Problem 19 - Thomas Calculus

Question

Maximizing the curvature of a helix In Example \(5,\) we found the curvature of the helix \(\mathbf{r}(t)=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}+b t \mathbf{k}\) \((a, b \geq 0)\) to be \(\kappa=a /\left(a^{2}+b^{2}\right) .\) What is the largest value \(\kappa\) can have for a given value of \(b ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

The largest value \( \kappa \) can have is \( \frac{1}{2b} \) when \( a = b \).

Step 1: Understanding the Problem

We need to find the maximum value of the curvature \( \kappa = \frac{a}{{a^2 + b^2}} \) of the helix given \( b \) as a constant. We are tasked with maximizing \( \kappa \) with respect to the variable \( a \).

Step 2: Setting Up the Optimization Problem

To maximize the curvature \( \kappa = \frac{a}{{a^2 + b^2}} \), we treat \( a \) as a variable and \( b \) as a constant parameter.

Step 3: Finding the Derivative

Find the derivative of \( \kappa \) with respect to \( a \). Using the quotient rule: if \( u = a \) and \( v = a^2 + b^2 \), then \( \kappa' = \frac{v u' - u v'}{v^2} = \frac{(a^2 + b^2)(1) - a(2a)}{(a^2 + b^2)^2} \). Simplify this to: \( \kappa' = \frac{b^2 - a^2}{(a^2 + b^2)^2} \).

Step 4: Finding Critical Points

Set the derivative \( \kappa' = \frac{b^2 - a^2}{(a^2 + b^2)^2} \) to zero to find the critical points: \( b^2 - a^2 = 0 \). Solving for \( a \), we get \( a = b \).

Step 5: Analyzing the Critical Point

Check the behavior of \( \kappa \) around \( a = b \) to confirm it is a maximum. Since \( \kappa' < 0 \) for \( a > b \) and \( \kappa' > 0 \) for \( a < b \), the function \( \kappa \) achieves a maximum at \( a = b \).

Step 6: Computing Maximum Curvature

Substitute \( a = b \) into the expression for \( \kappa \): \( \kappa = \frac{b}{b^2 + b^2} = \frac{b}{2b^2} = \frac{1}{2b} \). Thus, the maximum curvature is \( \frac{1}{2b} \).

Key Concepts

Helix

A helix is a fascinating geometric shape defined by a smooth and consistent twisting. It can be visualized as a curve that spirals around an axis, much like the shape of a spring or a corkscrew. In mathematics, the parametric representation of a helix can be utilized in coordinates, often involving trigonometric functions for the x and y components, while including a linear function for the z component.
For instance, the helix given by \( \mathbf{r}(t) = (a \cos t) \mathbf{i} + (a \sin t) \mathbf{j} + bt \mathbf{k} \) uses the parameter \( t \) to create this spiral effect. Here:

• \( a \) is a constant that affects the radius of the helix around the center axis.
• \( b \) is another constant that determines the pitch of the helix, or how tightly it coils.

Understanding the helix in this mathematical form helps in analyzing physical phenomena, such as waves or DNA structures, since such forms appear naturally or in designs inspired by nature.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. In the context of finding maximum curvature, taking the derivative of a function is crucial as it informs us about the rate of change of that function, which is expressed as \( \kappa(a) = \frac{a}{{a^2 + b^2}} \).
To find the derivative of \( \kappa \) with respect to the variable \( a \), we employ the quotient rule. This rule is used when differentiating a ratio of functions and states:
"The derivative of \( \frac{u}{v} \) is \( \frac{v u' - u v'}{v^2} \)".
Applying this to our curvature function where \( u = a \) and \( v = a^2 + b^2 \), the derived expression helps identify critical points. Calculating derivatives, this way allows us to set up the steps necessary to optimize functions and understand their behavior in terms of increase or decrease.

Critical Points

Critical points are values of the domain at which the derivative of a function is zero or undefined. These points are essential in calculus because they can represent locations where a function reaches a local maximum or minimum. By setting the derivative of the curvature \( \kappa' = \frac{b^2 - a^2}{(a^2 + b^2)^2} \) to zero, we find the critical points of the helix's curvature function.
Solving \( b^2 - a^2 = 0 \) reveals that \( a = b \) is a critical point. At this point, the curvature reaches its extremity for the given parameter \( b \).

• It's important to check the behavior of the function around these critical points to establish whether they are maxima or minima.

This validation involves considering the changes in sign of the derivative around these points, ensuring an accurate determination of the function's behavior.

Maximum Curvature

Maximum curvature refers to the largest value that the curvature function \( \kappa \) can achieve for a given set of parameters. In this exercise, calculating maximum curvature involves verifying the nature of the critical point identified. By substituting \( a = b \) into the original curvature equation \( \kappa = \frac{a}{{a^2 + b^2}} \), we simplify and find the value of maximum curvature.

It's derived that when \( a = b \), the curvature is \( \kappa = \frac{b}{2b^2} = \frac{1}{2b} \). This indicates that the largest possible curvature depends inversely on the parameter \( b \).

• This finding is crucial for optimization and is particularly relevant in fields like engineering and physics where the efficiency and integrity of structures or paths could rely on understanding and utilizing curvature.

This analysis allows better anticipation of conditions under which certain phenomena are likely to occur or to optimize designs for maximal performance.