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Chapter 7: Problem 47

Define a rectifiable curve.

Short Answer

Expert verified
A rectifiable curve is a curve that has finite length, and can be approximated with a sequence of straight line segments to an arbitrary degree of precision.

Step by step solution

01

Understand the Terminology

Firstly, get acquainted with the terms used. A 'curve' in mathematics is a line of an arbitrary shape that could be curved, straight or looped, and a 'rectifiable curve' is a term used to describe a special type of curve.
02

Define a Rectifiable Curve

A rectifiable curve in calculus is a curve that can be measured, i.e., it has a finite length. Specifically, if it's possible to approximate the curve with a sequence of straight line segments (also known as polygonal approximations) to an arbitrary degree of precision, then this curve is said to be rectifiable. A rectifiable curve鈥檚 length is equal to the limit of the lengths of these polygonal approximations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

curve
A curve is a fundamental concept in mathematics. It represents a one-dimensional object which can vary widely in form. Curves are not confined to simply straight lines. They can exhibit a vast range of shapes including bends, loops, or any arbitrary path.

Curves are also seen in higher dimensions; for instance, a curve on a plane may be familiar, but curves can reside in three-dimensional spaces as well.
  • In a strict sense, a curve is a continuous map from an interval of real numbers to a space. This means it assigns a position in space to each point on a number line, forming the idea of a path.
  • Curves can be closed, meaning they loop back on themselves, or open with definite starting and ending points.
Understanding curves is crucial as they form the basis for more complex geometric and physical systems.
polygonal approximation
Polygonal approximation is a vital concept when working with curves, particularly when assessing their lengths. It involves approximating a curve using a series of straight line segments.

This is important as it forms the foundation for calculating the length of a curve that may not initially have a defined length due to its intricate path.
  • To approximate a curve, one replaces the curved section with a number of small straight lines that follow the general path of the curve.
  • The more segments used, the closer the approximation is to the actual curve. In mathematical terms, as the number of segments approaches infinity, the approximation becomes exact.
This technique is especially useful in calculus to define what is known as a 'rectifiable curve', making the concept not just applicable, but also practical.
finite length
The notion of a finite length in curves is what makes them 'rectifiable'. If a curve has a finite length, it means that you can measure it, just like you would a straight line.

For practical purposes, it implies that you can, through polygonal approximation, determine how long a curve is and arrive at a specific, finite value.
  • Rectifiable curves are those which can be approximated by polygons to such a degree that, even as the count of line segments increases, the total length converges to a finite number.
  • This convergence property is critical, as it transforms an otherwise abstract mathematical concept into something tangible and measurable.
Essentially, when mathematicians say a curve is rectifiable, they're asserting that its length is a finite, real number that can be calculated, regardless of how complex or convoluted the curve may appear.

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Most popular questions from this chapter

The vertical cross section of an irrigation canal is modeled by \(f(x)=\frac{5 x^{2}}{x^{2}+4}\) where \(x\) is measured in feet and \(x=0\) corresponds to the center of the canal. Use the integration capabilities of a graphing utility to approximate the fluid force against a vertical gate used to stop the flow of water if the water is 3 feet deep.

Let \(V\) be the region in the cartesian plane consisting of all points \((x, y)\) satisfying the simultaneous conditions \(|x| \leq y \leq|x|+3 \quad\) and \(\quad y \leq 4\) Find the centroid \((\bar{x}, \bar{y})\) of \(V\).

Define fluid pressure.

Find the center of mass of the given system of point masses. $$ \begin{aligned} &\begin{array}{|l|c|c|} \hline m_{i} & 3 & 4 \\ \hline\left(x_{1}, y_{1}\right) & (-2,-3) & (5,5) \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|} \hline m_{i} & 2 & 1 & 6 \\ \hline\left(x_{1}, y_{1}\right) & (7,1) & (0,0) & (-3,0) \\ \hline \end{array} \end{aligned} $$

Find the center of mass of the given system of point masses. $$ \begin{array}{|l|c|c|c|} \hline m_{i} & 10 & 2 & 5 \\ \hline\left(x_{1}, y_{1}\right) & (1,-1) & (5,5) & (-4,0) \\ \hline \end{array} $$
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