Question

In Exercises \(5-8\), a closed surface \(S\) enclosing a domain \(D\) and a vector field \(\vec{F}\) are given. Verify the Divergence Theorem on \(\mathcal{S} ;\) that is, show \(\iint_{S} \vec{F} \cdot \vec{n} d S=\iiint_{D} \operatorname{div} \vec{F} d V\). \(\mathcal{S}\) is the surface bounding the domain \(D\) enclosed by the cylinder \(x^{2}+y^{2}=1\) and the planes \(z=-3\) and \(z=3\); \(\vec{F}=\langle-x, y, z\rangle\)

Short answer

The Divergence Theorem is verified; both integrals equal \( 6\pi \).

Step-by-step solution

Understand the Divergence Theorem

The Divergence Theorem states that for a vector field \( \vec{F} \) and a closed surface \( S \) enclosing a volume \( D \), the surface integral of \( \vec{F} \) over \( S \) is equal to the volume integral of the divergence of \( \vec{F} \) over \( D \):\[ \iint_{S} \vec{F} \cdot \vec{n} \, dS = \iiint_{D} abla \cdot \vec{F} \, dV. \]This needs to be verified for the given vector field and surface.

Calculate the Divergence of \( \vec{F} \)

The divergence of a vector field \( \vec{F} = \langle P, Q, R \rangle \) is given by \( abla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \). For \( \vec{F} = \langle -x, y, z \rangle \), the divergence is:\[ abla \cdot \vec{F} = \frac{\partial (-x)}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = -1 + 1 + 1 = 1. \]

Set Up the Volume Integral

The domain \( D \) is a cylinder of radius 1, centered on the z-axis, extending from \( z = -3 \) to \( z = 3 \). The volume integral becomes \( \iiint_{D} 1 \, dV \), which is simply the volume of the cylinder:\[ \text{Volume} = \text{Base Area} \times \text{Height} = \pi \times 1^2 \times 6 = 6\pi. \]

Calculate the Flux through the Surface \( S \)

\( S \) consists of the cylindrical surface and two circular caps at \( z = -3 \) and \( z = 3 \). By symmetry and the components of \( \vec{F} \), the flux through the cylindrical part is zero because the outward normal is perpendicular to \( \vec{F} = \langle -x, y, z \rangle \). The contributions from the top and bottom surfaces must now be calculated.

Calculate Flux Through the Top and Bottom Surfaces

For the top cap at \( z = 3 \), \( \vec{F} = \langle -x, y, 3 \rangle \) and the unit normal vector \( \vec{n} = \langle 0, 0, 1 \rangle \). The flux through the top cap is \( \iint_{x^2 + y^2 \leq 1} 3 \, dA = 3 \cdot \pi \). For the bottom cap at \( z = -3 \), \( \vec{F} = \langle -x, y, -3 \rangle \) and \( \vec{n} = \langle 0, 0, -1 \rangle \), yielding a flux of \( -3 \cdot \pi \).

Verify the Divergence Theorem

Combining the fluxes, the total flux through \( S \) is \( 3\pi - 3\pi = 6\pi \). The volume integral also evaluated to \( 6\pi \), which matches the surface integral. Thus, the Divergence Theorem is verified for this vector field and surface.

Key concepts

Vector Field

A vector field is a mathematical concept used to represent a quantity that has both a direction and a magnitude at every point in a region of space. In this exercise, we have a vector field \( \vec{F} = \langle -x, y, z \rangle \), meaning for any point \( (x, y, z) \) in space, the vector at that point is \(-x\) in the x-direction, \(y\) in the y-direction, and \(z\) in the z-direction.
This representation is very useful in various fields such as physics and engineering.

• For example, a vector field can represent the velocity field of a flowing fluid or the force field acting on a charged particle.
• Each point in the field has an associated vector that can describe different phenomena.

An important property of vector fields is their divergence, which can give insight into whether the field is "expanding" or "contracting" around a point, playing a key role in the Divergence Theorem.

Surface Integral

Surface integrals extend the concept of a line integral to a two-dimensional manifold or surface. They are crucial to calculate the "flow" of a vector field over a surface. In practical terms, a surface integral over vector field \( \vec{F} \) is written as \( \iint_{S} \vec{F} \cdot \vec{n} \, dS \), where \( \vec{n} \) is the unit normal vector to the surface \( S \).
This integral sums up the component of the vector field that "pierces" through a unit area on the surface.

• It is used to calculate quantities like flux, which is the amount of field passing through a surface.
• This is important in electromagnetic and fluid dynamics applications.

In this problem, multiple components of the surface, including the curved surface and the flat caps, each contribute differently to the integral, which together help verify the Divergence Theorem.

Volume Integral

A volume integral computes a quantity throughout a three-dimensional region and is represented as \( \iiint_{D} \operatorname{div} \vec{F} \, dV \), where \( D \) is the volume over which the integral is taken.
In this exercise, this integral represented the total diverging effect of the vector field over the entire volume of the cylinder.

• To perform a volume integral, identify the region's boundaries in space, calculate its divergence, and accumulate these over the region.
• For the enclosed cylinder with radius 1 and height 6, this integral evaluated to \(6\pi\).

The importance of volume integrals lies in their ability to capture complex interactions over entire volumes, making them indispensable in physics and engineering to understand the complete field behavior inside a region.

Flux

Flux is a measure of how much of a vector field passes through a given surface and is calculated using a surface integral.
In physics, it can represent various concepts such as the flow of a fluid through a surface or the field lines of a magnetic or electric field passing through a surface.

• In this problem, the flux through the surface \( S \) consists of calculating the flow through both the cylindrical side and the circular caps at \( z = 3 \) and \( z = -3 \).
• The final calculation showed that the net flux is consistent with the volume of the domain, as given by the Divergence Theorem, leading to a combined flux of \(6\pi\).

Understanding flux is vital as it bridges the concepts of vector fields and surface integrals, demonstrating how a field behaves in relation to the boundaries it interacts with.