Problem 31 - Thomas Calculus

Question

In Exercises \(31-34,\) find the circulation and flux of the field \(\mathbf{F}\) around and across the closed semicircular path that consists of the semicircular arch \(\mathbf{r}_{1}(t)=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq \pi,\) followed by the line segment \(\mathbf{r}_{2}(t)=t \mathbf{i},-a \leq t \leq a\) $$\mathbf{F}=x \mathbf{i}+y \mathbf{j}$$

Step-by-Step Solution

Answer

The circulation and flux of the field are both zero.

Step 1: Understand the Parametrization of the Path

The closed path consists of two parts. The semicircular arc can be expressed as \( \mathbf{r}_1(t) = (a \cos t) \mathbf{i} + (a \sin t) \mathbf{j} \) for \( 0 \leq t \leq \pi \). The straight line segment can be expressed as \( \mathbf{r}_2(t) = t \mathbf{i} \) for \( -a \leq t \leq a \).

Step 2: Define the Field and Calculate Derivatives

The vector field is \( \mathbf{F} = x \mathbf{i} + y \mathbf{j} \). For the semicircular path, using \( \mathbf{r}_1(t) \), derive \( \mathbf{r}_1'(t) = (-a \sin t) \mathbf{i} + (a \cos t) \mathbf{j} \). For the line segment, derive \( \mathbf{r}_2'(t) = \mathbf{i} \).

Step 3: Set Up the Circulation Integral

The circulation \( C \) of the field around the path is given by \( \oint_C \mathbf{F} \cdot d\mathbf{r} \). For the semicircular path, this becomes \( \int_0^\pi \mathbf{F}(\mathbf{r}_1(t)) \cdot \mathbf{r}_1'(t) \, dt \), and for the line segment, it is \( \int_{-a}^a \mathbf{F}(\mathbf{r}_2(t)) \cdot \mathbf{r}_2'(t) \, dt \).

Step 4: Compute Circulation Around the Semicircle

Evaluate the semicircle integral: substitute \( \mathbf{r}_1(t) \) into \( \mathbf{F} \) to get \( (a \cos t) \mathbf{i} + (a \sin t) \mathbf{j} \). Then calculate \((a \cos t) (-a \sin t) + (a \sin t)(a \cos t)\), which simplifies to 0. Thus, \( \int_0^\pi 0 \, dt = 0 \).

Step 5: Compute Circulation Along the Line Segment

Evaluate the line integral: substitute \( \mathbf{r}_2(t) \) into \( \mathbf{F} \) to get \( t \mathbf{i} \). Then calculate \( t \cdot 1 = t \), and integrate \( \int_{-a}^a t \, dt = 0 \), since the integrand is an odd function symmetric about 0.

Step 6: Set Up and Calculate the Flux Across the Path

The flux across the closed path is given by \( \oint_C \mathbf{F} \cdot \mathbf{n} \, ds \), where \( \mathbf{n} \) is the normal to the path. For the semicircular path, use the outward normal: \( (a \cos t) \mathbf{j} - (a \sin t) \mathbf{i} \). For the line segment, the normal can be the vertical component \( \mathbf{j} \). Calculate both integrals.

Step 7: Compute the Flux Across the Semicircle

For the semicircle, the normal is given by \( -\sin t \mathbf{i} + \cos t \mathbf{j} \). Therefore, computing the flux \( (a \cos t) (-a \sin t) + (a \sin t)(a \cos t) \) also results in 0. Hence, \( \int_0^\pi 0 \, dt = 0 \).

Step 8: Compute the Flux Across the Line Segment

For the line \( y = 0 \), the normal is \( \mathbf{j} \) and \( \mathbf{F} \cdot \mathbf{n} = 0 \). Integrating over \( t \), \( \int_{-a}^a 0 \, dt = 0 \).

Step 9: Conclusion of Circulation and Flux

Both the circulation and the flux for the given vector field \( \mathbf{F} = x \mathbf{i} + y \mathbf{j} \) around and across the described path are zero.

Key Concepts

Circulation

Circulation measures how much a vector field spins around a closed path. In this exercise, we calculated the circulation of the vector field \( \mathbf{F} = x \mathbf{i} + y \mathbf{j} \) around the semicircular path. To do this, we used the line integral of the field along the path.

The semicircular path was divided into two components: the semicircle and the line segment. For each, we computed the line integral by finding the dot product of \( \mathbf{F} \) with the derivative of the parametrized path \( d\mathbf{r} \). Ultimately, the calculations showed that the integral for both components equaled zero, implying no net circulation of the vector field around the path.

This result means that the flow was perfectly balanced in the clockwise and counterclockwise direction along the path. The concept of circulation is crucial in fields like fluid dynamics and electromagnetism, where it helps describe rotational effects.

Flux

Flux quantifies the flow of a vector field across a surface or path. In this exercise, we determined the flux of \( \mathbf{F} = x \mathbf{i} + y \mathbf{j} \) across the given semicircular path.

To calculate the flux, we utilized Green's Theorem, which relates the line integral around a closed path to a double integral over the region inside the path. However, given the problem requires line integrals, the flux was found by computing the line integrals of \( \mathbf{F} \) along segments of the semicircle and the linear section with respect to inflow (or normal vectors).

The normals along the semicircle were defined as \( -\sin t \mathbf{i} + \cos t \mathbf{j} \) while the line segment used \( \mathbf{j} \). From these computations, we found out that the flux was zero across both paths. This indicates that the vector field does not contribute any net flow either inwards or outwards, expressing a balance characteristic often notable in steady-state conditions.

Vector Fields

A vector field is a way of assigning a vector to every point in a space. In this exercise, the vector field \( \mathbf{F} = x \mathbf{i} + y \mathbf{j} \) assigns each point in the plane a vector based on its coordinates. Each vector generally shows a direction and possibly the magnitude of something, such as velocity.

Understanding vector fields is essential because they describe many physical phenomena, like gravitational fields, magnetic fields, and fluid flow. For instance, the vector field in this problem might represent a flow where at any given point, the velocity is directed along the radial lines and varies linearly with distance from the origin, suggestive of a source in the middle.

• Each point in space has a vector
• The behavior of the field can show rotational movement or linear flow
• Integral calculus, including line and surface integrals, is often needed to analyze vector fields properly

Overall, identifying how a vector field behaves helps understand underlying physical processes.

Line Integrals

Line integrals allow us to integrate a function along a curve or path. In this problem, we used line integrals to assess properties of the vector field \( \mathbf{F} = x \mathbf{i} + y \mathbf{j} \).

To find the circulation or flux, line integrals require determining the path parametrization. Then, the function is integrated with respect to this parametrization. In physical terms, a line integral can represent the work done by a force field in moving a particle along a path.

• To evaluate a line integral, parametrize each segment of the path
• Compute the dot product of the vector field with the differential of the path
• Perform the integration across the interval

By managing line integrals, one can obtain critical information regarding field interactions with specific paths, such as energy transfer, fluid flow, or boundary circulation, critical to numerous fields including engineering and physics.