Problem 16 - 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 $$\begin{array}{l}{\mathbf{F}=(x-y) \mathbf{i}+(y-z) \mathbf{j}+(z-x) \mathbf{k}} \\ {S : \quad \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+(5-r) \mathbf{k}}, \\ {0 \leq r \leq 5, \quad 0 \leq \theta \leq 2 \pi}\end{array}$$

Step-by-Step Solution

Answer

The flux of the curl of \( \mathbf{F} \) 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 around the boundary of the surface. The theorem is expressed as \( \int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} = \int_{C} \mathbf{F} \cdot d\mathbf{r} \), where \( S \) is a surface bounded by a closed curve \( C \).

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

Given \( \mathbf{F} = (x-y) \mathbf{i} + (y-z) \mathbf{j} + (z-x) \mathbf{k} \), calculate \( abla \times \mathbf{F} \) using the determinant method:\[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ (x-y) & (y-z) & (z-x) \end{vmatrix} \]This simplifies to \( abla \times \mathbf{F} = (-1, -1, -1) \).

Step 3: Describe the Surface \(S\)

The surface \( S \) is given by the parametric equations \( \mathbf{r}(r, \theta) = (r \cos \theta) \mathbf{i} + (r \sin \theta) \mathbf{j} + (5-r) \mathbf{k} \). It is a portion of a cone extending from \( r = 0 \) to \( r = 5 \) and for \( 0 \leq \theta \leq 2\pi \).

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

The surface element can be found using the cross product of the partial derivatives of \( \mathbf{r}(r, \theta) \):\[ \frac{\partial \mathbf{r}}{\partial r} = (\cos \theta) \mathbf{i} + (\sin \theta) \mathbf{j} - \mathbf{k}, \]\[ \frac{\partial \mathbf{r}}{\partial \theta} = (-r\sin \theta) \mathbf{i} + (r\cos \theta) \mathbf{j}. \]The cross product is \( \frac{\partial \mathbf{r}}{\partial r} \times \frac{\partial \mathbf{r}}{\partial \theta} = (-r\cos \theta, -r\sin \theta, -r) \).

Step 5: Set Up the Surface Integral

Substitute \( abla \times \mathbf{F} = (-1, -1, -1) \) and \( d\mathbf{S} = (-r\cos \theta, -r\sin \theta, -r) dr d\theta \) into the integral:\[ \int_{0}^{2\pi} \int_{0}^{5} (-1, -1, -1) \cdot (-r\cos \theta, -r\sin \theta, -r) \, dr \, d\theta. \]

Step 6: Evaluate the Integral

Calculate the dot product \((-1, -1, -1) \cdot (-r\cos \theta, -r\sin \theta, -r) = r\cos \theta + r\sin \theta + r = r(\cos \theta + \sin \theta + 1)\). The integral becomes:\[ \int_{0}^{2\pi} \int_{0}^{5} r(\cos \theta + \sin \theta + 1) \, dr \, d\theta. \]When evaluating, \( \int_{0}^{5} r \, dr = \frac{25}{2} \). Integrating with respect to \( \theta \), only the term \( r \) contributes, integrating to zero over the symmetric interval \( 0 \leq \theta \leq 2\pi \) due to \( \cos \theta \) and \( \sin \theta \). The remaining term evaluates to zero. Thus, the flux across the surface is 0.

Key Concepts

Surface Integral

In the context of Stokes' Theorem, the surface integral plays a crucial role in relating a line integral to the flux of the curl of a vector field across a surface. A surface integral of a vector field over a surface \( S \) is written as \( \int_{S} \mathbf{F} \cdot d\mathbf{S} \), where \( \mathbf{F} \) is the vector field, and \( d\mathbf{S} \) is the vector area element of the surface.
The key is that \( d\mathbf{S} \) points outward perpendicular from the surface and its magnitude equals the area of the infinitesimally small portion of the surface.

• The integral sums the component of \( \mathbf{F} \) normal to the surface over every point on \( S \).
• In this exercise, we compute \( \int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} \) to find the "flux" of the curl through \( S \).
• Stokes' Theorem connects this integral over a surface to a line integral around its boundary.

Understanding the accurate computation of surface integrals helps in visualizing how the vector field "flows" across the surface.

Curl of a Vector Field

The curl of a vector field is a measure of the rotation or "twisting" of the field at a point. It applies specifically to three-dimensional space and is denoted as \( abla \times \mathbf{F} \).
It can be intuitively thought of as the maximum amount of "rotation" per unit area.

• The computation of the curl uses the determinant method involving the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), and the partial derivatives of the vector field components with respect to their corresponding variables.
• In the given problem, \( \mathbf{F} = (x-y) \mathbf{i} + (y-z) \mathbf{j} + (z-x) \mathbf{k} \), has a curl of \( (-1, -1, -1) \), indicating a constant rotational effect across the vector field.

The resulting vector helps determine how the field is behaving in a rotation sense around any chosen point in the space.

Parametric Equations

Parametric equations allow for detailed descriptions of surfaces or curves using parameters, such as \( r \) and \( \theta \) here. They are essential in defining complex geometries and shapes in a more manageable form.
For surfaces like the one given in the original problem, parametric equations describe "how to walk along" the surface by giving the x, y, and z coordinates explicitly in terms of parameters.

• For the parametric equations \( \mathbf{r}(r, \theta) = (r \cos \theta) \mathbf{i} + (r \sin \theta) \mathbf{j} + (5-r) \mathbf{k} \), they beautifully outline a conical surface.
• The parameters \( r \) and \( \theta \) range between specified limits which represent unfolding the surface starting from a point \( r = 0 \) at the apex to \( r = 5 \) at the base, and \( \theta = 0 \) to \( \theta = 2\pi \) which completes the circle.

These equations are very useful when calculating integrals over surfaces, as they allow us to express surface area elements in terms of the parameters, which simplifies the integration process.