Finding the Curvature for the Vector-Valued Function r(t) = <4cos(t), t, 4sin(t)>

In the realm of calculus and vector analysis, the concept of curvature plays a crucial role in understanding the behavior and properties of vector-valued functions. Curvature is a measure of how quickly a curve is changing direction at a given point, and it is an important characteristic in various applications, such as in physics, engineering, and computer graphics.

In this article, we will focus on finding the curvature for the vector-valued function:

r(t) = <4cos(t), t, 4sin(t)>

This function represents a three-dimensional curve, and we will explore the steps to calculate its curvature.

Step 1: Differentiate the Vector-Valued Function

To begin, we need to find the first and second derivatives of the vector-valued function r(t).

The first derivative of r(t) is: r'(t) = <-4sin(t), 1, 4cos(t)>

The second derivative of r(t) is: r''(t) = <-4cos(t), 0, -4sin(t)>

Step 2: Calculate the Magnitude of the Cross Product of the Derivatives

The curvature of a vector-valued function is given by the formula:

κ(t) = ||r'(t) × r''(t)|| / ||r'(t)||^3

Where:

• κ(t) is the curvature at the point t on the curve
• r'(t) is the first derivative of the vector-valued function
• r''(t) is the second derivative of the vector-valued function
• || || represents the magnitude (length) of a vector

Let's calculate the cross product of the first and second derivatives:

r'(t) × r''(t) = <-4, -16sin(t), -16cos(t)>

Now, we need to find the magnitude of this cross product:

||r'(t) × r''(t)|| = sqrt((-4)^2 + (-16sin(t))^2 + (-16cos(t))^2) = sqrt(16 + 256sin^2(t) + 256cos^2(t)) = sqrt(16 + 256) = 20

Next, we need to find the magnitude of the first derivative:

||r'(t)|| = sqrt((-4sin(t))^2 + 1^2 + (4cos(t))^2) = sqrt(16sin^2(t) + 1 + 16cos^2(t)) = sqrt(16 + 1) = sqrt(17)

Step 3: Calculate the Curvature

Now, we can plug in the values into the curvature formula:

κ(t) = ||r'(t) × r''(t)|| / ||r'(t)||^3 = 20 / (sqrt(17))^3 = 20 / (17^(3/2))

Therefore, the curvature of the vector-valued function r(t) = <4cos(t), t, 4sin(t)> is:

κ(t) = 20 / (17^(3/2))

This curvature value represents how quickly the curve is changing direction at the point t on the curve. The higher the curvature, the more the curve is bending or turning at that point.