Problem 22 - Thomas Calculus

Question

In Exercises \(19-24,\) use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(F\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n}\) . \begin{equation} \begin{array}{l}{\mathbf{F}=(x-y) \mathbf{i}+(y-z) \mathbf{j}+(z-x) \mathbf{k}} \\ {S : \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+(5-r) \mathbf{k}} \\ {0 \leq r \leq 5, \quad 0 \leq \theta \leq 2 \pi}\end{array} \end{equation}

Step-by-Step Solution

Answer

The flux is zero.

Step 1: Understanding Stokes' Theorem

Stokes' Theorem relates a surface integral of the curl of a vector field over a surface \(S\) to a line integral of the vector field over the boundary curve of the surface. Mathematically, it can be expressed as: \(\int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_{C} \mathbf{F} \cdot d\mathbf{r}\), where \(C\) is the boundary curve of \(S\). Here, we need to find the flux of the curl of \(\mathbf{F}\) across \(S\).

Step 2: Calculate the Curl of F

Find the curl of the vector field \(\mathbf{F} = (x-y) \mathbf{i} + (y-z) \mathbf{j} + (z-x) \mathbf{k}\) using \(abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right)\mathbf{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \mathbf{k}\).\After performing the calculation, we find that \(abla \times \mathbf{F} = -3 \mathbf{i} + 3 \mathbf{j} + 3 \mathbf{k}\).

Step 3: Parameterized Description of the Surface S

The surface \(S\) is parameterized with \(\mathbf{r}(r, \theta) = (r \cos \theta) \mathbf{i} + (r \sin \theta) \mathbf{j} + (5-r) \mathbf{k}\), where \(0 \leq r \leq 5\) and \(0 \leq \theta \leq 2\pi\).

Step 4: Calculate Surface Element \(d\mathbf{S}\)

The surface element \(d\mathbf{S}\) is given by \(d\mathbf{S} = \left( \frac{\partial \mathbf{r}}{\partial r} \times \frac{\partial \mathbf{r}}{\partial \theta} \right) dr d\theta\). First, compute partial derivatives: \(\frac{\partial \mathbf{r}}{\partial r} = (\cos \theta) \mathbf{i} + (\sin \theta) \mathbf{j} - \mathbf{k}\) and \(\frac{\partial \mathbf{r}}{\partial \theta} = (-r \sin \theta) \mathbf{i} + (r \cos \theta) \mathbf{j}\). Calculate their cross product: \(\frac{\partial \mathbf{r}}{\partial r} \times \frac{\partial \mathbf{r}}{\partial \theta} = r \mathbf{i} + r \mathbf{j} + r \cos \theta \mathbf{k}\).

Step 5: Integrate to Find the Flux

The flux of \(abla \times \mathbf{F}\) through \(S\) is \(\int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S}\). Substituting values, we evaluate \(-3i+3j+3k \cdot (r \mathbf{i} + r \mathbf{j} + r \cos \theta \mathbf{k})\), which simplifies to \(-3r + 3r + 3r \cos \theta\). Integrate over \(r \in [0, 5]\) and \(\theta \in [0, 2\pi]\). Result yields 0 after integration because the non-zero part \(3r\cos\theta\) evaluates to zero over a complete period \([0, 2\pi]\).

Step 6: Conclusion

The flux of the curl of \(\mathbf{F}\) across the surface \(S\) is zero, as shown by integrating the simplified expression.

Key Concepts

Surface Integral

A surface integral is a way to calculate a two-dimensional integral over a surface in three-dimensional space. Think of it like summing up or "adding" a function over every tiny piece of that surface. In this exercise, we're focusing on surfaces that lie in three-dimensional spaces, like planes or curved surfaces.

In the context of Stokes' Theorem, the surface integral is used to relate the curl of a vector field over a surface to a line integral around the boundary of that surface. It essentially tells us how much of the vector field is "spinning" through the surface.

• The integral \(\int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S}\) measures this spin, often called the "circulation" density integrated over the surface.

Understanding surface integrals gives us the power to determine the flow (or flux) through surfaces in vector fields effectively.

Curl of a Vector Field

The curl of a vector field is a vector that describes the rotational or swirling behavior of the field at a given point. When we say that a vector field has a curl, we're looking at how the field "twists," just like water swirling around a drain.

In mathematical terms, if you have a vector field \(\mathbf{F} = (F_1, F_2, F_3)\), the curl is given by: \(abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right)\mathbf{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \mathbf{k}\).

• This operator gives us a new vector pointing in the direction of the maximum swirl, with a magnitude proportional to the strength of that swirl.
• In our example, the curl \(abla \times \mathbf{F} = -3\mathbf{i} + 3\mathbf{j} + 3\mathbf{k}\) was calculated for the vector field, showing the specific twisting pattern of the field we are working with.

Parameterized Surface

A parameterized surface is a way to describe a surface using parameters, usually denoted as \(u\) and \(v\), instead of just regular \(x\), \(y\), and \(z\). Essentially, it's like creating a map of a surface.

In this exercise, the surface \(S\) is described using parameters \(r\) and \(\theta\), with the vector \(\mathbf{r}(r, \theta) = (r \cos \theta) \mathbf{i} + (r \sin \theta) \mathbf{j} + (5-r) \mathbf{k}\). This parameterization helps us express and evaluate the surface integral more easily.

• The span for \(r\) is \[0, 5\]\ and \(\theta\) is \[0, 2\pi\]\, defining the complete extent of the surface over which we're integrating.
• Parameterized surfaces play a crucial role in converting a surface integral into a more manageable form using vector calculus.

Flux Calculation

Flux is a measure of how much of a vector field passes through a surface. It can be thought of like wind passing through a window. Are you measuring a light breeze or a strong gust?

The calculation of flux through a surface using Stokes' Theorem involves several steps. You first determine the curl of the vector field and then account for the surface's shape and orientation.

• The integral performed here, \(\int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S}\), captures how much the curl of a field penetrates the surface \(S\).
• In this setup, after calculating the cross product for \(d\mathbf{S}\) and plugging in the values, the remaining integral evaluates to zero, particularly due to symmetry over a full rotation (from \(0\) to \(2\pi\)). This indicates no net "spin" through the surface.

Understanding flux is invaluable in fields like fluid dynamics, electromagnetism, and field theory because it quantifies the total flow through a given surface.