Problem 391 - Calculus Volume 3

Question

Use the divergence theorem to compute flux integral \(\iint_{S} \mathbf{F} \cdot d \mathbf{S},\) where \(\mathbf{F}(x, y, z)=y \mathbf{j}-z \mathbf{k}\) and \(S\) consists of the union of paraboloid \(y=x^{2}+z^{2}, 0 \leq y \leq 1,\) and disk \(x^{2}+z^{2} \leq 1, y=1,\) oriented outward. What is the flux through just the paraboloid?

Step-by-Step Solution

Answer

The flux through the paraboloid is \(-\pi\).

Step 1: Understand the Divergence Theorem

The Divergence Theorem states that for a vector field \( \mathbf{F} \) and a closed surface \( S \), \( \iint_{S} \mathbf{F} \cdot d \mathbf{S} = \iiint_{V} abla \cdot \mathbf{F} \, dV \), where \( V \) is the volume enclosed by \( S \). To use the theorem, calculate the divergence of \( \mathbf{F} \).

Step 2: Compute Divergence of \( \mathbf{F} \)

The vector field is \( \mathbf{F}(x, y, z) = y \mathbf{j} - z \mathbf{k} \). The divergence is calculated as \( abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(0) + \frac{\partial}{\partial y}(y) + \frac{\partial}{\partial z}(-z) = 0 + 1 - 1 = 0 \).

Step 3: Analyze the Implications of Divergence

Since \( abla \cdot \mathbf{F} = 0 \), by the Divergence Theorem, the surface integral over a closed surface \( S \) is zero: \( \iint_{S} \mathbf{F} \cdot d \mathbf{S} = \iiint_{V} 0 \, dV = 0 \). This means the total flux through the closed surface is zero.

Step 4: Separate the Total Surface into Parts

The surface \( S \) consists of the paraboloid and the disk. To find the flux through just the paraboloid, we must consider the orientation and sum of flux through the closed surface \( S \).

Step 5: Compute Flux Through the Disk at \( y=1 \)

For the disk, \( y = 1 \), the outward unit normal vector is \( \mathbf{j} \). Here, the vector field \( \mathbf{F}(x, y, z) = \mathbf{j} - z \mathbf{k} \) gives flux \( \iint_{D} (\mathbf{j} \cdot \mathbf{j}) \, dA = \iint_{D} 1 \, dA \), where \( D \) is the projection of the disk on \( yz \)-plane, with area \( \pi \). Thus, the flux through the disk is \( \pi \).

Step 6: Apply Conservation of Flux

Since the total flux through the closed surface is zero and the flux through the disk is \( \pi \), the flux through the paraboloid surface is \( -\pi \) to ensure the conservation of flux as dictated by the Divergence Theorem.

Key Concepts

Flux Integral

A flux integral measures how much of a vector field passes through a given surface. Imagine a fluid flowing throughout a space, and you're interested in knowing how much fluid penetrates a particular barrier. This scenario represents a flux integral.
Flux integrals play a central role in physics, especially electromagnetism and fluid dynamics. When we calculate the flux integral, we evaluate how the vector field interacts with the surface it crosses.
In our case, the vector field is \( \mathbf{F}(x, y, z) = y \mathbf{j} - z \mathbf{k} \), which suggests a flow direction influenced by both \(y\) and \(z\) components of the field. These components determine how the field affects or interacts with the surface.

Paraboloid

A paraboloid is a 3D surface that can resemble a satellite dish or a bowl. It is a shape that's defined mathematically by equations like \( y = x^2 + z^2 \). The surface is symmetrical around its principal axis.
This particular paraboloid spans from \(y = 0\) to \(y = 1\), and can be visualized as a cup-shaped area opening upward along the y-axis. It acts as part of the closed surface in our flux integration problem.
This paraboloid forms a part of the surface \( S \), combining with the disk to form a closed shape when evaluating the vector field's flux.

Vector Field

A vector field assigns a vector to every point in space. In a vector field like \( \mathbf{F}(x, y, z) = y \mathbf{j} - z \mathbf{k} \), each point has a component of direction and magnitude.
Such fields are crucial in physics and engineering because they represent different quantities like velocity fields in fluid motion or electric fields in electromagnetism.
When considering a vector field's interaction with a surface, we're essentially interested in how these vectors penetrate the plane. The components \( y \mathbf{j} \) and \( -z \mathbf{k} \) in our vector field describe movement along the y-axis and z-axis respectively, indicating that flows are perpendicular to these axes at every point in the field.

Surface Integral

A surface integral extends the concept of integrating over curves to integrating over surfaces. It helps calculate the total of some field quantity passing through the surface.
This type of integration adds up contributions from all infinitesimally small areas on that surface. It's particularly helpful for understanding complex 3D surfaces and the flux through them.
The process involves

• Choosing an orientation of the surface (inward/outward),
• Dividing the surface into small patches,
• Computing the contribution of each patch to the integral using the dot product between the vector field and the normal to the surface.

In our problem, the total surface is divided into two parts: the paraboloid and the disk. We use surface integrals to evaluate flux through these parts separately, obeying the Divergence Theorem.