Question

Determine whether the following statements are true and give an explanation or counterexample. a. The position, unit tangent, and principal unit normal vectors ( \(\mathbf{r}, \mathbf{T},\) and \(\mathbf{N}\) ) at a point lie in the same plane. b. The vectors \(\mathbf{T}\) and \(\mathbf{N}\) at a point depend on the orientation of a curve. c. The curvature at a point depends on the orientation of a curve. d. An object with unit speed \((|\mathbf{v}|=1 \text { ) on a circle of radius } R\) has an acceleration of \(\mathbf{a}=\mathbf{N} / R\) e. If the speedometer of a car reads a constant \(60 \mathrm{mi} / \mathrm{hr}\), the car is not accelerating. f. A curve in the \(x y\) -plane that is concave up at all points has positive torsion. g. A curve with large curvature also has large torsion.

Short answer

## Short Answer True or False: Large curvature on a curve always implies large torsion. **Answer:** False A curve with large curvature doesn't necessarily have large torsion, as curvature and torsion are separate measures of a curve's properties. Curvature measures how sharply a curve bends, while torsion measures the twist of a curve out of its osculating plane.

Step-by-step solution

a. Position, unit tangent, and principal unit normal vectors lie in the same plane

We know that the unit tangent vector \(\mathbf{T}\) is tangent to the curve, and the principal normal vector \(\mathbf{N}\) always lies in the plane perpendicular to the unit tangent vector at that point. The position vector \(\mathbf{r}\) connects the origin to a point on the curve. Therefore, \(\mathbf{r}\), \(\mathbf{T}\), and \(\mathbf{N}\) will always lie in the same plane called the osculating plane. This statement is true.

b. Vectors \(\mathbf{T}\) and \(\mathbf{N}\) depend on the orientation of a curve

The unit tangent vector \(\mathbf{T}\) is related to the derivative of the position vector and represents the direction of the curve. If we change the orientation of the curve (i.e., move it around without changing its shape), the unit tangent vector will change accordingly as it depends on the direction. For the principal normal vector \(\mathbf{N}\), since it lies in the plane that is perpendicular to the unit tangent vector, it will also change whenever the orientation of the curve changes. This statement is true.

c. The curvature at a point depends on the orientation of a curve

Curvature is a scalar quantity that describes the rate of change of the unit tangent vector with respect to arc length and represents the 'tightness' or 'sharpness' of the curve. Mathematically, curvature is given by \(\kappa = \frac{|\mathbf{T'}|}{|\mathbf{v}|}\). Since \(\kappa\) is a scalar quantity, it remains same even if we change the orientation of the curve. This statement is false.

d. An object with unit speed on a circle of radius \(R\) has an acceleration of \(\mathbf{a}=\mathbf{N} / R\)

The centripetal acceleration of an object moving at unit speed \((|\mathbf{v}|=1)\) in a circle of radius R is given by \(\mathbf{a_c}=-\frac{1}{R} \mathbf{N}\). The acceleration vector of an object moving along a curve is the sum of the tangential acceleration and the centripetal acceleration. Now, if the object is moving with a unit speed, then there is no tangential acceleration (as the object is neither speeding up nor slowing down). Hence, the acceleration vector is equal to the centripetal acceleration, which is \(\mathbf{a}=-\frac{1}{R}\mathbf{N}\). This statement is false.

e. If the speedometer of a car reads a constant 60 mi/hr, the car is not accelerating

The derivative of velocity is acceleration. If a car is moving at a constant 60 mph (constant speed), it is not accelerating in the direction of the path (no tangential acceleration). However, if the car is moving in a curved path, there will still be centripetal acceleration (because of the change in the direction of the velocity vector) even if its speed is constant. Therefore, this statement is not necessarily true.

f. A curve in the \(xy\)-plane that is concave up has positive torsion

Torsion is a measure of how the curve twists out of its osculating plane. A curve in the \(xy\)-plane has no component in the \(z\) direction, so the curve remains fully inside the same plane at every point. Therefore, the torsion for such a curve is always zero, regardless of whether it is concave up or down. This statement is false.

g. A curve with large curvature also has large torsion

Curvature and torsion are separate measures of a curve's properties. Curvature is a measure of how sharply a curve bends, whereas torsion measures the twist of a curve out of its osculating plane. High curvature does not imply high torsion, and vice versa. Thus, this statement is false.

