Problem 48 - Essential Calculus Early Transcendentals

Question

Show that the curvature of a plane curve is \(\kappa=|d \phi / d s|,\) where \(\phi\) is the angle between \(T\) and \(\mathbf{i} ;\) that is, \(\phi\) is the angle of inclination of the tangent line.

Step-by-Step Solution

Answer

The curvature \( \kappa \) is \( \kappa = \left| \frac{d\phi}{ds} \right| \).

Step 1: Understand the curvature definition

The curvature \( \kappa \) of a curve in the plane is defined as the rate of change of the tangent angle \( \phi \) with respect to the arc length \( s \). This is mathematically given by \( \kappa = \left| \frac{d\phi}{ds} \right| \).

Step 2: Relate tangent vector to the angle \( \phi \)

The unit tangent vector \( T \) can be described in terms of \( \phi \) as \( \mathbf{T} = (\cos\phi, \sin\phi) \). \( \phi \) is the angle between the tangent line and the positive x-axis (\( \mathbf{i} \)).

Step 3: Express \( T \) in terms of arc length \( s \)

Since arc length \( s \) is parameterized by the curve, the tangent vector is \( \mathbf{T}(s) = \frac{d \mathbf{r}}{ds} \), where \( \mathbf{r}(s) \) is the position vector along the curve.

Step 4: Calculate tangent angle in terms of derivative

Given the unit tangent vector \( \mathbf{T} = (\cos\phi, \sin\phi) \), differentiate \( \phi \) with respect to \( s \) to find the rate of change of \( \phi \), i.e., \( \frac{d\phi}{ds} \).

Step 5: Establish the relationship through differentiation

The curvature \( \kappa \) is defined as \( \kappa = \left| \frac{d\phi}{ds} \right| \), showing that the curvature of the plane curve indeed equals the absolute value of the rate of change of angle \( \phi \) with respect to the arc length \( s \).

Key Concepts

Tangent Angle

The tangent angle, often denoted as \( \phi \), is a fundamental concept when exploring the curvature of a plane curve. It represents the angle between the tangent line of a curve and the positive x-axis, or the unit vector \( \mathbf{i} \). Understanding this angle is essential as it connects to how sharply a curve changes direction. In simpler terms, imagine tracing a path along a roller coaster track. The tangent angle tells you just how much your direction is changing as you move along the track. The steeper or sharper the turn, the greater the change in tangent angle. The relationship between the tangent angle and the curvature comes from the rate of change of this angle. As the angle changes while you move along the curve, the curve itself is changing as well. Curvature, defined as \( \kappa = \left| \frac{d\phi}{ds} \right| \), tells us how quickly the tangent angle is changing with respect to the arc length, \( s \). Thus, the tangent angle is crucial in determining a curve's curvature.

Arc Length

Arc length is a measure of the distance along a curve between two points. It helps us understand and quantify the size or extent of a curve, much like measuring a piece of ribbon by laying it straight. In mathematics, arc length \( s \) serves not only as a physical length but as a parameter to describe how one moves along a curve. As opposed to the standard coordinate system, using arc length as the parameter focuses on the journey along the curve itself. When discussing curvature, the arc length helps depict how tightly or loosely a curve bends. Since curvature is defined as \( \kappa = \left| \frac{d\phi}{ds} \right| \), where \( s \) is the arc length, it emphasizes the local bending behavior of the curve per unit distance traveled. Here, the change in \( \phi \) relative to changes in \( s \) is critical for a precise measurement of curvature.

Unit Tangent Vector

The unit tangent vector \( \mathbf{T} \) is a vector that points in the direction of the curve at any given point. Why is it important? It gives a consistent way to talk about direction as you travel along the curve, irrespective of the curve's curvature or length.Mathematically, the unit tangent vector is represented as \( \mathbf{T} = (\cos\phi, \sin\phi) \), where \( \phi \) is the tangent angle. This description links it directly to how much the curve's angle is changing. Additionally, the unit tangent vector is crucial for understanding the geometry of a curve. As \( \mathbf{T} \) is derived from \( \frac{d \mathbf{r}}{ds} \), where \( \mathbf{r}(s) \) is the position vector, it provides a normalized way of differentiating the curve's direction. Even if a curve loops around or twists, the unit tangent vector keeps consistency in describing each segment’s direction.