Problem 13 - Thomas Calculus

Question

In Exercises \(13-18,\) use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(\mathbf{F}\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n}\) . $$\begin{array}{l}{\mathbf{F}=2 \mathbf{z} \mathbf{i}+3 \mathbf{x} \mathbf{j}+5 y \mathbf{k}} \\ {S : \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+\left(4-r^{2}\right) \mathbf{k}}, \\ {0 \leq r \leq 2, \quad 0 \leq \theta \leq 2 \pi}\end{array}$$

Step-by-Step Solution

Answer

The flux of the curl across surface \(S\) is 0.

Step 1: Understand Stokes' Theorem

Stokes' Theorem relates a surface integral over a surface \(S\) to the line integral around its boundary \(C\). Mathematically, it is expressed as \( \int_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS = \oint_{C} \mathbf{F} \cdot d\mathbf{r} \). We need to compute the flux of the curl of \( \mathbf{F} \) across \( S \).

Step 2: Calculate the Curl of the Vector Field \(\mathbf{F}\)

Given \( \mathbf{F} = 2z \mathbf{i} + 3x \mathbf{j} + 5y \mathbf{k} \), calculate \( abla \times \mathbf{F} \):\[abla \times \mathbf{F} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\partial / \partial x & \partial / \partial y & \partial / \partial z \2z & 3x & 5y \\end{vmatrix}\]After calculating, we find:\[abla \times \mathbf{F} = (5 - 0) \mathbf{j} - (3 - 0) \mathbf{k} = 5 \mathbf{j} - 3 \mathbf{k}\]

Step 3: Set Up the Surface \(S\)

The surface \(S\) is given by the parameterization \( \mathbf{r}(r, \theta) = (r \cos \theta) \mathbf{i} + (r \sin \theta) \mathbf{j} + (4 - r^2) \mathbf{k} \). It is a paraboloid cap with boundaries: \(0 \leq r \leq 2\) and \(0 \leq \theta \leq 2 \pi\).

Step 4: Compute the Normal Vector \(\mathbf{n}\) and \(dS\)

Calculate the normal vector using the cross product of partial derivatives of \(\mathbf{r}(r, \theta)\): \(\mathbf{n} = \frac{\partial \mathbf{r}}{\partial r} \times \frac{\partial \mathbf{r}}{\partial \theta} \).After computation, find:\[\mathbf{n} = (2r \cos \theta) \mathbf{i} + (2r \sin \theta) \mathbf{j} + r \mathbf{k}\]The differential area element is \(|\mathbf{n}| \, dr \, d\theta = r \sqrt{5} \, dr \, d\theta\).

Step 5: Evaluate the Surface Integral

Now, evaluate the flux across the surface:\[\int_{0}^{2} \int_{0}^{2\pi} ((5 \mathbf{j} - 3 \mathbf{k}) \cdot ((2r \cos \theta) \mathbf{i} + (2r \sin \theta) \mathbf{j} + r \mathbf{k})) \, r \sqrt{5} \, dr \, d\theta\] Calculate the dot product and integrate:\[\int_{0}^{2} \int_{0}^{2\pi} (10r \sin \theta - 3r) \, r \sqrt{5} \, dr \, d\theta\]Solve the integral, using symmetry to simplify, find that the result is 0.

Step 6: Interpret the Result

The result of the surface integral is 0, indicating that the net flux of the curl of the vector field \(\mathbf{F}\) across the surface \(S\) is zero, as seen from the outward unit normal \(\mathbf{n}\).

Key Concepts

Understanding Surface Integrals in Stokes' Theorem

Surface integrals are a fundamental tool in vector calculus, often used to calculate the flow or flux of a vector field across a given surface. In the context of Stokes' Theorem, this surface integral can be translated to a line integral around the boundary of the surface. This powerful theorem provides a bridge between these two types of integrals, showing how the behavior of a vector field over a surface relates to its behavior along the boundary curve.
To set up a surface integral, particularly when using Stokes' Theorem, you must identify the surface, parameterize it correctly, and determine its orientation. For the given problem, the surface is a paraboloid cap defined parametrically through variables like polar coordinates \(r\) and \(\theta\).
The orientation of this surface is determined by the outward unit normal \(\mathbf{n}\), which you calculate by taking the cross product of the partial derivatives of the parameterization. This ensures that the orientation is consistent with Stokes' Theorem, allowing an accurate computation of the flux.

Flux Calculation Across a Surface

Flux calculation measures the quantity of a vector field flowing across a certain surface. In calculus, this is usually determined using the formula \( \int_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS \), where \( abla \times \mathbf{F} \) is the curl of the vector field and \(\mathbf{n}\) is the normal vector to the surface. The focus here is to evaluate how much of the vector field penetrates through the surface.
In our context, the vector field \(\mathbf{F}\) and its curl \( abla \times \mathbf{F} \) have been calculated to simplify the integral. By incorporating the normal vector and using the correct parameterization, we find that the flux through surface \(S\) is ultimately zero. This means there's no net flow of the field through the surface, which often suggests symmetries or particular properties of the vector field or surface.
Flux calculations are essential in fields such as electromagnetism and fluid dynamics, as they offer insight into the movement or flow of fields and fluids across boundaries.

Curl of a Vector Field: What it Tells Us

The curl of a vector field is a vector operation that describes the rotation or the swirling strength of the field at a particular point. Mathematically, for a vector field \(\mathbf{F} = 2z \mathbf{i} + 3x \mathbf{j} + 5y \mathbf{k}\), the curl is found using the determinant method, yielding \( abla \times \mathbf{F} = 5 \mathbf{j} - 3 \mathbf{k} \).
This result tells us that while there is no component in the \(\mathbf{i}\) direction, the field has rotational tendencies within the \(\mathbf{j}\) and \(\mathbf{k}\) directions. Understanding the curl is crucial because it indicates whether or not a vector field is conservative, and it plays a key role in applying Stokes' Theorem.
In practical applications, the curl helps engineers and scientists determine the presence or absence of vortices and eddies in fields such as fluid dynamics or electromagnetism. A zero curl suggests no rotation or a conservative field, while a non-zero curl often indicates rotation or swirling within the space.