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Chapter 14: Problem 2

Interpret the volume integral in the Divergence Theorem.

Short Answer

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Answer: The volume integral in the Divergence Theorem represents the triple integral of the divergence of a vector field (F) over a given region (V), summing the net outflow/inflow of the field's property over the entire volume enclosed by the surface. According to the Divergence Theorem, this volume integral is equal to the surface integral (outward flux), reflecting that the flux through the closed surface is solely determined by the net outflow or influx of the vector field's property inside the volume. This relationship provides a powerful tool to connect surface and volume information of a vector field and analyze its behavior in different contexts.

Step by step solution

01

Volume Integral Definition and Physical Interpretation

The volume integral in the Divergence Theorem represents the triple integral of the divergence of a vector field (F) over a given region (V). Mathematically, the volume integral is written as: 鈭埆鈭�(div F ) dV where div F is the divergence of the vector field F, and dV is the differential volume element. In physical terms, the divergence of a vector field at a point represents the net outflow (positive divergence) or influx (negative divergence) of the field's quantity/property per unit volume around that point. The volume integral of the divergence, then, essentially sums the net outflow/inflow of the vector field's property over the entire volume enclosed by the surface, considering the local behavior of the field.
02

Relate the Volume Integral to the Surface Integral

Recall that the Divergence Theorem states that the surface integral (outward flux) is equal to the volume integral (divergence over the region): 鈭�(F 路 dA) = 鈭埆鈭�(div F ) dV This equality demonstrates that the flux through the closed surface (the left-hand side) is solely determined by the net outflow or influx of the vector field's property inside the volume enclosed by the surface, which is calculated using the volume integral (the right-hand side). In conclusion, the volume integral in the Divergence Theorem sums the net outflow/inflow of a vector field's property over an enclosed volume, providing information about the local behavior of the field inside the volume. This summation, when equated to the surface integral according to the Divergence Theorem, gives a powerful tool to relate the surface and volume information of a vector field and draw conclusions about its behavior in different contexts.

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Most popular questions from this chapter

The potential function for the force field due to a charge \(q\) at the origin is \(\varphi=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{|\mathbf{r}|},\) where \(\mathbf{r}=\langle x, y, z\rangle\) is the position vector of a point in the field and \(\varepsilon_{0}\) is the permittivity of free space. a. Compute the force field \(\mathbf{F}=-\nabla \varphi\). b. Show that the field is irrotational; that is \(\nabla \times \mathbf{F}=\mathbf{0}\).

Suppose \(y=f(x)\) is a continuous and positive function on \([a, b] .\) Let \(S\) be the surface generated when the graph of \(f\) on \([a, b]\) is revolved about the \(x\) -axis. a. Show that \(S\) is described parametrically by \(\mathbf{r}(u, v)=\langle u, f(u) \cos v, f(u) \sin v\rangle,\) for \(a \leq u \leq b, 0 \leq v \leq 2 \pi\) b. Find an integral that gives the surface area of \(S\) c. Apply the result of part (b) to find the area of the surface generated with \(f(x)=x^{3},\) for \(1 \leq x \leq 2\) d. Apply the result of part (b) to find the area of the surface generated with \(f(x)=\left(25-x^{2}\right)^{1 / 2},\) for \(3 \leq x \leq 4\).

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 \(z=1\)

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 \(0 \leq \varphi \leq \pi / 2\) is a fixed angle. What is the area of \(S ?\) Find a vector normal to \(S\).

Find a vector field \(\mathbf{F}\) with the given curl. In each case, is the vector field you found unique? $$\operatorname{curl} \mathbf{F}=\langle 0,1,0\rangle$$
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