Problem 33 - Thomas Calculus

Question

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 \(\mathbf{r}_{2}(t)=t \mathbf{i},-a \leq t \leq a\) $$ \mathbf{F}=-y \mathbf{i}+x \mathbf{j} $$

Step-by-Step Solution

Answer

The circulation is \(a^2 \pi\) and the flux is \(\pi a^2\).

Step 1: Parameterize the semicircular path

The given semicircular arch is parameterized by \(\mathbf{r}_{1}(t) = (a \cos t) \mathbf{i} + (a \sin t) \mathbf{j}\) where \(0 \leq t \leq \pi\). This describes a semicircle on the upper half of the circle of radius \(a\), centered at the origin.

Step 2: Find the derivative of the path for circulation

To find the circulation, we need the derivative of \(\mathbf{r}_1(t)\). Thus, \(\mathbf{r}_1'(t) = -a\sin t\, \mathbf{i} + a\cos t\, \mathbf{j}\).

Step 3: Calculate the vector field along the semicircle

Substitute \(x = a\cos t\) and \(y = a\sin t\) into \(\mathbf{F} = -y \mathbf{i} + x \mathbf{j}\), we get \(\mathbf{F}(\mathbf{r}_1(t)) = -a\sin t \mathbf{i} + a\cos t \mathbf{j}\).

Step 4: Compute the circulation for the semicircle

The circulation is given by the line integral \(\int_{\mathbf{r}_1} \mathbf{F} \cdot d\mathbf{r}\). Substitute in \(\mathbf{F}(\mathbf{r}_1(t)) = -a\sin t \mathbf{i} + a\cos t \mathbf{j}\) and \(d\mathbf{r} = (-a\sin t\, \mathbf{i} + a\cos t\, \mathbf{j})\, dt\). Therefore, \(\mathbf{F} \cdot d\mathbf{r} = (-a\sin t)(-a\sin t) + (a\cos t)(a\cos t)\, dt = a^2\sin^2 t + a^2\cos^2 t\, dt = a^2\, dt\). The integral becomes \(\int_{0}^{\pi} a^2\, dt = a^2[ t]_0^{\pi} = a^2 \pi\).

Step 5: Parameterize the line segment for circulation

The line segment \(\mathbf{r}_{2}(t) = t \mathbf{i}\) for \(-a \leq t \leq a\) is parameterized over the interval as a straight line along the x-axis.

Step 6: Calculate the vector field along the line

For the line segment, \(x = t\) and \(y = 0\), so \(\mathbf{F}(\mathbf{r}_2(t)) = 0 \mathbf{i} + t \mathbf{j}\).

Step 7: Compute the circulation for the line segment

With \(\mathbf{r}'_2(t) = \mathbf{i}\), the line integral is \(\int_{-a}^{a} \mathbf{F}(\mathbf{r}_2(t)) \cdot d\mathbf{r} = \int_{-a}^{a} (0 \cdot 1 + t \cdot 0) = 0\).

Step 8: Find total circulation

The total circulation is the sum of the circulations over the semicircle and the line segment: \(a^2 \pi + 0 = a^2 \pi\).

Step 9: Calculate the flux using Green's Theorem

Green's Theorem relates the circulation around a closed curve to the flux across the area it encloses. For \(\mathbf{F} = (-y, x)\), the curl is \((\frac{\partial x}{\partial x} - \frac{\partial (-y)}{\partial y}) = 2\). The flux is thus \(\int_{R} 2\, dA\), where \(R\) is the semicircular region. The area \(A\) of the region is \(\frac{1}{2}\pi a^2\), so the flux is \(2 \times \frac{1}{2}\pi a^2 = \pi a^2\).

Step 10: Conclusion

The circulation around the closed path is \(a^2 \pi\) and the flux across the surface it encloses is also \(\pi a^2\). These results are consistent because they are derived from Green's Theorem.

Key Concepts

Circulation

Circulation is a critical concept in vector calculus and physics. It measures how much a vector field "circulates" or "moves around" a closed path. In simpler terms, it tells us about the fluid-like motion along a given loop.
In this exercise, we calculate the circulation of the vector field \( \mathbf{F} = -y \mathbf{i} + x \mathbf{j} \) along a semicircular path. We begin by parameterizing our path to express it in a form suitable for integration.
By using the equation \( \mathbf{r}_{1}(t) = (a \cos t) \mathbf{i} + (a \sin t) \mathbf{j} \), we described the semicircle, which is then differentiated to find \( \mathbf{r}_1'(t) \). This allows us to set up a line integral, measuring the circulation of \( \mathbf{F} \) along the semicircle. We cross-check by recomputing along the straight-line segment and the results aggregate to a total circulation of \( a^2 \pi \).
This result aligns with the properties articulated by Green's Theorem, demonstrating the robustness of the circulation calculation.

Flux

Flux, in the context of vector fields, is a measure of the field's flow across a surface. It tells us how much of the field is passing through a surface or area.
For our scenario, to find the flux through the semicircular region, we can use Green's Theorem. This theorem connects circulation around a closed curve to the area "swept" by the curve.
Using \( \mathbf{F} = (-y, x) \), the theorem helps us find the curl of \( \mathbf{F} \), calculated here as 2, which succinctly relates the vector field components. When calculating the flux, we integrate over the entire region of interest. In this case, the semicircle's area is \( \frac{1}{2} \pi a^2 \), and hence, the flux across this region is \( \pi a^2 \).
This nicely illustrates the complementary relationship with circulation and provides insights into how these quantities capture different aspects of vector fields.

Line Integral

The line integral plays an important role in calculating both circulation and flux. It allows us to integrate a function or vector field along a curve.
A line integral can compute the work done by a force field in moving an object along a path or, as in this exercise, the circulation of a vector field.
To evaluate the line integral for circulation, the path \( \mathbf{r}_1(t) \) and \( \mathbf{r}_2(t) \) must be parameterized. The line integral is then set up as \( \int \mathbf{F} \cdot d\mathbf{r} \), which is the dot product of the vector field \( \mathbf{F} \) with the path's differential element \( d\mathbf{r} \). Both the semicircle and line segment need to be considered separately, then their contributions are summed for total circulation.
The process demonstrates how integral calculus helps in evaluating physical phenomena through geometric paths.

Parametrization

Parametrization is a technique used to express a path or curve in terms of one or more variables. It allows for the transformation of complex geometric shapes into manageable algebraic forms.
In this exercise, the semicircular path is represented by \( \mathbf{r}_{1}(t) = (a \cos t) \mathbf{i} + (a \sin t) \mathbf{j} \), where \( t \) varies from \( 0 \) to \( \pi \). This converts the semicircle into a smooth curve whose properties simplify the integration process during line integrals.
Similarly, the line segment is parameterized as \( \mathbf{r}_{2}(t) = t \mathbf{i} \) for \(-a \leq t \leq a\). This makes calculations straightforward for assessing path-related attributes like circulation and helps integrate \( \mathbf{F} \) effectively. Understanding parametrization helps us visualize paths and curves and describe them mathematically, crucial for applying techniques to find physical and geometrical attributes across varied paths.