Question

Radial fields and zero circulation Consider the radial vector fields \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p},\) where \(p\) is a real number and \(\mathbf{r}=\langle x, y, z\rangle\) Let \(C\) be any circle in the \(x y\) -plane centered at the origin. a. Evaluate a line integral to show that the field has zero circulation on \(C\) b. For what values of \(p\) does Stokes' Theorem apply? For those values of \(p,\) use the surface integral in Stokes' Theorem to show that the field has zero circulation on \(C\).

Short answer

In conclusion, the radial vector field \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^p}\) has zero circulation on any circle centered at the origin in the \(xy\)-plane. This result is shown through two approaches: by calculating the line integral around a circle and by using Stokes' Theorem for values of \(p < 2\).

Step-by-step solution

Line Integral Calculation

First, we need to parameterize the circle \(C\) in the \(xy\)-plane. Let the radius of the circle be \(R\). The parameterization of the circle is \(\mathbf{r}(t) = (R\cos{t}, R\sin{t}, 0)\) with \(0 \leq t \leq 2\pi\). Now, we have to evaluate the circulation along \(C\) and is given by $$\oint_C \mathbf{F} \cdot d\mathbf{r} = \oint_C \frac{\mathbf{r}}{|\mathbf{r}|^p} \cdot d\mathbf{r}$$ where \(d\mathbf{r} = \frac{d\mathbf{r}}{dt} dt\). From the parameterization, the radius vector is \(\mathbf{r} = \langle R \cos t, R \sin t, 0 \rangle\). Therefore, the magnitude of the radius vector is \(R\), and the vector \(\frac{\mathbf{r}}{|\mathbf{r}|^p} = \frac{1}{R^{p-1}}\langle \cos t, \sin t, 0 \rangle\). Now let's find the tangent vector \(\frac{d\mathbf{r}}{dt}\), which is \(\langle -R\sin{t}, R\cos{t}, 0 \rangle\). Now take the dot product between the vector field \(\frac{1}{R^{p-1}}\langle \cos t, \sin t, 0 \rangle\) and the tangent vector \(\langle -R\sin{t}, R\cos{t}, 0 \rangle\), and then integrate over the parameter \(t\) from \(0\) to \(2\pi\): $$\oint_C \mathbf{F} \cdot d\mathbf{r} = \int_0^{2\pi}\frac{1}{R^{p-1}}\langle \cos t, \sin t, 0 \rangle \cdot \langle -R\sin{t}, R\cos{t}, 0 \rangle dt$$ $$= \int_0^{2\pi}\frac{1}{R^{p-1}}(-R\sin{t}\cos{t} + R\sin{t}\cos{t}) dt = 0$$ Since the line integral is zero, the circulation around the circle \(C\) in the \(xy\)-plane is zero.

Applying Stokes' Theorem

Stokes' Theorem states that \(\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}\), where \(S\) is a surface enclosed by the curve \(C\). To apply the theorem, the vector field \(\mathbf{F}\) must be differentiable, which means that all partial derivatives of its components must exist and be continuous. First, let's find the curl of the vector field \(\mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|^p}\): $$\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ \frac{x}{(x^2 + y^2 + z^2)^{p/2}} & \frac{y}{(x^2 + y^2 + z^2)^{p/2}} & \frac{z}{(x^2 + y^2 + z^2)^{p/2}} \end{vmatrix}$$ The curl and its partial derivatives must exist and be continuous. Since dividing by zero or taking the derivative of singularities would result in discontinuities, we need to find the values of \(p\) that ensure the curl's continuity. By analyzing the entries of the matrix above, one can observe that singularities appear when \(p \geq 2\), as the denominators become equal to zero at some locations (e.g., the origin). Therefore, Stokes' Theorem applies for values of \(p < 2\). For these values, the vector field is differentiable, and the line integral is equal to the surface integral representing the curl of the vector field. Since the line integral was found to be zero for any circle in the \(xy\)-plane, it follows that the curl of the vector field is also zero for those values of \(p\). Thus, the radial vector field \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^p}\) has zero circulation on any circle \(C\) in the \(xy\)-plane for \(p < 2\).

Key concepts

Vector Fields

Imagine a space filled with arrows, each arrow representing a force acting at a point. Mathematically, this is known as a vector field. Vector fields assign a vector to every point in space, which can symbolically be represented as \( \mathbf{F}(x,y,z) \). These vectors can represent various physical concepts like the velocity of fluid at a point or the force of gravity.

For instance, if we have a radial vector field expressed as \( \mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|^p} \), this means that the strength and direction of the field vector depend on the position vector \( \mathbf{r} \) and the power \( p \), giving us varied field behaviors at different points in the space.

Line Integral

A line integral is a type of integral where we 'add up' the vectors of a vector field along a path or curve. Visually think of it as tracking along a path and collecting or measuring the vector field's influence along this journey. Mathematically, it's often written as \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) for a curve \( C \) and vector field \( \mathbf{F} \).

In the case of our radial vector field, the line integral along a circular path helps us understand the field's 'effort' around that path, which leads to the concept of circulation – in this instance, revealed to be zero.

Circulation

Circulation refers to the total 'turning effect' created by a vector field along a closed loop. If you picture a river curving through a landscape, the circulation would be the total flow of the river along a closed loop that follows the river's path. In technical terms, it's the line integral over a closed path. A circulation of zero, as shown in the exercise's radial vector field, implies there is no net turning effect – there is a kind of balance with no net flow of the vector field's 'fluid' around the path.

Surface Integral

Moving from lines to surfaces, a surface integral extends the idea of integration over a two-dimensional surface. If the line integral is about walking a path and adding up vectors, the surface integral is akin to covering a patch of ground and accounting for how much the vector field 'pokes' through it. This is often needed in physics to calculate flux, which essentially measures the amount of field penetrating a surface. Surface integrals are key in relating a vector field's behavior on a boundary to its behavior over an entire surface via Stokes' Theorem.

Curl of a Vector Field

The curl of a vector field measures the tendency of the field to rotate or curl around a point – like water swirling around a drain. Mathematically, the curl of \( \mathbf{F} \) is denoted as \( abla \times \mathbf{F} \) and involves taking partial derivatives of the field's components. In fluid dynamics, a nonzero curl might indicate a whirlpool or eddy. In our radial vector field example, evaluating the curl shows whether there are any intrinsic swirls in the field characteristic across a surface.

Parametric Equations

To work with complex shapes in calculus, we often use parametric equations to describe paths. These equations use one or more parameters to define a curve's coordinates. For example, a circle in the plane can be described using \( x = R\cos t \) and \( y = R\sin t \) with \( t \) being the parameter. This technique is neatly applicable in line integrals, where we express the path using parametric form and calculate the integral along the parameterized curve.

Differentiability

In calculus, differentiability is a property that determines if a function is complex enough to have a derivative at each point in its domain. A differentiable function essentially has no sharp corners or discontinuities and has a tangent line at every point. This concept is essential when applying the modern toolbox of calculus, as seen in Stokes' Theorem, which requires the vector field to be differentiable across the domain of interest. In the problem, differentiability is a condition for applying Stokes' Theorem to connect the line and surface integrals, thus determining the circulation for our vector field.