Problem 406 - Calculus Volume 3

Question

Find the net outward flux of field \(\mathbf{F}=\langle b z-c y, c x-a z, a y-b x\rangle\) across any smooth closed surface in \(\mathbf{R}^{3},\) where \(a, b,\) and \(c\) are constants.

Step-by-Step Solution

Answer

The net outward flux is zero.

Step 1: Define Divegence of Vector Field

The net outward flux of a vector field through a closed surface can be found using the Divergence Theorem. The theorem states that the net outward flux is equal to the integral of the divergence of the vector field over the volume enclosed by the surface. First, we'll find the divergence of the given vector field \( \mathbf{F} = \langle b z - c y, c x - a z, a y - b x \rangle \).

Step 2: Apply the Divergence Operator

The divergence of a vector field \( \mathbf{F} = \langle P , Q , R \rangle \) is given by \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \). Calculate each partial derivative: \( \frac{\partial}{\partial x}(b z - c y) = 0 \), \( \frac{\partial}{\partial y}(c x - a z) = 0 \), and \( \frac{\partial}{\partial z}(a y - b x) = 0 \) . Thus, the divergence is \( abla \cdot \mathbf{F} = 0 + 0 + 0 = 0 \).

Step 3: Evaluate the Volume Integral

According to the Divergence Theorem, the net outward flux is \( \iiint_V (abla \cdot \mathbf{F}) \,dV \). Since we have found \( abla \cdot \mathbf{F} = 0 \), the integral becomes \( \iiint_V 0 \,dV \), which evaluates to 0 regardless of the volume \( V \).

Step 4: Conclude the Result

Because the divergence of \( \mathbf{F} \) is zero, the net outward flux through any closed surface is also zero. This makes use of the Divergence Theorem, stating that the flux through a closed surface only depends on the divergence within the enclosed volume, which is zero in this case.

Key Concepts

Vector Field

A vector field is a function that assigns a vector to every point in a space. Imagine it as a set of arrows, where each arrow has a direction and magnitude depending on its position. In the context of the given problem, our vector field is \( \mathbf{F} = \langle b z - c y, c x - a z, a y - b x \rangle \).

• The components of the vector field depend linearly on the coordinates \( x, y, \) and \( z \).
• Each term involves constants \( a, b, \) and \( c \), which determine the weight or influence of each coordinate on the vector field.

Understanding the behavior of these vectors in space can help in visualizing how they flow through surfaces, affecting calculations such as flux.

Flux Calculation

Flux is a measure of how much of something (often envisioned as fluid or gas) is passing through a surface. In mathematics, when dealing with vector fields, the flux through a surface gives insight into how vectors are penetrating or exiting through the surface. To compute the net outward flux of a vector field through a closed surface, we use the Divergence Theorem.
This powerful tool integrates the divergence of the field over the volume enclosed by the surface to find the flux across the entire surface.

• Step 1 in our problem was to find the divergence of the vector field.
• This involves taking the sum of partial derivatives of each component of \( \mathbf{F} \).
• A divergence of zero, as found, means no net flow across any closed surface.

The divergence specifically informs us how a field behaves internally with regard to its expansion or contraction in space.

Closed Surface

Understanding a closed surface is key to applying the Divergence Theorem.
A closed surface is like a wrapper around a volume in space, such as a sphere or a cube.
It completely encloses a region without any gaps. Consider why closed surfaces are crucial:

• They allow us to use the Divergence Theorem, which relates the flow of a vector field through a surface to the behavior within the volume.
• When the flux through a closed surface is calculated, it helps in determining whether there's any "source" or "sink" activity within the volume enclosed.
• A divergence of zero means no net production or absorption of the vector field content inside.

In the exercise, no matter what closed surface you choose, since the divergence of the vector field \( \mathbf{F} \) is zero, the net outward flux remains zero.