Problem 209 - Calculus Volume 3

Question

Sketch the curves for the following vector equations. Use a calculator if needed. \(\mathbf{r}(t)=\left\langle\sin (20 t) e^{-t}, \cos (20 t) e^{-t}, e^{-t}\right\rangle\)

Step-by-Step Solution

Answer

The curve is a rapidly spiraling helix in 3D, shrinking towards the origin.

Step 1: Analyzing the Vector Equation

The vector equation given is \( \mathbf{r}(t)=\left\langle \sin (20 t) e^{-t}, \cos (20 t) e^{-t}, e^{-t}\right\rangle \). This equation represents a 3D space curve, where each component is a function of parameter \( t \). The first component is \( x(t) = \sin (20 t) e^{-t} \), the second component is \( y(t) = \cos (20 t) e^{-t} \), and the third component is \( z(t) = e^{-t} \).

Step 2: Understanding the Exponential Decay

All components include the exponential decay factor \( e^{-t} \). This means as \( t \) increases, all components and thus the magnitude of the vector \( \mathbf{r}(t) \) will decrease exponentially, causing the vector to approach the origin, \( \mathbf{0} \).

Step 3: Examining the Oscillatory Behavior

The terms \( \sin(20t) \) and \( \cos(20t) \) suggest rapid oscillations in the \( x \) and \( y \) directions. The frequency of 20 implies the curve will make 20 oscillations over an interval where \( t \) ranges from 0 to a small positive value. As \( t \) increases, the effects of \( e^{-t} \) will cause these oscillations to dampen and shrink towards the z-axis.

Step 4: Combining Factors to Sketch the Curve

Considering both the oscillatory and exponential behaviors, the curve starts at a point on the unit circle in the \( xy \)-plane when \( t = 0 \) (namely \( \langle 0, 1, 1 \rangle \)) and spirals inward and downward exponentially as \( t \) increases, forming a tight spiral because of the rapid oscillations in the \( xy \)-plane.

Key Concepts

Vector Equations

In vector calculus, a vector equation represents a curve in which each coordinate corresponds to a function of a parameter, often denoted as \( t \). Here, the vector equation is \( \mathbf{r}(t) = \left\langle \sin(20t) e^{-t}, \cos(20t) e^{-t}, e^{-t} \right\rangle \). Each function of \( t \) describes how the curve evolves in 3D space.

Breaking it down:

• The first component \( x(t) = \sin(20t) e^{-t} \) shows how the curve moves along the x-axis.
• The second component \( y(t) = \cos(20t) e^{-t} \) controls movement along the y-axis.
• The third component \( z(t) = e^{-t} \) indicates fluctuation in the z-axis.

These components together graph a trajectory in space, revealing intricate paths through the 3D coordinate system. Each point on the curve represents a specific moment in time, determined by \( t \). This type of mathematical description is fundamental in understanding the motion of particles and fields in physics.

3D Space Curves

A 3D space curve is a path in three-dimensional space described by a vector equation. In our case, the curve is formed by combining all three components in the vector \( \mathbf{r}(t) \).

These curves often represent physical phenomena such as the trajectory of an object, the path of a fluid particle, or the electromagnetic field lines. To visualize the given equation:

• Initially, the curve could start on the unit circle on the \( xy \)-plane, at a height of \( e^{-0}=1 \) at \( t=0 \).
• The path then notably spirals inward due to the combined effect of the sine and cosine components.

With the exponential decay, as time progresses, the spiral increasingly tightens and draws towards the origin. Understanding these curves is crucial in fields like computer graphics and robotics, helping to simulate complex movements and interactions.

Exponential Decay

Exponential decay is a mathematical concept describing the process of quantities decreasing at a rate proportional to their current value. In the vector equation \( \mathbf{r}(t) = \left\langle \sin(20t) e^{-t}, \cos(20t) e^{-t}, e^{-t} \right\rangle \), all components share the exponential decay factor \( e^{-t} \).

This means:

• Initially, the magnitude of the vector is relatively higher.
• However, as \( t \) becomes larger, \( e^{-t} \) shrinks all components exponentially.
• The vector approaches zero, or the origin, indicative of the curve folding into itself.

Exponential decay signifies the diminishing influence of functions over time. In real-world applications, it helps model natural processes like radioactive decay, cooling of hot objects, or depreciation in finance. These models closely follow a similar mathematical trajectory over time.

Oscillatory Behavior

Oscillatory behavior refers to a repeating pattern or fluctuation occurring in the values of a function or system. In the vector equation given, the expressions \( \sin(20t) \) and \( \cos(20t) \) dictate oscillations.

Key features include:

• The frequency component of 20, indicating rapid oscillations in the x and y directions.
• The oscillation period can be calculated based on the standard function properties of sine and cosine: each cycle repeats frequently as noted.
• Initial large oscillations gradually reduce due to the damping effect of \( e^{-t} \).

This behavior explains why the path doesn't smoothly rest towards the axis but spirals downward in a shrinking manner. Understanding oscillations is beneficial not only in math, but also in areas like acoustics, signal processing, and wave dynamics. It is instrumental in deciphering complex systems where repetitive dynamics occur.