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Chapter 14: Problem 3

Explain the meaning of the Divergence Theorem.

Short Answer

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#tag_title# Short Answer #tag_content# The net outward flux of the vector field \(\vec{F}(x, y, z) = x \hat{i} + y^2 \hat{j} - z^3 \hat{k}\) through the closed surface \(S\) (a sphere with radius \(R\) and center at the origin) is \(\frac{8}{3} R^3 \pi^2\). This is obtained using the Divergence Theorem, which relates the net outward flux of a vector field through a closed surface to the sum of the divergences within the enclosed volume.

Step by step solution

01

Understanding the Divergence Theorem

The Divergence Theorem, also known as Gauss' Theorem, is a theorem in vector calculus that states that the net outward flux of a vector field through a closed surface is equal to the sum of the divergences at each point within the enclosed volume. Mathematically, it is expressed as: $$\oint_{\partial V} \vec{F} \cdot \hat{n} \,dS = \iiint_{V} (\nabla \cdot \vec{F}) \, dV$$ Where \(\vec{F}\) is a vector field, \(\partial V\) represents the boundary of the volume \(V\), \(\hat{n}\) is the unit normal vector to the surface, and \(\nabla \cdot \vec{F}\) denotes the divergence of the vector field. For a better understanding, let's consider an example problem and use the Divergence Theorem to solve it.
02

Example problem

Let \(\vec{F}(x, y, z) = x \hat{i} + y^2 \hat{j} - z^3 \hat{k}\) be a vector field, and let the closed surface \(S\) be the sphere with radius \(R\) and center at the origin. Find the net outward flux of the vector field \(\vec{F}\) through the surface \(S\).
03

Find the divergence of the vector field

The first step is to calculate the divergence of the given vector field \(\vec{F}\). By definition, the divergence is given by: $$\nabla \cdot \vec{F} = \frac{\partial}{\partial x}(x) + \frac{\partial}{\partial y}(y^2) - \frac{\partial}{\partial z}(z^3)$$ Now, compute the partial derivatives with respect to \(x\), \(y\), and \(z\): $$\nabla \cdot \vec{F} = 1 + 2y - 3z^2$$
04

Convert the problem into spherical coordinates

Since we are dealing with a sphere, it is more convenient to work with spherical coordinates. The following coordinate transformation can be applied: $$x = r\sin{\theta}\cos{\phi}, \quad y = r\sin{\theta}\sin{\phi}, \quad z = r\cos{\theta}$$ Now, express the divergence in terms of spherical coordinates: $$\nabla \cdot \vec{F} = 1 + 2r\sin{\theta}\sin{\phi} - 3(r\cos{\theta})^2$$
05

Apply the Divergence Theorem

According to the Divergence Theorem, we can relate the net outward flux of \(\vec{F}\) to the sum of the divergences within the sphere. For spherical coordinates, the volume element is given by \(dV=r^2\sin{\theta} dr\,d\theta\,d\phi\). Then, we can rewrite the Divergence Theorem as: $$\text{Net outward flux} = \iiint_{V} (\nabla \cdot \vec{F}) \, dV = \int_{0}^{R} \int_{0}^{\pi} \int_{0}^{2\pi} r^2(1 + 2r\sin{\theta}\sin{\phi} - 3r^2\cos^2{\theta})\sin{\theta} \, d\phi \, d\theta \, dr$$
06

Evaluate the integral

Now, this is a triple integral that we need to compute. It can be separated into three one-dimensional integrals. Take the outer integral first with respect to \(r\): $$\int_{0}^{R} r^2(1 - 3r^2)dr = R^3 - \frac{3}{5}R^5$$ Next, take the middle integral with respect to \(\theta\): $$\int_{0}^{\pi} (R^3 \sin{\theta} - \frac{3}{5} R^5 \cos^2{\theta}\sin{\theta})d\theta = \frac{4}{3}R^3\pi$$ Finally, take the inner integral with respect to \(\phi\): $$\int_{0}^{2\pi} (1 + R^2\sin{\theta}\sin{\phi})d\phi = 2\pi$$ Now, multiply all the results together to obtain the net outward flux: $$\text{Net outward flux} = \frac{4}{3}R^3\pi \times 2\pi = \frac{8}{3} R^3 \pi^2$$

