Problem 22 - Thomas Calculus

Question

The base of the closed cubelike surface shown here is the unit square in the \(x y\) -plane. The four sides lie in the planes \(x=0\) , \(x=1, y=0,\) and \(y=1 .\) The top is an arbitrary smooth surface whose identity is unknown. Let \(\mathbf{F}=x \mathbf{i}-2 y \mathbf{j}+(z+3) \mathbf{k}\) and suppose the outward flux of \(\mathbf{F}\) through Side \(A\) is 1 and through Side \(B\) is \(-3 .\) Can you conclude anything about the outward flux through the top? Give reasons for your answer.

Step-by-Step Solution

Answer

The outward flux through the top is 2, determined using the Divergence Theorem.

Step 1: Understand The Given Information

We are given a closed cubelike surface with a base that is a unit square in the \( xy \)-plane, and vertical sides that lie in the planes \(x=0\), \(x=1\), \(y=0\), and \(y=1\). The top is an unknown smooth surface. The vector field \(\mathbf{F} = x \mathbf{i} - 2y \mathbf{j} + (z+3) \mathbf{k}\) is defined over this closed surface. We know the outward flux through Side \(A\) is 1 and through Side \(B\) is -3.

Step 2: Apply the Divergence Theorem

The Divergence Theorem tells us that the total flux through a closed surface \(\Sigma\) is equal to the integral of the divergence of the vector field \(\mathbf{F}\) over the volume \(V\) enclosed by \(\Sigma\). Mathematically, \( \iint_{\Sigma} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{V} (abla \cdot \mathbf{F}) \, dV \). Here, \(\mathbf{n}\) is the outward unit normal vector on \(\Sigma\).

Step 3: Compute the Divergence of \(\mathbf{F}\)

Calculate the divergence of \(\mathbf{F}\):\[abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x) + \frac{\partial}{\partial y}(-2y) + \frac{\partial}{\partial z}(z+3)\]\[= 1 - 2 + 1 = 0\]The divergence of \(\mathbf{F}\) in this case is zero.

Step 4: Conclude Using the Divergence Theorem

Since the divergence of \(\mathbf{F}\) is zero, the integral of \(abla \cdot \mathbf{F}\) over the volume \(V\) is zero:\[\iiint_{V} (abla \cdot \mathbf{F}) \, dV = 0\]By the Divergence Theorem, this implies that the total flux through the closed surface \(\Sigma\) must also be zero.

Step 5: Calculate Outward Flux Through the Top

The total flux through the closed surface \(\Sigma\) is the sum of the fluxes through the base, four sides, and the top. If the total flux is zero, then:\[\text{Flux}_{\text{Base}} + \text{Flux}_{x=0} + \text{Flux}_{x=1} + \text{Flux}_{y=0} + \text{Flux}_{y=1} + \text{Flux}_{\text{Top}} = 0\]Given: \(\text{Flux}_{x=0} = 1\) and \(\text{Flux}_{x=1} = -3\). Since the vector field \(\mathbf{F}\) has no \(z\)-component at \(z=0\) (on the base), \(\text{Flux}_{\text{Base}}\) is zero. Remaining sides contribute zero due to symmetry in \(y\)-components. Thus, \(1 - 3 + \text{Flux}_{\text{Top}} = 0\).

Step 6: Final Calculation of Outward Flux Through the Top

By solving \(1 - 3 + \text{Flux}_{\text{Top}} = 0\), we obtain:\[\text{Flux}_{\text{Top}} = 2\].The outward flux through the top of the surface is 2.

Key Concepts

Outward Flux

Outward flux refers to the measure of how much of a vector field is flowing out of a closed surface. When we talk about outward flux, it's essential to understand the direction in which we are measuring the flow. For a given vector field, the term 'outward' indicates the flux calculated considering the outward normal vector on the surface. This can be visualized by imagining a balloon surface where the air inside represents the vector field. The outward flux would signify how much air is moving out of the balloon's surface.
The Divergence Theorem connects the concept of outward flux over a closed surface to the behavior of the entire volume enclosed by that surface. Evaluating outward flux gives insights into how a vector field interacts with boundaries, playing a key role in various physical and engineering applications.

Vector Field

A vector field is a mathematical construct where a vector is assigned to every point in space. Typically denoted by symbols such as \( \mathbf{F} \) or \( \mathbf{v} \), they provide a way to describe a quantity that has both magnitude and direction, like the flow of a fluid or electromagnetic fields. In our exercise, the vector field is given as \( \mathbf{F}=x \mathbf{i}-2y \mathbf{j}+(z+3) \mathbf{k} \), where each component represents the vector direction in 3D space.
Understanding a vector field involves recognizing how these vectors change as we move through space. This understanding includes the concepts of divergence and curl, which measure how a vector field 'spreads out' or 'rotates' around a point, respectively.

Closed Surface

A closed surface is a boundary that fully encloses a volume with no openings. Consider it as analogous to the exterior of a balloon or a basketball, which encapsulates an entire space within. In mathematical terms, closed surfaces are crucial in applying the Divergence Theorem, which relates properties within the volume to those across its boundary.
In our exercise, the cubelike structure forms a closed surface composed of a unit square base in the xy-plane, four sides defined by various x and y planes, and a smooth unknown top surface. This setup allows for the total outward flux to be evaluated across all these surfaces as part of the Divergence Theorem application, offering insights into the net flow of the vector field through the entire structure.

Integration over Volumes

Integration over volumes involves calculating the integral of a function through a three-dimensional space. This operation is central in applying the Divergence Theorem, where we integrate the divergence of a vector field over the volume enclosed by a closed surface. The resulting value helps relate the total outward flux across the enclosing surface.
In the specific exercise, integration over the volume involves computing the divergence of the vector field \( \mathbf{F} \). When this divergence is zero, as in our example, the total outward flux is zero, indicating balance or steady-state conditions across the unit cube structure. By solving the problem guided by the principles of integration over volumes, we accurately determine the outward flux through unknown parts of the closed surface, like the smooth top in this case.