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Magazine Chapter 4: Problem 11
Determine which of the following paths are regular: (a) \(\mathbf{c}(t)=(\cos t, \sin t, t)\). (b) \(\mathbf{c}(t)=\left(t^{3}, t^{5}, \cos t\right)\). (c) \(\mathbf{c}(t)=\left(t^{2}, e^{t}, 3 t+1\right)\).
Short Answer
Expert verified
All paths (a), (b), and (c) are regular.
Step by step solution
01
Understand the Definition of a Regular Path
A path \( \mathbf{c}(t) = (x(t), y(t), z(t)) \) is regular if its velocity vector is not zero, i.e., \( \dot{\mathbf{c}}(t) = (\dot{x}(t), \dot{y}(t), \dot{z}(t)) eq \mathbf{0} \) for all \( t \in \mathbb{R} \). The dot denotes the derivative with respect to \( t \).
02
Determine Derivative for Path (a)
For \( \mathbf{c}(t) = (\cos t, \sin t, t) \), calculate the derivative: \( \dot{\mathbf{c}}(t) = (-\sin t, \cos t, 1) \). This vector is not zero for all \( t \), as the third component is always 1.
03
Determine Derivative for Path (b)
For \( \mathbf{c}(t) = (t^{3}, t^{5}, \cos t) \), calculate the derivative: \( \dot{\mathbf{c}}(t) = (3t^2, 5t^4, -\sin t) \). This vector is not the zero vector for any \( t \), as at least one component is non-zero.
04
Determine Derivative for Path (c)
For \( \mathbf{c}(t) = (t^{2}, e^{t}, 3t+1) \), calculate the derivative: \( \dot{\mathbf{c}}(t) = (2t, e^{t}, 3) \). This vector cannot be zero for any \( t \) because \( e^{t} > 0 \) for all \( t \).
05
Verify Regularity of Paths
All derivatives calculated are non-zero for any \( t \). Therefore, each path is regular.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Regular Path
In vector calculus, a path is essentially a function that describes the position of a point in space as a function of a parameter, often denoted as \( t \). The concept of a regular path is vital because it ensures that the path is well-behaved and doesn't do anything unexpected, like stop suddenly or backtrack. A path \( \mathbf{c}(t) = (x(t), y(t), z(t)) \) is considered regular if its velocity vector is never zero for any value of \( t \).
- This means the point continues to move in some direction, never stopping.
- It is not irregular or stationary at any moment desired within its parameter's scope.
Velocity Vector
Understanding the velocity vector is key to determining if a path is regular. The velocity vector of a path \( \mathbf{c}(t) = (x(t), y(t), z(t)) \) is given by its derivative with respect to time \( t \), denoted as \( \dot{\mathbf{c}}(t) = (\dot{x}(t), \dot{y}(t), \dot{z}(t)) \).
In the context of a physical situation, this means an object is always moving and never at rest, thus making its modeling possible with differential equations.
- This vector tells us the direction and speed of the point as it moves along the path.
- If this vector is ever zero, it means that the point has stopped moving, which would imply the path is not regular.
In the context of a physical situation, this means an object is always moving and never at rest, thus making its modeling possible with differential equations.
Derivative Calculation
The derivative calculation is a crucial step when checking the regularity of a path. Calculating the derivative of the path \( \mathbf{c}(t) \) involves differentiating each component of the path with respect to \( t \).
- In the exercise, for path \( (a) \) given by \( \mathbf{c}(t) = (\cos t, \sin t, t) \), the derivative is \( \dot{\mathbf{c}}(t) = (-\sin t, \cos t, 1) \).
- For path \( (b) \) \( \mathbf{c}(t) = (t^{3}, t^{5}, \cos t) \), we have \( \dot{\mathbf{c}}(t) = (3t^2, 5t^4, -\sin t) \).
- For path \( (c) \) \( \mathbf{c}(t) = (t^{2}, e^{t}, 3t+1) \), the derivative is \( \dot{\mathbf{c}}(t) = (2t, e^{t}, 3) \).
Parametric Equations
Parametric equations are equations where the set of quantities are expressed as explicit functions of a parameter. In the realm of vector calculus, parametric equations help in defining paths in a three-dimensional space.
- Each component of a vector equation \( (x(t), y(t), z(t)) \) is a separate function of the independent variable \( t \), telling us how each dimension behaves as \( t \) changes.
- They provide a convenient way to describe complex curves and motions, particularly when the motion is more naturally expressed with parameters rather than traditional Cartesian coordinates.
