Problem 49 - Essential Calculus Early Transcendentals

Question

(a) Show that \(d \mathbf{B} / d s\) is perpendicular to \(\mathbf{B}\) . (b) Show that \(d \mathbf{B} / d s\) is perpendicular to \(\mathbf{T}\) . (c) Deduce from parts (a) and (b) that \(d \mathbf{B} / d s=-\tau(s) \mathbf{N}\) for some number \(\tau(s)\) called the torsion of the curve. (The torsion measures the degree of twisting of a curve.) (d) Show that for a planc curve the torsion is \(\tau(s)=0\) .

Step-by-Step Solution

Answer

\(\frac{d \mathbf{B}}{d s}\) is perpendicular to both \(\mathbf{B}\) and \(\mathbf{T}\), hence \(\frac{d \mathbf{B}}{d s} = -\tau(s) \mathbf{N}\) with \(\tau(s) = 0\) for planar curves.

Step 1: Derivative of a Vector

Given a unit vector \(\mathbf{B}\), take its derivative with respect to the arc length \(s\). The derivative is \(\frac{d \mathbf{B}}{d s}\). Since \(\mathbf{B}\) is a unit vector, this derivative must be perpendicular to \(\mathbf{B}\) itself.

Step 2: Use Dot Product for Perpendicularity

The dot product of two perpendicular vectors is zero. Thus, prove \(\frac{d \mathbf{B}}{d s} \cdot \mathbf{B} = 0\) to show that \(\frac{d \mathbf{B}}{d s}\) is perpendicular to \(\mathbf{B}\).

Step 3: Derivative and Tangent Vector

The tangent vector \(\mathbf{T}\) is orthogonal to \(\mathbf{B}\) by definition in a frame. Use similar reasoning as in Step 2 to prove \(\frac{d \mathbf{B}}{d s} \cdot \mathbf{T} = 0\) to show that \(\frac{d \mathbf{B}}{d s}\) is perpendicular to \(\mathbf{T}\).

Step 4: Combination of Orthogonality

Since \(\frac{d \mathbf{B}}{d s}\) is perpendicular to both \(\mathbf{B}\) and \(\mathbf{T}\), it must be aligned along \(\mathbf{N}\), the normal vector, within the Frenet-Serret frame. Thus, \(\frac{d \mathbf{B}}{d s} = -\tau(s) \mathbf{N}\).

Step 5: Torsion for Planar Curve

A planar curve lies entirely in one plane, making its binormal \(\mathbf{B}\) constant. So, the torsion \(\tau(s) = 0\) since there is no twisting (no change in \(\mathbf{B}\)).

Key Concepts

Frenet-Serret formulas

The Frenet-Serret formulas are fundamental equations in the study of curves in three-dimensional space. These formulas comprise three key equations that describe how three essential vector fields—tangent (\( \mathbf{T} \)), normal (\( \mathbf{N} \)), and binormal (\( \mathbf{B} \))—change along a space curve. A curve can be intuitively understood as a smooth line drawn in space, and these vector fields live "on" the curve, giving it structure.

The tangent vector \( \mathbf{T} \) points in the direction of the curve and shows its path. The normal vector \( \mathbf{N} \) points towards the curve's curvature, indicating where the curve bends. Lastly, the binormal vector \( \mathbf{B} \) is perpendicular to the other two, forming a right-handed orthonormal triad.

These vectors help understand complex curves, with the speed of change in angles and directions being quantified by quantities called curvature and torsion. The formulas themselves, named after French mathematicians Jean Frenet and Joseph Serret, are

• \( \frac{d\mathbf{T}}{ds} = \kappa \mathbf{N} \)
• \( \frac{d\mathbf{N}}{ds} = -\kappa \mathbf{T} + \tau \mathbf{B} \)
• \( \frac{d\mathbf{B}}{ds} = -\tau \mathbf{N} \)

where \( \kappa \) is the curvature and \( \tau \) is the torsion of the curve. Together, these equations provide a comprehensive description of the geometric and dynamic properties of a curve.

arc length parameterization

Arc length parameterization simplifies the description of a curve by considering the arc length, \( s \), along the curve as the parameter. Instead of using arbitrary parameters like time, arc length gives a direct measure of distance along the curve.

In this parameterization, a point on the curve at \( s \) units from a starting point is denoted by \( \mathbf{r}(s) \). Using arc length ensures the unit tangent vector \( \mathbf{T} \) remains of unit length everywhere along the curve. This makes the mathematics cleaner and more intuitive.

With arc length parameterization:

• The derivative of a position vector, \( \frac{d\mathbf{r}}{ds} \), gives the unit tangent vector, \( \mathbf{T} \).
• It provides a natural and intuitive perspective to measure distances, facilitating the application of the Frenet-Serret formulas.

Arc length parameterization thus serves as a powerful tool for studying curves, making analyses simpler and calculations more straightforward.

unit vectors

Unit vectors are vectors with a magnitude of one. They are crucial in mathematics and physics because they help describe directions without concerning themselves with magnitude. In the study of curves, unit vectors are used to define the basic character of the curve.

The tangent vector \( \mathbf{T} \), normal vector \( \mathbf{N} \), and binormal vector \( \mathbf{B} \) associated with a curve are all unit vectors. Collectively, they form an orthonormal basis that provides a standard reference frame for analyzing the curve's properties.

The key features of unit vectors include:

• Having a magnitude of one, which means \( \|\mathbf{v}\| = 1 \).
• Maintaining direction without regard to length or size.

The beauty of unit vectors is that they provide a simple way to understand complex geometrical structures by focusing solely on direction, leading to insights about curvature, torsion, and other properties of curves.

perpendicular vectors

Perpendicular vectors, also known as orthogonal vectors, are pairs of vectors whose dot product equals zero. This relationship tells us that they meet at a right angle, which is a cornerstone concept in vector analysis used frequently in three-dimensional geometry.

In the context of the Frenet-Serret frame, the concept of perpendicular vectors is vital. The tangent vector \( \mathbf{T} \) is perpendicular to the binormal \( \mathbf{B} \), and the normal vector \( \mathbf{N} \) is perpendicular to both. This relationship forms an orthonormal triad.

Key aspects of perpendicular vectors include:

• Their dot product is zero, for instance, \( \mathbf{A} \cdot \mathbf{B} = 0 \).
• They define the plane and directionality around a curve through the right-angle intersections.

Understanding how vectors relate perpendicularly allows deeper insight into the curvature and torsion properties of curves, leading to effective problem-solving and analysis.