Question

If \(f:(a, b) \rightarrow \mathbb{R}^{n}\) is a parametrization of a smooth curve (i.e., a 1-dimensional \(C^{1}\) submanifold of \(\mathbb{R}^{n}\) ), the Hausdorff 1-dimensional measure of the curve is \(\int_{a}^{b}\left|f^{\prime}(t)\right| d t .\)

Short answer

Compute \( \int_{a}^{b} |f'(t)| \, dt \).

Step-by-step solution

Understanding the Problem

We need to find the Hausdorff 1-dimensional measure of a curve given by a parametrization \( f: (a, b) \rightarrow \mathbb{R}^{n} \). This measure involves computing an integral over the interval \((a, b)\) of the magnitude of the derivative of \( f \).

Identifying the Derivative

First, we need to calculate the derivative of the parametrization \(f\), denoted as \( f'(t) \), with respect to the parameter \( t \). This derivative represents the tangent vector to the curve at each point \( t \).

Calculating the Magnitude

Once we have \( f'(t) \), calculate its magnitude, \( |f'(t)| \). This gives us the speed of the curve at each point \( t \), which is necessary for measuring the length of the curve.

Setting Up the Integral

The integral that measures the Hausdorff 1-dimensional measure of the curve is given by \( \int_{a}^{b} |f'(t)| \, dt \). This integral represents the total length of the curve over the interval \((a, b)\).

Evaluating the Integral

Finally, evaluate the integral \( \int_{a}^{b} |f'(t)| \, dt \). This will provide the measure of the length of the curve, also known as the Hausdorff measure.

Key concepts

Parametrization

In mathematics, parametrization involves representing a curve or surface using parameters. This creates a mapping from an interval or domain, onto a curve or surface in space. Consider a function \( f:(a, b) \to \mathbb{R}^{n} \) which defines a curve. Here, \( t \in (a, b) \) acts as the parameter. Each value of \( t \) corresponds to a point \( f(t) \) on the curve.

Parametrization helps in describing complex geometric shapes in simple terms. By using parameters, one can easily manipulate and analyze shapes. For example, the parametric representation of a circle in -dimensional space might use the angle \( t \) to define every point along the circle's path. This technique is essential in both theoretical and applied aspects in fields like physics and computer graphics, as well as for evaluating measures and performing integrals over curves.

Smooth Curve

A smooth curve is a type of curve where mathematical definitions are aimed at being as 'nice' as possible. Smoothness means the curve is continuously differentiable. Specifically, a curve \( f:(a, b) \to \mathbb{R}^{n} \) is smooth if it is a \( C^1 \) function. This means the function is at least once differentiable, and its derivative is continuous.

Smooth curves do not have sharp corners or cusps, and their tangent vectors exist at every point along the curve. These properties enable precise and meaningful analyses along the entire curve without encountering undefined behavior. Such curves often arise in natural settings, and in abstract mathematics, they provide a foundation for calculus and differential equations.

Tangent Vector

A tangent vector is a vector that 'touches' a curve at a single point and points in the direction of the curve's immediate path. For a parametric curve \( f(t) \), the tangent vector at a point is given by the derivative \( f'(t) \). The vector indicates the velocity and direction of the curve as you move along it.

If you imagine zooming into a point on a curve, the tangent vector represents how the curve behaves at that point. In other words, the tangent vector describes the instantaneous rate of change of the curve's position \( f(t) \) with respect to the parameter \( t \).

• Magnitude of \( f'(t) \): This is the speed of the curve at point \( t \).
• Direction of \( f'(t) \): Demonstrates which direction the curve is moving toward.

These aspects are critical in fields like physics for motion analysis, as well as in geometry when studying the properties and behaviors of shapes and paths.

Integral Calculus

Integral calculus is a branch of mathematics focused on accumulation and net change. It is the mathematical tool used for finding areas, volumes, central points, and lengths pertaining to functions and curves. In the context of a curve defined by \( f(t) \), integral calculus helps measure the Hausdorff 1-dimensional measure, which tells you the curve's length.

To obtain this measure, we compute the integral \( \int_{a}^{b} |f'(t)| \, dt \). This expression calculates the total length by summing the magnitudes of tangent vectors across the range \( (a, b) \).

Key points to note:

• The integral adds up tiny lengths \( |f'(t)| \, dt \) of the curve. This is akin to stretching a piece of string straight and measuring it.
• The technique can help solve real-world problems in engineering and physics, where determining path lengths or transformations is crucial.

Overall, integral calculus provides a framework for continuous addition, which is necessary for accurately determining the length of curves like those defined by parametric equations.