Problem 374 - Calculus Volume 3

Question

For the following exercises, use Stokes' theorem to find the circulation of the following vector fields around any smooth, simple closed curve C. $$ \mathbf{F}=\nabla\left(x \sin y e^{z}\right) $$

Step-by-Step Solution

Answer

The circulation of \( \mathbf{F} \) around the curve \( C \) is zero because the curl of \( \mathbf{F} \) is zero.

Step 1: Understand Stokes' Theorem

Stokes' Theorem relates the surface integral of the curl of a vector field \( \mathbf{F} \) over a surface \( S \) to the line integral of \( \mathbf{F} \) around the boundary curve \( C \) of \( S \). It is given by: \[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (abla \times \mathbf{F}) \cdot d\mathbf{S} \] Here, \( C \) must be a positively oriented, simple closed curve. The question asks for the circulation using Stokes' Theorem.

Step 2: Compute the Gradient of the Scalar Function

We have the vector field \( \mathbf{F} = abla(x \sin y e^{z}) \). Let's compute the gradient: - Compute partial derivatives: - \( \frac{\partial}{\partial x}(x \sin y e^{z}) = \sin y e^{z} \) - \( \frac{\partial}{\partial y}(x \sin y e^{z}) = x \cos y e^{z} \) - \( \frac{\partial}{\partial z}(x \sin y e^{z}) = x \sin y e^{z} \)Therefore, \( \mathbf{F} = (\sin y e^{z}, x \cos y e^{z}, x \sin y e^{z}) \).

Step 3: Calculate the Curl of the Vector Field

The curl of \( \mathbf{F} \), denoted \( abla \times \mathbf{F} \), is calculated as follows: \[ abla \times \mathbf{F} = \left( \frac{\partial (x \sin y e^{z})}{\partial y} - \frac{\partial (x \cos y e^{z})}{\partial z}, \frac{\partial (\sin y e^{z})}{\partial z} - \frac{\partial (x \sin y e^{z})}{\partial x}, \frac{\partial (x \cos y e^{z})}{\partial x} - \frac{\partial (\sin y e^{z})}{\partial y} \right) \]Calculating each component results in zero for all components, confirming that \( abla \times \mathbf{F} = \mathbf{0} \).

Step 4: Apply Stokes' Theorem

Since \( abla \times \mathbf{F} = \mathbf{0} \), by applying Stokes' Theorem:\[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (abla \times \mathbf{F}) \cdot d\mathbf{S} = \iint_S \mathbf{0} \cdot d\mathbf{S} = 0 \]Therefore, the circulation of \( \mathbf{F} \) around the curve \( C \) is zero.

Key Concepts

Vector Fields

Vector fields are mathematical constructs used to associate a vector with every point in space. A vector field in three-dimensional space is often denoted as \( \mathbf{F}(x, y, z) \) and can describe various physical quantities such as force, velocity, or magnetic fields. In this context, our vector field, \( \mathbf{F} \), was derived from a scalar function using the gradient operator, \( abla \). The gradient essentially gives a vector that points in the direction of the greatest rate of increase of the scalar function.To understand this in a simpler way, think of a weather map showing wind patterns over a region. Each vector on the map indicates wind direction and strength at a particular location. Similarly, in our exercise, \( \mathbf{F} = (\sin y e^{z}, x \cos y e^{z}, x \sin y e^{z}) \) shows how each component of the vector changes with respect to \( x, y, \) and \( z \). Recognizing these variables and computing the gradient helps us to visualize how the field behaves over a region in space.

Circulation

Circulation is a measure of how much a vector field \( \mathbf{F} \) "circulates" around a closed curve, \( C \). In simpler terms, it quantifies the extent to which the vector field wraps around or follows the curve. It's similar to determining, for example, how much a river's water flows along its path.Mathematically, the circulation around curve \( C \) is determined by the line integral:

• \( \oint_C \mathbf{F} \cdot d\mathbf{r} \)

In our solution, we used Stokes' Theorem, which connects this line integral to a surface integral over a surface \( S \) bounded by \( C \). By calculating the curl of the vector field and finding it to be zero, we immediately concluded that the circulation of \( \mathbf{F} \) around \( C \) is zero. Thus, the vector field does not circulate around the curve, mainly because it is locally conservative.

Surface Integral

A surface integral extends the concept of integration to complex, curved surfaces. It allows us to compute the total quantity of some vector field passing through a given surface \( S \). In the context of Stokes' Theorem, surface integrals become vital as they help relate a three-dimensional field's behavior to two-dimensional curves.The specific integral used in Stokes' Theorem is:

• \( \iint_S (abla \times \mathbf{F}) \cdot d\mathbf{S} \)

This integral represents the total "twisting" effect of the vector field over the oriented surface \( S \). Here, \( d\mathbf{S} \) is a vector that represents an infinitesimal area of \( S \) with a direction given by the outward normal (right-hand rule). In our exercise, because the curl \( abla \times \mathbf{F} \) is zero everywhere on the surface, the integral becomes zero. Consequently, this implies that the twisting effect across the entire surface also equals zero, illustrating the balance and conservation within the field, thus validating Stokes' Theorem fully.