Question

Let \(\gamma_{1}\) be a curve in \(R^{*}\), defined on \([a, b] ;\) let \(\phi\) be a continuous \(1-1\) mapping of \([c, d]\) onto \([a, b]\), such that \(\phi(c)=a\); and define \(\gamma_{2}(s)=\gamma_{1}(\phi(s))\). Prove that \(\gamma_{2}\) is an arc, a closed curve, or a rectifiable curve if and only if the same is true of \(\gamma_{1}\). Prove that \(\gamma_{2}\) and \(\gamma_{1}\) have the same length.

Short answer

In summary, we have shown that the properties of being an arc, a closed curve, or a rectifiable curve are preserved in the curve \(\gamma_2(s) = \gamma_1(\phi(s))\), given a continuous \(1-1\) mapping \(\phi\). Moreover, we demonstrated that the lengths of \(\gamma_1\) and \(\gamma_2\) are equal. This is achieved by first defining the properties of arcs, closed curves, and rectifiable curves and then showing that these properties hold for \(\gamma_2\) if they hold for \(\gamma_1\) under the given conditions. Finally, we proved the equality of the lengths of \(\gamma_1\) and \(\gamma_2\) through integration and change of variables.

Step-by-step solution

Define \(\gamma_2(s)\) and the properties of interest

Define \(\gamma_2(s) = \gamma_1(\phi(s))\), where \(\phi\) is a continuous \(1-1\) mapping from \([c, d]\) onto \([a, b]\). The properties we are interested in are: 1. Arc - A curve that does not cross itself (injective). 2. Closed curve - A curve that has the same starting and end point (e.g., a loop). 3. Rectifiable curve - A curve that can be approximated by a finite number of line segments, i.e., it has a finite length.

Establishing the conditions for \(\gamma_2\) to have the same properties as \(\gamma_1\)

Since \(\phi\) is continuous and \(1-1\), it must also be injective. Thus, if \(\gamma_1\) is an arc, \(\gamma_2\) would also be an arc because the composition of two injective mappings is injective. If \(\gamma_1\) is a closed curve, it means that its starting and end points are the same: \(\gamma_1(a) = \gamma_1(b)\). Thus, we have \(\gamma_2(c) = \gamma_1(\phi(c)) = \gamma_1(a)\) and \(\gamma_2(d) = \gamma_1(\phi(d)) = \gamma_1(b)\). So, \(\gamma_2(c) = \gamma_2(d)\), which implies that \(\gamma_2\) is also a closed curve. We know that \(\gamma_1\) is a rectifiable curve if its length is finite. In the following step, we will prove that the lengths of the two curves are the same.

Prove that the length of \(\gamma_1\) is the same as the length of \(\gamma_2\)

We can calculate the lengths of \(\gamma_1\) and \(\gamma_2\) as follows: Let \[L(\gamma_1) = \int_{a}^{b} \|\gamma_1'(t)\| dt\] and \[L(\gamma_2) = \int_{c}^{d} \|\gamma_2'(s)\| ds\] Using the Chain Rule, we have: \[\gamma_2'(s) = \frac{d\gamma_1(\phi(s))}{ds} = \gamma_1'(\phi(s)) \phi'(s)\] Thus, we can find the length of \(\gamma_2\) using the same method: \[L(\gamma_2) = \int_{c}^{d} \|\gamma_1'(\phi(s)) \phi'(s)\| ds\] Now, we can change the integral from \([c, d]\) to \([a, b]\) by using the change of variables method: \[ L(\gamma_2) = \int_{a}^{b} \|\gamma_1'(t) \phi'(s_{(\phi^{-1}(t))})\| dt\] Since \(\phi\) is a continuous, 1-to-1 mapping, \(\phi'(s) \neq 0\), and we have: \[ L(\gamma_2) = \int_{a}^{b} \|\gamma_1'(t)\| dt= L(\gamma_1)\] Based on these conclusions, we have proven that if \(\gamma_1\) is an arc, a closed curve, or a rectifiable curve, then the same is true for \(\gamma_2\), and that the lengths of \(\gamma_1\) and \(\gamma_2\) are the same.

Key concepts

Arcs in Mathematics

Arcs play a fundamental role in the study of mathematical analysis and geometry. An arc is a section of a smooth curve that does not intersect itself, also described as injective or non-self-intersecting. When analyzing arcs, it's crucial to differentiate them from other curve types because their unique properties are essential for various proofs and mathematical concepts.

For example, the exercise involves proving that if \(\gamma_1\) is an arc, then so is \(\gamma_2(s)=\gamma_1(\phi(s))\), under the assumption that \(\phi\) is a continuous and one-to-one mapping. This fact is significant because the non-intersecting nature of arcs ensures that they can be used in contexts where distinctness of points on the curve is required, such as in topological proofs or when defining curve integrals.

Closed Curves

Closed curves are those that form a loop, meaning that the starting point is the same as the endpoint. Closed curves are ubiquitous in geometry and topology, forming the basis for discussing shapes like circles and ellipses.

In the solution provided, the exercise demonstrates that if \(\gamma_1\) defines a closed curve, the resulting curve \(\gamma_2(s)\) will also be closed. This is derived from the way \(\gamma_2\) is mapped through a continuous, injective function \(\phi\), preserving the property of the curve being ‘closed’. Understanding the behavior of closed curves under different mappings is fundamental in complex analysis and geometric functions.

Rectifiable Curves

Rectifiable curves, which can be approximated by a finite number of line segments, and consequently, have a finite length, are of great importance in mathematical analysis, particularly in the study of curve length and integrals. A curve is rectifiable if we can ascribe a definite length to it.

The exercise asserts that if \(\gamma_1\) is a rectifiable curve, \(\gamma_2\) must be one too. This proof relies on the property that \(\phi\) doesn’t distort the length of \(\gamma_1\) when it is used to produce \(\gamma_2\), a consequence of being a continuous and one-to-one mapping. Proving rectifiability is crucial when dealing with physical applications like calculating the length of a path in space or the perimeter of an object.

Continuous Mapping

Continuous mapping is a core concept in many areas of mathematics, including analysis and topology. It’s characterized by the idea that small changes in the input lead to small changes in the output. In the context of curves, a continuous mapping is often needed to ensure that curve properties like connectivity and boundaries are maintained.

In the given exercise, the mapping \(\phi\) is continuous and one-to-one. This allows for the properties of \(\gamma_1\) to be inherited by \(\gamma_2\), ensuring that the essence of the curve is not lost during the transformation. Since continuity is vital to keeping the integrity of many curve properties, it’s no surprise that it is highlighted in the step-by-step solution.

Chain Rule

The Chain Rule is a fundamental theorem in calculus, often used when composing functions. It provides a way to compute the derivative of a composite function. In the practice of evaluating curve lengths, the Chain Rule allows us to connect the derivatives of mapped functions, thereby providing a method to relate the lengths of two curves.

The step-by-step solution uses the Chain Rule to express the derivative of \(\gamma_2\) in terms of the derivative of \(\gamma_1\) and \(\phi\), and thus, relate their lengths. This is key to proving that both \(\gamma_1\) and \(\gamma_2\) have the same length, which seals the argument that \(\gamma_2\) shares the properties of being an arc, a closed curve, or a rectifiable curve, just as \(\gamma_1\) is.