Problem 19 - Advanced Engineering Mathematics

Question

Show that the curvature of a straight line is the constant \(\kappa=0\).

Step-by-Step Solution

Answer

The curvature of a straight line is 0 because the second derivative is zero.

Step 1: Define Curvature

The curvature of a curve is a measure of how quickly it changes direction. For a curve given by a vector function \( \mathbf{r}(t) \), the curvature \( \kappa \) at any point is defined by \( \kappa = \frac{|\mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3} \), assuming \( \mathbf{r}'(t) eq 0 \).

Step 2: Represent a Straight Line

A straight line can be expressed in vector form as \( \mathbf{r}(t) = \mathbf{a} + t \mathbf{b} \), where \( \mathbf{a} \) is a point on the line and \( \mathbf{b} \) is a direction vector.

Step 3: Compute the First Derivative

Differentiate the line equation to find \( \mathbf{r}'(t) \). For \( \mathbf{r}(t) = \mathbf{a} + t \mathbf{b} \), the derivative is \( \mathbf{r}'(t) = \mathbf{b} \), a constant vector representing the direction of the line.

Step 4: Compute the Second Derivative

Differentiate \( \mathbf{r}'(t) \) to find \( \mathbf{r}''(t) \). Since \( \mathbf{b} \) is constant, \( \mathbf{r}''(t) = \mathbf{0} \).

Step 5: Calculate Curvature Using the Formula

Substitute \( \mathbf{r}'(t) = \mathbf{b} \) and \( \mathbf{r}''(t) = \mathbf{0} \) into the curvature formula: \( \kappa = \frac{|\mathbf{0}|}{|\mathbf{b}|^3} = 0 \). The numerator is zero because the second derivative is zero, which makes the curvature \( \kappa = 0 \).

Step 6: Conclusion

Since the curvature \( \kappa = 0 \), we see that the curvature of a straight line is indeed constant and equal to zero, confirming that straight lines do not curve.

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector spaces and the differentiation and integration of vector functions. These concepts are used extensively in physics and engineering to describe fields such as electromagnetic fields, and in the case of our problem, geometry of curves.
A vector function involves a mapping from a set of real numbers to a set of vectors in a vector space. For a given parameter, such as time or any other variable, a vector function will output a vector, which can describe a position, velocity, or force.

• One of the central themes of vector calculus is to explore how these vectors change with respect to one another.
• It includes various operations such as differentiation and integration applied to vectors—key for understanding motion and change.
• Vector calculus forms the basis for more complex fields like tensor calculus, which is used in general relativity.

Understanding vector calculus is crucial for analyzing how lines and curves behave in space.

Curvature Calculation

Curvature quantifies how much a curve deviates from being straight at a particular point. It gives us an understanding of the bend of the curve, which is zero for straight lines, confirming that they do not bend. This is an essential property in geometry and physics.

Formula for Curvature

The curvature \( \kappa \) of a curve defined by a vector function \( \mathbf{r}(t) \) is given by:\[ \kappa = \frac{|\mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3} \]Here is how we compute it:

• The numerator \( |\mathbf{r}''(t)| \) is the magnitude of the second derivative. It indicates the rate at which the direction of the tangent vector to the curve is changing.
• The denominator \( |\mathbf{r}'(t)|^3 \) normalizes the rate of change by the cube of the velocity vector's magnitude, making this formula applicable to parametric curves like circles or helices.

For a straight line, since the second derivative is zero, the curvature becomes zero, affirming that straight lines have no curvature.

Differentiation of Vector Functions

Differentiation in vector calculus is essential for finding quantities like velocity and acceleration, which are vectors. When we differentiate a vector function, we do it component-wise, considering each vector component separately. This process helps understand the behavior of the curve.

Steps in Differentiation

• The first derivative \( \mathbf{r}'(t) \) is obtained by differentiating each component of the vector function individually. It represents the tangent to the curve and indicates the direction of motion.
• The second derivative \( \mathbf{r}''(t) \) provides information about the curve's acceleration and indicates changes in the direction of motion. For a straight line, this would be a zero vector indicating no change in direction.

Applying differentiation to vector functions like a straight line \( \mathbf{r}(t) = \mathbf{a} + t \mathbf{b} \), you find that the direction vector \( \mathbf{b} \) remains constant, as seen in the solution. This reflects how straight lines maintain a constant direction, confirming their zero curvature.