Problem 13 - Calculus Early Transcendentals

Question

Use the Divergence Theorem to find the flux of F across the surface ? with outward orientation. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=x^{3} \mathbf{i}+y^{3} \mathbf{j}+z^{3} \mathbf{k} ; \sigma \text { is the surface of the cylin- }} \\ {\text { drical solid bounded by } x^{2}+y^{2}=4, z=0, \text { and } z=3}\end{array} $$

Step-by-Step Solution

Answer

The flux of \( \mathbf{F} \) across the surface \( \sigma \) is \( 240\pi \).

Step 1: Understand the problem

You are given a vector field \( \mathbf{F}(x, y, z) = x^3 \mathbf{i} + y^3 \mathbf{j} + z^3 \mathbf{k} \) and a cylindrical solid defined by \( x^2 + y^2 = 4 \), bounded by \( z = 0 \) and \( z = 3 \). You need to find the outward flux of \( \mathbf{F} \) across this surface using the Divergence Theorem.

Step 2: State the Divergence Theorem

The Divergence Theorem states that \( \iint_{\sigma} \mathbf{F} \cdot d\mathbf{S} = \iiint_{V} abla \cdot \mathbf{F} \, dV \), where \( \sigma \) is the closed surface of volume \( V \), and \( abla \cdot \mathbf{F} \) is the divergence of the vector field \( \mathbf{F} \).

Step 3: Calculate the divergence of \( \mathbf{F} \)

The divergence of \( \mathbf{F} \) is \( abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x^3) + \frac{\partial}{\partial y}(y^3) + \frac{\partial}{\partial z}(z^3) \). Compute this to get \( abla \cdot \mathbf{F} = 3x^2 + 3y^2 + 3z^2 \).

Step 4: Set up the triple integral over the volume

The volume is described by the cylinder with \( x^2 + y^2 \leq 4 \), from \( z=0 \) to \( z=3 \). Express the triple integral in cylindrical coordinates: \( x = r \cos \theta, y = r \sin \theta, z = z \) with \( r \), \( \theta \), and \( z \) having the limits: \( 0 \leq r \leq 2 \), \( 0 \leq \theta \leq 2\pi \), \( 0 \leq z \leq 3 \).

Step 5: Evaluate the triple integral

Compute \( \iiint_{V} (3r^2 + 3z^2) \, r \, dr \, d\theta \, dz \). Start by integrating with respect to \( r \) from 0 to 2, \( \theta \) from 0 to \( 2\pi \), and \( z \) from 0 to 3. The integrand simplifies to \( 3r^3 + 3rz^3 \). Perform the integration: \( \int_0^{2} \int_0^{2\pi} \int_0^{3} (3r^3 + 3rz^3) \, r \, dz \, d\theta \, dr \).

Step 6: Final computations and result

Perform integration sequentially. First integrate \( dz \), then \( dr \), and finally \( d\theta \) and evaluate to get the final value of the flux. You should get a result of \( 240\pi \) for the total outward flux across the surface \( \sigma \).

Key Concepts

vector field

In this exercise, we need to understand what a vector field is before applying the Divergence Theorem. A vector field is a function that assigns a vector to every point in space. In our case, the vector field is \( \mathbf{F}(x, y, z) = x^3 \mathbf{i} + y^3 \mathbf{j} + z^3 \mathbf{k} \). This means for any point \( (x, y, z) \) in space, the vector is composed of \( x^3 \) in the \( x \)-direction, \( y^3 \) in the \( y \)-direction, and \( z^3 \) in the \( z \)-direction.
Understanding the nature of these vectors helps us visualize how they "flow" through space. In physical terms, this vector field can model various phenomena, such as fluid flow or electromagnetic fields. Each component of the vector depends solely on its respective coordinate, indicating radial symmetry from the origin. This characteristic is crucial, as it helps simplify our calculations when applying the Divergence Theorem. By understanding this, we acknowledge that as we move farther from the origin, the influence of the vector field increases with the cube of their respective coordinates.

flux calculation

Calculating the flux through a surface is key in applying the Divergence Theorem. Flux measures how much of a vector field passes through a surface, which is critical in understanding how the field behaves across different boundaries.
The Divergence Theorem provides a convenient way to calculate this flux by converting a surface integral into a volume integral. Instead of calculating complex surface integrals directly, this method allows us to work with a volume integral over a simpler region, as long as the divergence of the field is known.
In this scenario, the vector field’s divergence is given by: \( abla \cdot \mathbf{F} = 3x^2 + 3y^2 + 3z^2 \). This makes the calculation more straightforward. We calculate the triple integral of the divergence over the volume of the cylindrical solid, thus finding the total flux across the surface. This transformation highlights the core power of the Divergence Theorem, offering an efficient way to deal with complex flux calculations. By ensuring the surface is closed and the field differentiable, this theorem provides certainty that the inward and outward flux are balanced.

cylindrical coordinates

Cylindrical coordinates are essential for problems involving cylindrical symmetry, like the one presented. In Cartesian coordinates, dealing with cylinders can be mathematically intensive; however, by switching to cylindrical coordinates, we align the coordinate system with the symmetry of the shape. This simplifies expressing and calculating integrals significantly.
For this exercise, the cylindrical solid is defined by the cylinder \( x^2 + y^2 = 4 \) and bounded by \( z = 0 \) and \( z = 3 \). We can translate these boundaries into cylindrical coordinates: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( z = z \), where \( r \), \( \theta \), and \( z \) have limits: \( 0 \leq r \leq 2 \), \( 0 \leq \theta \leq 2\pi \), and \( 0 \leq z \leq 3 \).
This conversion inherently accounts for rotational symmetry around the z-axis, which is why it is preferable for integration in such shapes. These adjustments let us evaluate the triple integral set for the divergence over the volume efficiently. Using cylindrical coordinates reduces the computational complexity and allows for easier integration over the defined limits, delivering precise solutions with less effort.