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Chapter 17: Problem 36

Let S be the disk enclosed by the curve \(C: \mathbf{r}(t)=\langle\cos \varphi \cos t, \sin t, \sin \varphi \cos t\rangle,\) for \(0 \leq t \leq 2 \pi,\) where \(\overrightarrow{0} \leq \varphi \leq \pi / 2\) is a fixed angle. Use Stokes' Theorem and a surface integral to find the circulation on \(C\) of the vector field \(\mathbf{F}=\langle-y, x, 0\rangle\) as a function of \(\varphi .\) For what value of \(\varphi\) is the circulation a maximum?

Short Answer

Expert verified
Answer: The circulation on C of the vector field is 0, meaning that the circulation does not have a maximum value for any value of \(\varphi\).

Step by step solution

01

Find the curl of the vector field \(\mathbf{F}\)

To find the curl of \(\mathbf{F}=\langle-y, x, 0\rangle\), we use the formula for the curl in Cartesian coordinates: $$\text{curl}(\mathbf{F}) = \left\langle \frac{\partial}{\partial x} (0) - \frac{\partial}{\partial z} (x), \frac{\partial}{\partial z} (-y) - \frac{\partial}{\partial x} (0), \frac{\partial}{\partial x} (x) - \frac{\partial}{\partial y} (-y) \right\rangle = \langle 0, 0, 1+1 \rangle = \langle 0, 0, 2 \rangle.$$
02

Find the differential vector \(d\mathbf{S}\)

To find the differential vector \(d\mathbf{S}\), we need to find the partial derivatives of \(\mathbf{r}(t)\) with respect to \(t\) and \(\varphi\). The given curve is \(\mathbf{r}(t)=\langle\cos \varphi \cos t, \sin t, \sin \varphi \cos t\rangle,\) where \(\varphi\) is a fixed angle and \(0 \leq t \leq 2\pi\). Now it's time to compute the cross product of \(\frac{\partial \mathbf{r}}{\partial t}\) and \(\frac{\partial \mathbf{r}}{\partial \varphi}\): $$\frac{\partial \mathbf r}{\partial t} = \langle -\cos \varphi \sin t, \cos t, -\sin \varphi \sin t \rangle$$ $$\frac{\partial \mathbf r}{\partial \varphi} = \langle -\sin \varphi \cos t, 0, \cos \varphi \cos t \rangle$$ $$d\mathbf{S} = \frac{\partial \mathbf r}{\partial t} \times \frac{\partial \mathbf r}{\partial \varphi} = \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k \\ -\cos \varphi \sin t & \cos t & -\sin \varphi \sin t \\ -\sin \varphi \cos t & 0 & \cos \varphi \cos t \end{vmatrix} = \langle \cos^2 \varphi \cos t, \sin \varphi \cos^2t , \cos \varphi \sin t \rangle$$
03

Compute the surface integral

Now we can compute the surface integral using the curl of \(\mathbf{F}\) and the differential vector \(d\mathbf{S}\) we have found: $$\iint_S \text{curl} (\mathbf{F}) \cdot d\mathbf{S} = \iint_S \langle 0,0,2 \rangle \cdot \langle \cos^2 \varphi \cos t, \sin \varphi \cos^2t , \cos \varphi \sin t \rangle dtd\varphi$$ $$= \iint_S (2 \cos \varphi \sin t) dtd\varphi$$ We know that \(\varphi\) varies between 0 and \(\frac{\pi}{2}\), and \(t\) varies between 0 and \(2\pi\). Therefore, the surface integral becomes: $$\int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi} (2 \cos \varphi \sin t) dtd\varphi = 2 \cos \varphi \int_{0}^{\frac{\pi}{2}} \int_{0}^{2\pi} \sin t dtd\varphi$$ We can integrate with respect to \(t\) first: $$2 \cos \varphi \int_{0}^{\frac{\pi}{2}} (-\cos t \Big|_{0}^{2\pi})d\varphi = 2 \cos \varphi \int_{0}^{\frac{\pi}{2}} (0)d\varphi = 0$$ Therefore, the circulation of the vector field on C is 0, meaning that the circulation does not have a maximum value for any value of \(\varphi\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface Integral
A surface integral allows us to sum up a scalar function or a vector field over a surface in space. It's like adding up tiny pieces of a surface's influence on a vector field. Much like how we calculate area, but in a more advanced way by considering how the field interacts with the surface.

