Problem 419 - Calculus Volume 3

Question

& \text { Evaluate } & \iint_{S} \mathbf{F} \cdot d \mathbf{S}\end{array}\( where \)\mathbf{F}(x, y, z)=b x y^{2} \mathbf{i}+b x^{2} y \mathbf{j}+\left(x^{2}+y^{2}\right) z^{2} \mathbf{k}\( and \)S\( is a closed surface bounding the region and consisting of solid cylinder \)x^{2}+y^{2} \leq a^{2}\( and \)0 \leq z \leq b$.

Step-by-Step Solution

Answer

Use the Divergence Theorem to solve the surface integral.

Step 1: Identify the Given Data

The vector field is given by \( \mathbf{F}(x, y, z) = bxy^2 \mathbf{i} + bx^2y \mathbf{j} + (x^2 + y^2)z^2 \mathbf{k} \). The surface \( S \) is the closed surface of a cylinder defined by \( x^2 + y^2 \leq a^2 \) and \( 0 \leq z \leq b \).

Step 2: Understand the Surface

The surface \( S \) is a closed cylinder with its side surface, top surface at \( z=b \), and bottom surface at \( z=0 \). We need to compute the surface integral over all three parts.

Step 3: Use the Divergence Theorem

To simplify the calculation, apply the Divergence Theorem, which states \( \iint_{S} \mathbf{F} \cdot d \mathbf{S} = \iiint_{V} abla \cdot \mathbf{F} \, dV \), where \( V \) is the volume inside the surface \( S \). First, calculate the divergence \( abla \cdot \mathbf{F} \).

Step 4: Calculate Divergence of F

The divergence of \( \mathbf{F} \) is \( abla \cdot \mathbf{F} = \frac{\partial }{\partial x}(bxy^2) + \frac{\partial }{\partial y}(bx^2y) + \frac{\partial }{\partial z}((x^2 + y^2)z^2) \). Calculating these derivatives gives \( by^2 + b2xy + 2z(x^2 + y^2)z \).

Step 5: Simplify the Divergence

Simplifying the divergence, we have \( abla \cdot \mathbf{F} = by^2 + 2bxy + 2x^2z + 2y^2z \).

Step 6: Set Up Triple Integral

We set up the triple integral \( \iiint_{V} abla \cdot \mathbf{F} \, dV \) over the cylindrical volume. This is \( \int_{0}^{b} \int_{0}^{2\pi} \int_{0}^{a} (br^2\sin^2\theta + 2br^3\cos^2\theta\sin\theta + 2r^2z ) r \, dr \, d\theta \, dz \).

Step 7: Evaluate the Integrals

First, integrate with respect to \( r \), then \( \theta \), and finally \( z \). Calculate each integral step by step using cylindrical coordinates where \( x = r\cos\theta \) and \( y = r\sin\theta \).

Step 8: Final Calculation

After evaluating all the integrals and simplifying, you obtain the result of the triple integral which equals the surface integral \( \iint_{S} \mathbf{F} \cdot d \mathbf{S}\).

Key Concepts

Surface Integrals

A surface integral helps calculate the flow of a vector field across a surface in three-dimensional space. It works similarly to a line integral but is extended to two dimensions, where we explore how much vectors 'penetrate' a region.
Surface integrals are important when you need to understand phenomena like fluid flow across boundaries or electromagnetic field interaction with objects.
Typically, surface integrals involve evaluating the dot product of a vector field and a vector normal to the surface. This expression helps capture how the vector field behaves relative to the surface's orientation.
In the given problem, the surface is a closed cylinder composed of its side, top, and bottom parts, making it a composite region that must be considered entirely for evaluation.
The method first requires computing the surface integral in three parts, one for each surface segment. However, using the Divergence Theorem can simplify this calculation.

Vector Field

A vector field assigns a vector to every point in space, providing information like direction and magnitude across a region. Such fields are used extensively in physics and engineering, often representing force fields, velocity fields, or electromagnetic fields.
In mathematical terms, a vector field in three dimensions can be expressed as \( \mathbf{F}(x, y, z) = P(x, y, z)\mathbf{i} + Q(x, y, z)\mathbf{j} + R(x, y, z)\mathbf{k} \), where \( P, Q, \) and \( R \) are functions of the spatial variables.
The problem provides a vector field \( \mathbf{F}(x, y, z) = bxy^2 \mathbf{i} + bx^2y \mathbf{j} + (x^2 + y^2)z^2 \mathbf{k} \), which varies with position. Understanding its components is crucial as each part has its effect on a given region.
For instance, the component \( bxy^2 \mathbf{i} \) describes part of the flow in the \( x \)-direction, affected by both \( x \) and \( y \). This field influences how we calculate physical quantities, like flux, across surfaces.

Cylindrical Coordinates

Cylindrical coordinates are a powerful coordinate system that simplifies the evaluation of problems with cylindrically symmetric geometries, like the ones seen in this problem.
In cylindrical coordinates, we replace Cartesian coordinates \( (x, y, z) \) with \( (r, \theta, z) \), where \( r \) is the radial distance, \( \theta \) is the angular position, and \( z \) remains the vertical height. The transformations are \( x = r\cos\theta \), \( y = r\sin\theta \), and these are particularly useful in describing cylinders and computing integrals over cylindrical volumes.
For the integration, the limits for \( r \), \( \theta \), and \( z \) are set according to the cylinder's properties. \( r \) varies from 0 to \( a \), \( \theta \) from 0 to \( 2\pi \), and \( z \) from 0 to \( b \), corresponding to the bounds of the cylinder defined in the problem.
This coordinate system significantly simplifies the integral evaluation because it aligns naturally with the shape of the region, making calculations more straightforward.

Triple Integrals

Triple integrals extend the idea of integration to three dimensions, allowing us to compute a volume or integrate a function over a 3D region. Generally, a triple integral of a function \( f(x, y, z) \) is written as \( \iiint_{V} f(x, y, z) \, dV \), where \( V \) is the region of integration.
For the problem at hand, once the divergence is computed, the triple integral \( \iiint_{V} abla \cdot \mathbf{F} \, dV \) is set up over the cylindrical region. Using cylindrical coordinates, the integral is expressed as \( \int_{0}^{b} \int_{0}^{2\pi} \int_{0}^{a} (br^2\sin^2\theta + 2br^3\cos^2\theta\sin\theta + 2r^2z ) r \, dr \, d\theta \, dz \).
This setup involves evaluating a series of integrals concerning \( r, \theta, \) and \( z \), each representing a layer of the problem's complexity.
Approaching it step-by-step ensures that the effects of each dimension are properly considered, eventually yielding the total result combining all these aspects.