Problem 22 - Calculus Early Transcendentals

Question

Let \(\mathbf{F}(x, y, z)=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}\) be a constant vector field and let \(\sigma\) be the surface of a solid \(G .\) Use the Divergence Theorem to show that the flux of \(\mathbf{F}\) across \(\sigma\) is zero. Give an informal physical explanation of this result.

Step-by-Step Solution

Answer

The flux of \(\mathbf{F}\) across \(\sigma\) is zero because its divergence is zero, indicating no net movement through \(\sigma\).

Step 1: Understand the Divergence Theorem

The Divergence Theorem states that for a vector field \(\mathbf{F}\) and a region \(G\) with boundary surface \(\sigma\): \[ \iint_{\sigma} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{G} abla \cdot \mathbf{F} \, dV \] where \(\mathbf{n}\) is the unit normal to the surface.

Step 2: Calculate the Divergence of \(\mathbf{F}\)

For the constant vector field \(\mathbf{F}(x, y, z) = a \mathbf{i} + b \mathbf{j} + c \mathbf{k}\), the divergence is defined as \(abla \cdot \mathbf{F} = \frac{\partial a}{\partial x} + \frac{\partial b}{\partial y} + \frac{\partial c}{\partial z}\). Since \(\mathbf{F}\) is constant, the partial derivatives are zero, i.e., \(abla \cdot \mathbf{F} = 0\).

Step 3: Integrate the Divergence Over the Volume

Since \(abla \cdot \mathbf{F} = 0\), the volume integral becomes \(\iiint_{G} abla \cdot \mathbf{F} \, dV = \iiint_{G} 0 \, dV = 0\).

Step 4: Apply the Divergence Theorem Result

By the Divergence Theorem: \( \iint_{\sigma} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{G} abla \cdot \mathbf{F} \, dV \). Substituting the volume integral, we have: \( \iint_{\sigma} \mathbf{F} \cdot \mathbf{n} \, dS = 0 \).

Step 5: Provide a Physical Explanation

Physically, the flux through the surface represents the 'flow' of the vector field out of the volume \(G\). Since the vector field \(\mathbf{F}\) is constant and its divergence is zero, it implies there is no 'source' or 'sink' within \(G\), meaning no net flow out of the surface.

Key Concepts

Vector Field

A vector field is a function that assigns a vector to every point in space. In our exercise, the vector field is represented by \[\mathbf{F}(x, y, z) = a \mathbf{i} + b \mathbf{j} + c \mathbf{k}\] where \(a\), \(b\), and \(c\) are constants. This particular vector field is uniform, meaning that its direction and magnitude are the same at every point in the space.

• Each vector in the field indicates a direction and a magnitude.
• The constant vector field suggests that vectors are not changing over space, making things simpler to analyze.

Understanding vector fields is crucial when dealing with physical phenomena like fluid flow or electromagnetic fields, where it represents various attributes such as velocity or force acting at every point in space.

Flux

Flux refers to the quantity of a vector field passing through a given surface. Imagine water flowing through a net; the amount of water penetrating the net is similar to the concept of flux in a vector context. In the case of our vector field \(\mathbf{F}\), the flux across a surface \(\sigma\) can be calculated using a surface integral.

• The flux is calculated as \( \iint_{\sigma}\mathbf{F} \cdot \mathbf{n} \, dS \), where \(\mathbf{n}\) is the unit normal to the surface.
• Positive flux means that more field lines are exiting the surface than entering it.
• The physical significance of flux gives insight into how much of a field passes through a particular area, helping understand field interactions in spaces like magnetic fields or heat flow.

In this exercise, we find that the flux is zero, which means that the amount of the vector field entering and leaving the surface is equal, indicating no net flow.

Surface Integral

A surface integral is a tool used to calculate the flux of a vector field across a surface. It extends the concept of integrals to two-dimensional surfaces and is particularly handy in physics and engineering for integrating over complex, curved surfaces. To compute the surface integral of a vector field \(\mathbf{F}\) over a surface \(\sigma\), we use \[\iint_{\sigma} \mathbf{F} \cdot \mathbf{n} \, dS\],where \(\mathbf{n}\) is the unit normal vector to the surface. Here's how it works:

• The integrand \(\mathbf{F} \cdot \mathbf{n}\) considers the component of the vector field perpendicular to the surface.
• The surface \(\sigma\) over which we're integrating must be smooth and well-defined.
• The integration essentially "sums up" the field's contribution over the entire surface.

In this particular problem, computing the surface integral tells us about the net flow across the surface, which, due to the Divergence Theorem, is zero because the divergence inside the volume is zero.

Divergence

Divergence is a scalar measure of a vector field's tendency to originate from or converge into a point. You can think of divergence as measuring the 'expansiveness' or 'congestiveness' of a vector field at any point. In mathematical terms, for the vector field \(\mathbf{F}(x, y, z) = a \mathbf{i} + b \mathbf{j} + c \mathbf{k}\), the divergence \(abla \cdot \mathbf{F}\) is given by \[abla \cdot \mathbf{F} = \frac{\partial a}{\partial x} + \frac{\partial b}{\partial y} + \frac{\partial c}{\partial z}\].

• In the exercise provided, since each \(a\), \(b\), and \(c\) are constants, the partial derivatives are zero, leading to a zero divergence.
• Divergence provides us insights into whether the field has any "sources" (positive divergence) or "sinks" (negative divergence).
• Zero divergence indicates that there are no sources or sinks in the vector field, implying a uniform field throughout.

Understanding divergence is key to applying the Divergence Theorem, which links the flux across a closed surface to the divergence within the enclosed volume.