To compute a surface integral involving vector fields, we first find a vector element on the surface, denoted as \(d\mathbf{S}\). This element gives the orientation and infinitesimal area of the surface where the field interacts with it. In our example, the disk \(S\) is described parametrically by \(\mathbf{r}(t)\), and the element is derived by taking cross products of its partial derivatives.

The surface integral is computed using the curl of the vector field and the surface element. We perform this operation by evaluating \(\iint_S \text{curl} \mathbf{F} \cdot d\mathbf{S}\). Essentially, we're summarizing how a vector field circulates around the surface.
Vector Field
Vector fields are mathematical constructions that return vectors at every point in a given space. They help us visualize how forces like winds or electric fields might behave across regions.

In our example, we're dealing with the vector field \(\mathbf{F} = \langle-y, x, 0 \rangle\). This means at any point \((x, y, z)\), the vector aligns with the negative \(y\) and positive \(x\) directions, while having no component in the \(z\) direction.

Understanding vector fields is crucial because they help describe physical phenomena in mathematics and physics. Visualizing vectors as arrows sprouting from each point helps make sense of the complex interactions like those encountered in fluid dynamics or electromagnetism.
Curl of a Vector Field
The curl of a vector field is a measurement of the rotation of a field around a point. Think of it as the "twist" a field might exert locally.

In the Cartesian coordinate plane, the curl of a vector field \(\mathbf{F} = \langle F_1, F_2, F_3 \rangle\) is given by:\[ \text{curl} (\mathbf{F}) = \left\langle \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right\rangle \]For the vector field \(\mathbf{F} = \langle-y, x, 0\rangle\), calculating the curl yields \(\langle 0, 0, 2\rangle\). This vector suggests a constant twist around the \(z\)-axis throughout the field.

Recognizing the significance of the curl helps in applications across electromagnetism, fluid flow, and even the use of Stokes' Theorem in converting surface integrals into line integrals.

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Most popular questions from this chapter

Find a vector field \(\mathbf{F}\) with the given curl. In each case, is the vector field you found unique? $$\operatorname{curl} \mathbf{F}=\langle 0, z,-y\rangle$$

Radial fields in \(\mathbb{R}^{3}\) are conservative Prove that the radial field \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}},\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(p\) is a real number, is conservative on any region not containing the origin. For what values of \(p\) is \(\mathbf{F}\) conservative on a region that contains the origin?

Inverse square fields are special Let \(F\) be a radial ficld \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p},\) where \(p\) is a real number and \(\mathbf{r}=\langle x, y, z\rangle .\) With \(p=3, \mathbf{F}\) is an inverse square field. a. Show that the net flux across a sphere centered at the origin is independent of the radius of the sphere only for \(p=3\) b. Explain the observation in part (a) by finding the flux of \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p}\) across the boundaries of a spherical box \(\left\\{(\rho, \varphi, \theta): a \leq \rho \leq b, \varphi_{1} \leq \varphi \leq \varphi_{2}, \theta_{1} \leq \theta \leq \theta_{2}\right\\}\) for various values of \(p\)

Prove the following identities. Assume \(\varphi\) is a differentiable scalar- valued function and \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields, all defined on a region of \(\mathbb{R}^{3}\). $$\nabla \times(\nabla \times \mathbf{F})=\nabla(\nabla \cdot \mathbf{F})-(\nabla \cdot \nabla) \mathbf{F}$$

Let \(\mathbf{F}=\langle z, 0,-y\rangle\) a. Find the scalar component of curl \(\mathbf{F}\) in the direction of the unit vector \(\mathbf{n}=\langle 1,0,0\rangle\). b. Find the scalar component of curl \(\mathbf{F}\) in the direction of the unit vector \(\mathbf{n}=\left\langle\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right\rangle\). c. Find the unit vector \(\mathbf{n}\) that maximizes \(\operatorname{scal}_{\mathbf{n}}\langle-1,1,0\rangle\) and state the value of \(\operatorname{scal}_{\mathbf{n}}\langle-1,1,0\rangle\) in this direction.
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