Problem 400 - Calculus Volume 3

Question

For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions \(D\). \(\quad\) ITI \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|}=\frac{\langle x, y, z\rangle}{\sqrt{x^{2}+y^{2}+z^{2}}} ; \quad D\) is the region between spheres of radius 1 and 2 centered at the origin.

Step-by-Step Solution

Answer

The net outward flux is 0.

Step 1: Understanding the Divergence Theorem

The divergence theorem relates the flux across a closed surface to the divergence of a vector field inside the volume bounded by the surface. Specifically, it states that: \[ \iint_{\partial D} \mathbf{F} \cdot d\mathbf{S} = \iiint_{D} abla \cdot \mathbf{F} \, dV \] where \( abla \cdot \mathbf{F} \) is the divergence of the vector field \( \mathbf{F} \), and \( D \) is the volume enclosed by the surface \( \partial D \). In this exercise, \( D \) is the region between two spheres of radii 1 and 2 centered at the origin.

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

We start by computing the divergence \( abla \cdot \mathbf{F} \) for the given vector field \( \mathbf{F} = \frac{\langle x, y, z \rangle}{\sqrt{x^2 + y^2 + z^2}} \). The divergence in spherical coordinates is given by: \[ abla \cdot \mathbf{F} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 F_r) + \text{terms for } \theta \text{ and } \phi \] Given that \( \mathbf{F} = \hat{\mathbf{r}} \), the divergence simplifies to \( abla \cdot \mathbf{F} = \frac{2}{r} - \frac{2}{r} = 0 \) upon differentiation.

Step 3: Apply the Divergence Theorem

Using the divergence theorem, the net outward flux across the boundary \( \partial D \) is: \[ \iint_{\partial D} \mathbf{F} \cdot d\mathbf{S} = \iiint_{D} (abla \cdot \mathbf{F}) \, dV \] Since we found that \( abla \cdot \mathbf{F} = 0 \), the integral over the volume \( D \) will also be zero: \[ \iint_{\partial D} \mathbf{F} \cdot d\mathbf{S} = \iiint_{D} 0 \, dV = 0 \] regardless of the volume \( D \), as long as the vector field is well-defined within the region.

Step 4: Conclude the Flux Calculation

The application of the divergence theorem shows that the net outward flux of the vector field \( \mathbf{F} \) through the boundary of region \( D \) (the region between two spheres of radii 1 and 2) is zero. This result is based on the fact that the divergence is zero throughout the region \( D \).

Key Concepts

Vector Fields

A vector field is a mathematical construction where each point in space is associated with a vector. Imagine a field of arrows, where each arrow has both direction and magnitude.
These arrows can represent various physical quantities like velocity, magnetic force, or in our exercise, the directional derivative of a position vector.

• The given vector field is \( \mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|} = \frac{\langle x, y, z \rangle}{\sqrt{x^2+y^2+z^2}} \).
• This specific vector field represents unit vectors pointing radially outwards from the origin.

Understanding how vector fields interact with surfaces can allow us to calculate flux, which is essentially how much of the vector field passes through a given surface.
A crucial part of handling these interactions is using the divergence theorem, which relates the flow (flux) through a closed surface to the behavior of the vector field inside it.

Spherical Coordinates

Spherical coordinates are a system of curvilinear coordinates that is natural for describing positions and movements in three-dimensional space, especially in cases involving symmetry around a point, like spheres.
This system uses three parameters:

• \( r \): the radial distance from the origin.
• \( \theta \): the polar angle measured from the positive z-axis (0 to \( \pi \)).
• \( \phi \): the azimuthal angle in the xy-plane from the positive x-axis (0 to \( 2\pi \)).

For the exercise, the vector field \( \mathbf{F} = \hat{\mathbf{r}} \) in spherical coordinates was easier to deal with because the symmetry simplifies the divergence computation.
Recognizing this symmetry not only makes calculations more straightforward but also aligns with the principle of exploiting natural coordinate systems to resolve physics problems.

Net Outward Flux

The net outward flux through a surface in a vector field measures how much of the field is moving out through the surface's boundary. This is crucial in fields like fluid dynamics to understand fluid behavior around objects.
According to the divergence theorem, the net outward flux through the surface of a volume \( D \) can be simplified with the expression:

• \( \iint_{\partial D} \mathbf{F} \cdot d\mathbf{S} = \iiint_{D} (abla \cdot \mathbf{F}) \, dV \)

Since in our case study, the divergence \( abla \cdot \mathbf{F} = 0 \), the integral over the volume \( D \) equals zero, showing no net flux.
Understanding and calculating net outward flux with the divergence theorem can reveal properties of the vector field over specific regions, especially when fields have simplifications (like zero divergence in particular zones).