Problem 14 - 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 $$\begin{aligned} \mathbf{F} &=(y-z) \mathbf{i}+(z-x) \mathbf{j}+(x+z) \mathbf{k} \\ S : & \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+\left(9-r^{2}\right) \mathbf{k}, \\ 0 & \leq r \leq 3, \quad 0 \leq \theta \leq 2 \pi \end{aligned}$$

Step-by-Step Solution

Answer

The flux of the curl \( \nabla \times \mathbf{F} \) across the surface \( S \) is zero.

Step 1: Understanding Stokes' Theorem

Stokes' Theorem states that for a surface \( S \) with boundary curve \( C \), the flux of the curl of a vector field \( \mathbf{F} \) across \( S \) is equal to the line integral of \( \mathbf{F} \) around \( C \). Mathematically, \( \int_S (abla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r} \). The theorem can simplify computing the flux of \( abla \times \mathbf{F} \) across surfaces.

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

First, compute the curl of \( \mathbf{F} = (y-z) \mathbf{i} + (z-x) \mathbf{j} + (x+z) \mathbf{k} \). The curl \( abla \times \mathbf{F} \) is given by the determinant:\[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ y-z & z-x & x+z \end{vmatrix} \]This evaluates to \( \mathbf{i}(0 - (-1)) - \mathbf{j}(0 - 1) + \mathbf{k}(1 - 1) = \mathbf{i} - \mathbf{j} \). Thus, \( abla \times \mathbf{F} = \mathbf{i} - \mathbf{j} \).

Step 3: Parametrize Surface \( S \) and Compute Normal Vector

The surface \( S \) is parametrized by \( \mathbf{r}(r, \theta) = (r \cos \theta) \mathbf{i} + (r \sin \theta) \mathbf{j} + (9 - r^2) \mathbf{k} \). Compute the partial derivatives \( \frac{\partial \mathbf{r}}{\partial r} = (\cos \theta) \mathbf{i} + (\sin \theta) \mathbf{j} - 2r \mathbf{k} \) and \( \frac{\partial \mathbf{r}}{\partial \theta} = (-r \sin \theta) \mathbf{i} + (r \cos \theta) \mathbf{j} \). The normal vector \( \mathbf{N} \) is obtained from their cross product.

Step 4: Compute Cross Product for Normal Vector \( \mathbf{N} \)

Find \( \mathbf{N} = \frac{\partial \mathbf{r}}{\partial r} \times \frac{\partial \mathbf{r}}{\partial \theta} \). The determinant gives:\[ \mathbf{N} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \cos \theta & \sin \theta & -2r \ -r \sin \theta & r \cos \theta & 0 \end{vmatrix} \]Calculating, we have \[ \mathbf{N} = (2r^2 \cos \theta) \mathbf{i} + (2r^2 \sin \theta) \mathbf{j} + (r) \mathbf{k} \].

Step 5: Evaluate Surface Integral using Stokes' Theorem

The surface integral is \( \iint_S (abla \times \mathbf{F}) \cdot \mathbf{N} \, dS \). Substitute \( abla \times \mathbf{F} = \mathbf{i} - \mathbf{j} \) and \( \mathbf{N} = (2r^2 \cos \theta) \mathbf{i} + (2r^2 \sin \theta) \mathbf{j} + (r) \mathbf{k} \):\[ (\mathbf{i} - \mathbf{j}) \cdot ((2r^2 \cos \theta) \mathbf{i} + (2r^2 \sin \theta) \mathbf{j} + (r) \mathbf{k}) \] \[= 2r^2 \cos \theta - 2r^2 \sin \theta \] Integrate with respect to \( r \) from \( 0 \) to \( 3 \) and \( \theta \) from \( 0 \) to \( 2\pi \). The integral over \( \theta \) no longer depends on \( \theta \), because \( \cos \theta \) and \( \sin \theta \) cancel out symmetrically over the interval; therefore, the remaining integral is zero.

Key Concepts

Surface Integral

A surface integral in the context of vector fields allows us to measure the flow of a vector field across a surface within a three-dimensional space. Think of it as measuring the "stuff" (could be fluid, air, etc.) that passes through the surface. This measurement depends on both the vector field and the geometry of the surface itself.
The surface integral involves calculating how the vector field impacts or interacts with the surface at every point. This might sound complicated, but in simple terms, it's tallying up the small parts of the surface impacted by the field to get the integral in whole. Surface integrals are key in physics and engineering, and they help describe phenomena like flux through a surface.
In Stokes' Theorem context, calculating a surface integral helps us determine the flux of the curl of a vector field, which essentially provides the total rotational effect crossing through the surface.

Curl of a Vector Field

The curl is a vector operator that describes the rotational motion or "twisting" nature of a vector field. When you hear about the curl, imagine small whirlpools or eddies in a vector field that exhibits rotation.
In mathematical terms, if you have a vector field \( \mathbf{F} \), the curl is defined through the cross-product and can be calculated using determinants. It's designed to tell us how much and in what direction the field is curling or swirling at any given point.
In applications like fluid mechanics, the curl is a measure of the circulation density at a point, which makes it essential for understanding fluid flows. In our example, once you find \( abla \times \mathbf{F} = \mathbf{i} - \mathbf{j} \), you're essentially saying the field has a curling effect in those directions.

Parametrization of a Surface

Parametrization is a way to define a surface using parameters, usually through two variables, such as \( r \) and \( \theta \) for polar coordinates, or any other suitable pairs. This helps in expressing complex surfaces in a structured manner, making calculations like finding surface integrals manageable.
Take a surface like a paraboloid as in this context; it can be parametrized in polar coordinates to simplify interactions. It turns \( heta \) and \( r \) into building blocks that map every part of the surface in 3D space.
Through parametrization, we can work with linear combinations of the coordinate basis vectors, thus bridging the geometric surface with analytic calculations like integration. It simplifies the process of calculating an integral over a surface, translating geometric concepts into algebraic methods.

Normal Vector

A normal vector to a surface is an important concept in calculus and geometry since it points perpendicularly away from the surface. Imagine it as an arrow sticking straight out of the surface, indicating its direction. This helps to define orientation and is crucial for various calculations, such as determining the flux through a surface.
To find the normal vector mathematically, especially when dealing with parametrized surfaces, you take the cross product of the partial derivatives of the parametrization. The result gives a vector that points in the direction perpendicular to the surface at that point.
In the context of a surface integral, the orientation provided by the normal vector is essential, as it defines "outward" or "inward" flux, which impacts calculations like those carried out with Stokes' Theorem. The normal vector ensures that your integration accounts for the geometry of the surface accurately.