(3 \times 2\pi = 6\pi\).

Explore the variation measure \(V_h[I]\) for \(h(x)\)

The curve described by \(h(x)\) undergoes infinite oscillations as \(x\) gets closer to 0. Thus, the total length along the curve from the variation measure \(V_h[I]\) results in \(+\infty\) because \(h(x)\) densely populates the circle, creating infinite length.

Key concepts

Parametrization of Curves

When we talk about parametrization of curves, we're referring to a method of describing a curve using equations that express the coordinates of points on the curve as functions of a single parameter. In our exercise, functions like \(f(x) = (\sin x, \cos x)\) provide a typical example, where the parameter is \(x\). This parameter \(x\) typically ranges over an interval, such as \([0, 2\pi]\),
which then maps to specific coordinates on the unit circle in the plane.
- **Function \(f(x)\):** One revolution around the unit circle.- **Function \(g(x)\):** Three complete revolutions.- **Function \(h(x)\):** A curve that densely fills the circle.Understanding this concept is crucial in mathematical analysis as it allows us to represent complex geometrical shapes and paths using simpler mathematical expressions.

Arc Length

The arc length of a curve is essentially the distance along the curve from one endpoint to another. It's calculated by integrating the infinitesimal distances across the interval you're interested in.
For a curve parametrized by \(f(x) = (\sin x, \cos x)\), where it traces the unit circle exactly once, the length, denoted as \(\ell_f A\), is the circumference of a circle with radius 1, which is \(2\pi\).
In contrast, the function \(g(x) = (\sin 3x, \cos 3x)\) traces the same circle three times, tripling the length to \(6\pi\).
This differentiation in arc length highlights how the number of times a curve revolutions influences its total length. Such calculations are fundamental in mathematical analysis as they help in understanding the properties of the curve and its geometry.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate system. It serves as a fundamental concept in trigonometry and calculus.
In this exercise, each function \(f(x), g(x),\) and \(h(x)\) maps intervals onto the unit circle.
This mapping showcases how different parametrizations can still encapsulate the same geometric shape, answering why \(f[I] = g[I] = h[I]\) equals the unit circle set \(A\).
- **Key Properties:** - Every point \((x, y)\) satisfies \(x^2 + y^2 = 1\). - It provides a simple way to study trigonometric functions and their relationships.Understanding the unit circle is vital as it forms the groundwork of Fourier transforms, signal processing, and complex number analysis.

Function Mapping

Function mapping describes how each input from one set (the domain) is assigned to an output in another set (the co-domain). In the context of this exercise, it indicates how each value of \(x\) in \([0, 2\pi]\) is assigned a point on the unit circle.
- \(f(x)\) maps \(x\) linearly onto the unit circle once.- \(g(x)\), on the other hand, maps each \(x\) value onto the circle thrice, effectively increasing the path traveled.- Finally, \(h(x)\) maps \(x\) in such a way that it results in dense and infinite filling as \(x\) approaches zero.This topic helps in understanding how mathematical functions can manipulate and transform spaces, which is fundamental in analysis and geometry.

Variation Measure

The variation measure of a curve is a mathematical concept used to describe the degree of 'wiggling' or oscillation of a curve in its parametrization.
In this example, function \(h(x)\) fills the unit circle in a peculiar way with \(V_h[I] = +\infty\).
- This infinite variation occurs because close to zero, \(h(x)\) induces numerous oscillations, causing the path to be extremely dense.This measure shows us that even if curves might outline the same shape, their behavior can vastly differ depending on the parametrization they follow. Understanding variation measure aids in studying the behavior of functions and their ability to describe more complex features of curves and surfaces.