Problem 18 - Calculus Early Transcendentals

Question

(a) Let \(\sigma\) denote the surface of a solid \(G\) with \(\mathbf{n}\) the outward unit normal vector field to \(\sigma\). Assume that \(\mathbf{F}\) is a vector field with continuous first-order partial derivatives on \(\sigma .\) Prove that $$ \iint_{\sigma}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d S=0 $$ [Hint: Let \(C\) denote a simple closed curve on \(\sigma\) that separates the surface into two subsurfaces \(\sigma_{1}\) and \(\sigma_{2}\) that share \(C\) as their common boundary. Apply Stokes 'Theorem to \(\sigma_{1}\) and to \(\sigma_{2}\) and add the results. (b) The vector field curl(F) is called the curl field of F. In words, interpret the formula in part (a) as a statement about the flux of the curl field.

Step-by-Step Solution

Answer

The integral \( \iint_{\sigma} (\text{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = 0 \) confirms no net flux of the curl field through a closed surface.

Step 1: Understand the Problem and Given Information

We need to show that the surface integral of the curl of a vector field \( \mathbf{F} \) over a closed surface \( \sigma \) is zero. We are given a closed surface \( G \) with \( \sigma \) as its boundary and \( \mathbf{n} \) as the outward unit normal vector. We consider \( \mathbf{F} \) to have continuous first-order partial derivatives.

Step 2: Use Stokes' Theorem

Invoke Stokes' Theorem for both subsurfaces \( \sigma_1 \) and \( \sigma_2 \) which share a common boundary \( C \). Stokes' Theorem states that: \[ \iint_{\sigma_1} (\text{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = \oint_{C} \mathbf{F} \cdot d\mathbf{r} \] \[ \iint_{\sigma_2} (\text{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = \oint_{C} \mathbf{F} \cdot d\mathbf{r} \] where both integrals over \( C \) can be oriented consistently.

Step 3: Add the Stokes' Theorem Results

Since \( C \) is oriented such that on \( \sigma_1 \) it is positive in the direction of \( \mathbf{n} \), and on \( \sigma_2 \) as negative, their line integrals cancel each other out because they have opposite directions: \[ \oint_{C} \mathbf{F} \cdot d\mathbf{r} - \oint_{C} \mathbf{F} \cdot d\mathbf{r} = 0 \] Thus, adding both surface integrals gives: \[ \iint_{\sigma_1} (\text{curl} \mathbf{F}) \cdot \mathbf{n} \, dS + \iint_{\sigma_2} (\text{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = 0 \] which means: \[ \iint_{\sigma} (\text{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = 0 \].

Step 4: Interpret the Result in Context of Flux

The equation \( \iint_{\sigma} (\text{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = 0 \) suggests there's no net 'twisting' effect extending through a closed surface in static scenarios, therefore the net flux of the curl field across a closed surface is zero. This implies the circulating flow of \( \mathbf{F} \) is entirely internal.

Key Concepts

Curl of a Vector Field

In the realm of vector calculus, the curl is a vital concept that helps us understand the local rotation, or spinning, of a vector field. If we think of a vector field as a field of arrows that can vary in magnitude and direction at each point in space, then the curl tells us how much those arrows twist around a point.
It's a way of encapsulating the rotation it experiences. For a vector field \( \mathbf{F} = P\hat{i} + Q\hat{j} + R\hat{k} \), the curl is defined mathematically as:

• \( \text{curl} \, \mathbf{F} = abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \hat{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \hat{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \hat{k} \)

This formula uses partial derivatives to calculate how the vector field rotates locally. By assessing the curl, we can identify points where the field exhibits a circular motion, much like how a whirlpool behaves. If the curl is zero everywhere, the field is said to be irrotational, indicating no local spinning at any point.

Surface Integral

A surface integral is a type of integral that allows us to compute a summation over a surface area. Surface integrals allow us to extend the concept of integration from lines and areas into higher dimensions. They are crucial for analyzing fields across surfaces, such as electromagnetic fields.
When performing a surface integral, we usually deal with functions defined on surfaces in three-dimensional space. The surface integral of a vector field \( \mathbf{F} \) across a surface \( \sigma \) is represented as:

• \( \iint_{\sigma} \mathbf{F} \cdot \mathbf{n} \, dS \)

In this equation, \( \mathbf{n} \) stands for the unit normal vector to the surface \( \sigma \), and \( dS \) is a small element of the surface area. This process effectively allows us to sum up how much the vector field \( \mathbf{F} \) is passing through the surface.
Working with surface integrals is essential in fields such as fluid dynamics and electromagnetism, where it often describes how much of the field "flows" across a surface.

Flux of the Curl Field

The term 'flux' refers to the amount of a field passing through a surface, like water flowing through a net. When we talk about the flux of the curl field of a vector field \( \mathbf{F} \), we're interested in the net amount of twisting or rotation that goes through a given surface. In our specific context, we're considering how the rotation contained in the curl field associates with a closed surface \( \sigma \).
The intriguing part of the exercise comes when Stokes' Theorem shows that the surface integral of the curl over a closed surface is always zero:

• \( \iint_{\sigma} (\text{curl} \, \mathbf{F}) \cdot \mathbf{n} \, dS = 0 \)

This result is significant because it tells us that whatever rotation happens inside the closed surface, it does not escape or enter through the boundary. It implies there is no external influence of twist or spin across a closed surface, reflecting the conservation of this rotating attribute inside.
In practical terms, if you're looking at a fluid flow or electromagnetic field, this indicates that within a closed boundary, all rotational activity is self-contained, maintaining equilibrium without any net external driving of rotation.