Question

What's wrong? Consider the rotation field \(\mathbf{F}=\frac{\langle-y, x\rangle}{x^{2}+y^{2}}\) a. Verify that the two-dimensional curl of \(F\) is zero, which suggests that the double integral in the circulation form of Green's Theorem is zero. b. Use a line integral to verify that the circulation on the unit circle of the vector field is \(2 \pi\) c. Explain why the results of parts (a) and (b) do not agree.

Short answer

Question: Verify that the two-dimensional curl of F is zero, calculate the circulation on the unit circle using the line integral, and explain the discrepancy between the two results. Answer: The two-dimensional curl of F is indeed zero. However, the circulation around the unit circle is found to be 2π using the line integral. The discrepancy between the two results arises because F has a singularity at the origin, making Green's theorem inapplicable. The correct circulation value is 2π.

Step-by-step solution

Calculate the two-dimensional curl of F

To calculate the curl in two dimensions, we perform the following operation: \[\text{Curl}(F) = \frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}\] Where F1 represents the x-component (-y) and F2 represents the y-component (x). Apply partial derivatives on F1 and F2. \[\frac{\partial F_{2}}{\partial x} = \frac{\partial}{\partial x} \left(\frac{x}{x^{2}+y^{2}}\right) = \frac{y^{2} - x^{2}}{(x^{2} + y^{2})^{2}}\] \[\frac{\partial F_{1}}{\partial y} = \frac{\partial}{\partial y} \left(\frac{-y}{x^{2}+y^{2}}\right) = \frac{y^{2} - x^{2}}{(x^{2} + y^{2})^{2}}\] Now, subtract these two derivatives to get the curl: \[\text{Curl}(F) = \frac{y^{2} - x^{2}}{(x^{2} + y^{2})^{2}} - \frac{y^{2} - x^{2}}{(x^{2} + y^{2})^{2}} = 0\]

Compute the circulation using a line integral

To compute the circulation using the line integral, we need to parameterize the unit circle: \[\mathbf{r}(t) = \langle\cos{t}, \sin{t}\rangle\] Now, compute the derivative of r(t) with respect to t: \[\frac{d\mathbf{r}}{dt} = \langle -\sin{t}, \cos{t} \rangle\] Next, evaluate F(r(t)): \[\mathbf{F}(r(t)) = \frac{\langle -\sin{t}, \cos{t} \rangle}{\cos^2{t} + \sin^2{t}} = \langle -\sin{t}, \cos{t} \rangle\] Finally, compute the circulation on the unit circle using the line integral: \[C=\oint_C \mathbf{F} \cdot d\mathbf{r} = \int_0^{2\pi} \langle -\sin{t}, \cos{t} \rangle \cdot \langle -\sin{t}, \cos{t} \rangle dt = \int_0^{2\pi} \sin^2{t} + \cos^2{t} dt = \int_0^{2\pi} dt = 2\pi\]

Explain the discrepancy between the results

The discrepancy between the results in parts (a) and (b) arises from the fact that F has a singularity at the origin, i.e., the vector field is not defined at the point (0,0), while Green's theorem requires the vector field to be defined and continuous on the region enclosed by the curve. Since the two-dimensional curl of F is zero, Green's theorem would suggest that the circulation of F around the unit circle should also be zero. However, Green's theorem is not applicable here due to the singularity. The calculated circulation, 2π, is the correct result.

Key concepts

Two-dimensional Curl

In the world of vector calculus, the concept of curl helps us understand how a vector field rotates about a point. For a two-dimensional field, calculating curl is simpler and focuses on how much the field twists around a certain area. Given a vector field \(\mathbf{F}(x,y) = \langle -y, x \rangle\), we identify the x-component as \(F_1 = -y\) and the y-component as \(F_2 = x\).

The two-dimensional curl is calculated using partial derivatives:

• First, differentiate \(F_2\) with respect to \(x\), which gives \(\frac{\partial F_2}{\partial x}\).
• Then, differentiate \(F_1\) with respect to \(y\), which gives \(\frac{\partial F_1}{\partial y}\).

These derivatives determine how the components of the field change, resulting in:\[\text{Curl}(\mathbf{F}) = \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\]For our particular field, both derivatives are \(\frac{y^2 - x^2}{(x^2 + y^2)^2}\), leading to a curl of zero:\[\text{Curl}(\mathbf{F}) = 0 \]

This indicates no net rotation in any part of the field on a two-dimensional plane under normal circumstances.

Line Integral

The line integral is a fundamental concept used to find the circulation of a vector field across a given path, which is crucial in real-life scenarios, such as fluid flow or electromagnetic fields. To compute a line integral over a path, we parameterize the path and calculate the contribution of the vector field along it.

For instance, consider the unit circle, parameterized by \(\mathbf{r}(t) = \langle \cos{t}, \sin{t} \rangle\), where \(t\) ranges from 0 to \(2\pi\). The derivative \(\frac{d\mathbf{r}}{dt}\) represents the direction of movement along the circle, \(\langle -\sin{t}, \cos{t} \rangle\). The vector field evaluated along the path is \(\mathbf{F}(\mathbf{r}(t)) = \langle -\sin{t}, \cos{t} \rangle\).
Combining these, the line integral for the circulation is computed as: \[C = \int_0^{2\pi} \left(\sin^2{t} + \cos^2{t}\right) \, dt\]The integral \(\int_0^{2\pi} dt\) sums up to \(2\pi\), indicating that the total circulation around the circle is \(2\pi\). The result provides the total twist or rotation along the path, an essential insight that contradicts initial expectations based on the curl only.

Vector Field Singularity

Singularity in a vector field is akin to a hole or undefined point where the field behaves unpredictably. In the exercise, the singularity exists at the origin \((0,0)\), where the given vector field \(\mathbf{F}(x, y) = \frac{\langle -y, x \rangle}{x^2 + y^2}\) becomes undefined. Green's Theorem offers a powerful method for connecting the curl over an area to the circulation along its boundary, under the assumption that the field is continuous within.

Due to the singularity, the field lacks continuity at the origin, and thus Green's Theorem does not apply in its standard form. Despite the zero curl computed, the presence of a singularity permits a non-zero circulation, calculated as \(2\pi\). When a field contains such singularities, careful analysis is required since it affects the otherwise expected outcomes.
This scenario underpins the need for singularity awareness in physics and engineering, where overlooking them could lead to incorrect conclusions about force or flow in a system.