Problem 18 - 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 $$\begin{aligned} \mathbf{F} &=y^{2} \mathbf{i}+z^{2} \mathbf{j}+x \mathbf{k} \\ S : & \mathbf{r}(\phi, \theta)=(2 \sin \phi \cos \theta) \mathbf{i}+(2 \sin \phi \sin \theta) \mathbf{j}+(2 \cos \phi) \mathbf{k}, \\ 0 & \leq \phi \leq \pi / 2, \quad 0 \leq \theta \leq 2 \pi \end{aligned}$$

Step-by-Step Solution

Answer

The flux of the curl of the field \( \mathbf{F} \) across the surface \( S \) is 0.

Step 1: Understand Stokes' Theorem

Stokes' Theorem relates a surface integral of the curl of a vector field to a line integral over the boundary of the surface. The theorem is stated 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 the surface \(S\).

Step 2: Compute the Curl of \( \mathbf{F} \)

The vector field is \( \mathbf{F} = y^2 \mathbf{i} + z^2 \mathbf{j} + x \mathbf{k} \). Compute the curl of \( \mathbf{F} \): \( abla \times \mathbf{F} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \y^2 & z^2 & x\end{vmatrix} \).This evaluates to: \( abla \times \mathbf{F} = (-2z)\mathbf{i} - \mathbf{j} + 0\mathbf{k} = -2z \mathbf{i} - \mathbf{j} \).

Step 3: Parametrize the Surface \( S \)

The given surface \( S \) is parametrized by \( \mathbf{r}(\phi, \theta) = 2 \sin \phi \cos \theta \mathbf{i} + 2 \sin \phi \sin \theta \mathbf{j} + 2 \cos \phi \mathbf{k} \). This representation is akin to a hemisphere of radius 2.

Step 4: Find the Normal Vector \( d\mathbf{S} \)

Calculate the normal vector \( d\mathbf{S} \) using the cross product of the partial derivatives with respect to \( \phi \) and \( \theta \):\( \frac{\partial \mathbf{r}}{\partial \phi} \times \frac{\partial \mathbf{r}}{\partial \theta} \).This gives:\[ \frac{\partial \mathbf{r}}{\partial \phi} = (2 \cos \phi \cos \theta, 2 \cos \phi \sin \theta, -2 \sin \phi) \]\[ \frac{\partial \mathbf{r}}{\partial \theta} = (-2 \sin \phi \sin \theta, 2 \sin \phi \cos \theta, 0) \]Compute:\[ \frac{\partial \mathbf{r}}{\partial \phi} \times \frac{\partial \mathbf{r}}{\partial \theta} = 4 \sin^2 \phi \cos \theta \mathbf{i} + 4 \sin^2 \phi \sin \theta \mathbf{j} + 4 \sin \phi \cos \phi \mathbf{k} \].

Step 5: Surface Integral for Flux

Now calculate the surface integral:\[ \int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} = \int_{0}^{\pi/2} \int_{0}^{2\pi} \begin{bmatrix}-2z \mathbf{i} - \mathbf{j}\end{bmatrix} \cdot \begin{bmatrix}4 \sin^2 \phi \cos \theta \mathbf{i} + 4 \sin^2 \phi \sin \theta \mathbf{j} + 4 \sin \phi \cos \phi \mathbf{k}\end{bmatrix} \, d\theta \, d\phi \].Simplifying, evaluate this to 0 as the integrand terms integrate to zero over the symmetrical domain.

Step 6: Verify with Line Integral

To confirm, note the boundary of \( S \) leads to a circle at \( \phi = \pi/2 \), integrating \( \mathbf{F} \) over this boundary \( C \) results in zero, echoing how symmetry simplifies the integral—specifically as the entire net contribution from the field across the hemisphere results from zero net contributions.

Key Concepts

Surface Integrals

A surface integral extends the concept of a line integral to a surface. Rather than summing infinitesimal contributions along a curve, a surface integral sums values over a two-dimensional surface.
In Stokes' Theorem, we are interested in the flux of a vector field through a surface. The flux measures how much of the field 'flows' through the surface. This can be thought of as how much the vector field passes through every tiny piece of the surface.

• For a surface parametrized by a vector function \(\mathbf{r}(u, v)\), the differential surface element \(d\mathbf{S}\) is found with the cross product of the partial derivatives \(\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}\).
• The total surface integral is calculated by integrating the dot product of the curl of the vector field and this normal vector over the whole parametric domain.

Integrating over symmetrical regions often results in simplifications, as certain terms can cancel out or integrate to zero.

Curl of a Vector Field

The curl of a vector field is a vector that describes the rotation of the field at a point. It answers the question of how much and in what direction the field tends to rotate around a small loop at a given point.
For a vector field \(\mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\), the curl is given by:
\[abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k}\]In the original example, the curl was computed as:
\[abla \times \mathbf{F} = (-2z)\mathbf{i} - \mathbf{j}\]This computation helps determine how the vector field behaves near the surface by quantifying the local spinning motion.

• The direction of this vector explains the axis of rotation and the magnitude gives its strength.

In applications, this concept is crucial for understanding fluid dynamics and electromagnetic theories, where the idea of rotational flow and fields is fundamental.

Parametrization of Surfaces

Parametrization is a way to represent a surface using parameters, often making complex surfaces easier to work with. By using parameters, typically denoted as \(u\) and \(v\), we can describe any point on a surface in terms of simpler variables.
For instance, in the exercise, the surface \(S\) was parametrized using \(\phi\) and \(\theta\) as:
\[\mathbf{r}(\phi, \theta) = (2 \sin \phi \cos \theta) \mathbf{i} + (2 \sin \phi \sin \theta) \mathbf{j} + (2 \cos \phi) \mathbf{k}\]This is akin to mapping a spherical surface or hemisphere by transforming spherical coordinates into Cartesian coordinates:

• \(\phi\) is often the angle from the positive z-axis, similar to the zenith angle.
• \(\theta\) is the azimuthal angle, similar to longitude in spherical polar coordinates.

By manipulating the parameters, you can visualize how the surface deforms or how different points map onto the surface.
This makes tasks like integration and differentiation over surfaces more manageable.