Key concepts

Vectors

In calculus, vectors are essential in understanding motion and properties of curves in various dimensions. A vector is a mathematical entity that has both magnitude and direction. It can point from one location to another in a space, making it vital to describe motion or spatial positions. For example, in a plane, a vector is often represented as \( \mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle \) for some parameter \( t \).

• Position Vector: This vector (\( \mathbf{r} \)) connects the origin of a coordinate system to a point on the curve. It's the primary element for describing the location of an object in space.


• Unit Tangent Vector: Denoted by \( \mathbf{T} \), this vector is a normalized derivative of the position vector \( \mathbf{r} \), defined as \( \mathbf{T} = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|} \). It indicates the direction of the curve at any given point.


• Principal Unit Normal Vector: The principal normal vector \( \mathbf{N} \) is perpendicular to \( \mathbf{T} \) and lies in the same plane as \( \mathbf{T} \). It points towards the center of curvature of the curve.

Understanding these vectors is crucial to exploring how different properties, like curvature and acceleration, change along a path.

Curvature

Curvature is a measure of how sharply a curve deviates from being straight. It's denoted by the symbol \( \kappa \) and gives insight into the "tightness" or "sharpness" of a curve at a particular point.

• Mathematical Definition: Curvature is mathematically described as \( \kappa = \frac{|\mathbf{T}'(s)|}{|rac{d\mathbf{r}}{ds}|} \), where \( \mathbf{T}'(s) \) is the derivative of the unit tangent vector with respect to the arc length \( s \).


• Conceptual Understanding: Curvature explains how much a curve bends at a given point. A high curvature means a tight turn, while a low curvature indicates a gentle bend.


• Independence from Orientation: Curvature is a scalar quantity. This means it does not depend on the direction in which you traverse the curve. Thus, it remains constant regardless of the orientation.

This concept allows us to differentiate between curves that look smooth versus those that exhibit tight loops or bends.

Acceleration

Acceleration in calculus refers to the rate at which the velocity of an object changes over time. It is crucial for understanding motions along curves and is especially insightful when analyzing circular or oscillatory movement.

• Basic Concept: Acceleration, denoted as \( \mathbf{a} \), is the derivative of velocity with respect to time, \( \mathbf{a} = \frac{d\mathbf{v}}{dt} \).


• Tangential and Normal Components: Total acceleration can be split into tangential and centripetal components. The tangential component gives the rate of change of speed along the curve, while the centripetal component (\( \mathbf{a}_c \)) accounts for changes in the direction of motion, as seen in curves or circular paths.


• Unit Speed and Circular Motion: For an object moving at unit speed around a circle, the acceleration arises purely from the centripetal force pulling it towards the center of the circle, described as \( \mathbf{a} = -\frac{1}{R} \mathbf{N} \), where \( R \) is the radius of the circle.

Understanding acceleration in various forms helps explain not only straightforward motion but also complex dynamics like spinning or oscillating.

Tangent and Normal Vectors

Tangent and normal vectors are vital notions when studying the geometry of curves in calculus. They provide a foundational framework for analyzing how curves behave in different spaces.

• Unit Tangent Vector \( \mathbf{T} \): This vector is tangential to the curve and runs in the direction of motion. It's computed as a normalized version of the derivative of the position vector, showing where the curve is headed at each point.


• Principal Unit Normal Vector \( \mathbf{N} \): Perpendicular to \( \mathbf{T} \), the normal vector points toward the curve's center of curvature. It's crucial for understanding the bending properties and is computed by differentiating \( \mathbf{T} \).


• Osculating Plane: This is the plane formed by the tangent and normal vectors along with the position vector. It's where much of the curve's immediate "action" takes place.

These vectors help parse the two-dimensional nature of a curve's immediate behavior, offering a method to distinguish between movement trajectories and analyze their path-related properties.