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Most popular questions from this chapter

One of Maxwell's equations for electromagnetic waves is \(\nabla \times \mathbf{B}=C \frac{\partial \mathbf{E}}{\partial t},\) where \(\mathbf{E}\) is the electric field, \(\mathbf{B}\) is the magnetic field, and \(C\) is a constant. a. Show that the fields $$\mathbf{E}(z, t)=A \sin (k z-\omega t) \mathbf{i} \quad \mathbf{B}(z, t)=A \sin (k z-\omega t) \mathbf{j}$$ satisfy the equation for constants \(A, k,\) and \(\omega,\) provided \(\omega=k / C\). b. Make a rough sketch showing the directions of \(\mathbf{E}\) and \(\mathbf{B}\).

Recall the Product Rule of Theorem \(14.11: \nabla \cdot(u \mathbf{F})=\nabla u \cdot \mathbf{F}+u(\nabla \cdot \mathbf{F})\) a. Integrate both sides of this identity over a solid region \(D\) with a closed boundary \(S\) and use the Divergence Theorem to prove an integration by parts rule: $$\iiint_{D} u(\nabla \cdot \mathbf{F}) d V=\iint_{S} u \mathbf{F} \cdot \mathbf{n} d S-\iiint_{D} \nabla u \cdot \mathbf{F} d V$$ b. Explain the correspondence between this rule and the integration by parts rule for single-variable functions. c. Use integration by parts to evaluate \(\iiint_{D}\left(x^{2} y+y^{2} z+z^{2} x\right) d V\) where \(D\) is the cube in the first octant cut by the planes \(x=1\) \(y=1,\) and \(z=1\)

The Navier-Stokes equation is the fundamental equation of fluid dynamics that models the flow in everything from bathtubs to oceans. In one of its many forms (incompressible, viscous flow), the equation is $$\rho\left(\frac{\partial \mathbf{V}}{\partial t}+(\mathbf{V} \cdot \nabla) \mathbf{V}\right)=-\nabla p+\mu(\nabla \cdot \nabla) \mathbf{V}.$$ In this notation, \(\mathbf{V}=\langle u, v, w\rangle\) is the three-dimensional velocity field, \(p\) is the (scalar) pressure, \(\rho\) is the constant density of the fluid, and \(\mu\) is the constant viscosity. Write out the three component equations of this vector equation. (See Exercise 40 for an interpretation of the operations.)

Consider the vector field \(\mathbf{F}=\langle y, x\rangle\) shown in the figure. a. Compute the outward flux across the quarter circle \(C: \mathbf{r}(t)=\langle 2 \cos t, 2 \sin t\rangle,\) for \(0 \leq t \leq \pi / 2\) b. Compute the outward flux across the quarter circle \(C: \mathbf{r}(t)=\langle 2 \cos t, 2 \sin t\rangle,\) for \(\pi / 2 \leq t \leq \pi\) c. Explain why the flux across the quarter circle in the third quadrant equals the flux computed in part (a). d. Explain why the flux across the quarter circle in the fourth quadrant equals the flux computed in part (b). e. What is the outward flux across the full circle?

a. Show that a torus with radii \(R>r\) (see figure) may be described parametrically by \(r(u, v)=\langle(R+r \cos u) \cos v,(R+r \cos u) \sin v, r \sin u\rangle\) for \(0 \leq u \leq 2 \pi, 0 \leq v \leq 2 \pi\) b. Show that the surface area of the torus is \(4 \pi^{2} R r\).
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