Problem 378 - Calculus Volume 3

Question

For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral \(\int_{S} \mathbf{F} \cdot \mathbf{n} d s\) for the given choice of \(\mathbf{F}\) and the boundary surface \(S\). For each closed surface, assume \(\mathbf{N}\) is the outward unit normal vector. \(\quad\) [T] \(\mathbf{F}(x, y, z)=\left(x^{2}+y^{2}-x^{2}\right) \mathbf{i}+x^{2} y \mathbf{j}+3 z \mathbf{k} ; \quad\) S is the surface of the five faces of unit cube \(0 \leq x \leq 1,0 \leq y \leq 1,0

Step-by-Step Solution

Answer

The surface integral \( \int_{S} \mathbf{F} \cdot \mathbf{n} \, ds = \frac{10}{3} \).

Step 1: Express Given Information

The vector field is given as \( \mathbf{F}(x, y, z) = (x^2 + y^2 - x^2) \mathbf{i} + x^2 y \mathbf{j} + 3z \mathbf{k} = y^2 \mathbf{i} + x^2 y \mathbf{j} + 3z \mathbf{k} \). The surface \( S \) is the surface of a cube defined by \( 0 \leq x \leq 1 \), \( 0 \leq y \leq 1 \), and \( 0 < z \leq 1 \).

Step 2: Set Up Divergence Theorem Formula

The Divergence Theorem states that \( \int_{S} \mathbf{F} \cdot \mathbf{n} \, ds = \iiint_{V} abla \cdot \mathbf{F} \, dV \), where \( V \) is the volume of the region enclosed by \( S \) and \( abla \cdot \mathbf{F} \) is the divergence of \( \mathbf{F} \).

Step 3: Calculate the Divergence of F

The divergence of \( \mathbf{F} \), denoted as \( abla \cdot \mathbf{F} \), is calculated as follows: \(abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(y^2) + \frac{\partial}{\partial y}(x^2 y) + \frac{\partial}{\partial z}(3z) \). Calculating these partial derivatives gives: \( 0 + x^2 + 3 = x^2 + 3 \).

Step 4: Set Up Triple Integral for Volume

With \( abla \cdot \mathbf{F} = x^2 + 3 \), set up the triple integral: \( \iiint_{V} (x^2 + 3) \, dV = \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} (x^2 + 3) \, dz \, dy \, dx \).

Step 5: Evaluate the Triple Integral

Calculate the integral \( \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} (x^2 + 3) \, dz \, dy \, dx \). Integrate with respect to \( z \): \( \int_{0}^{1} (x^2 + 3) \, dz = (x^2 + 3)z \bigg|_{0}^{1} = x^2 + 3 \).

Step 6: Continue Integration Over y and x

Next, integrate with respect to \( y \): \( \int_{0}^{1} (x^2 + 3) \, dy = (x^2 y + 3y) \bigg|_{0}^{1} = x^2 + 3 \). Finally, integrate with respect to \( x \): \( \int_{0}^{1} (x^2 + 3) \, dx = \left( \frac{x^3}{3} + 3x \right) \bigg|_{0}^{1} = \frac{1}{3} + 3 = \frac{10}{3} \).

Key Concepts

Vector Field

In mathematics, a vector field assigns a vector to every point in space. It describes the distribution of vectors in a particular region and is an essential concept in calculus and physics.

• Consider the vector field given in the problem: \( \mathbf{F}(x, y, z) = y^2 \mathbf{i} + x^2 y \mathbf{j} + 3z \mathbf{k} \).
• This field comprises three components:
  • \( y^2 \mathbf{i} \) along the x-axis direction,
  • \( x^2 y \mathbf{j} \) along the y-axis, and
  • \( 3z \mathbf{k} \) along the z-axis.
• Each component functionally depends on the coordinates \(x, y, \text{and } z\).
• Vector fields are visualized as arrows with magnitudes and directions, indicating both the strength and the orientation of the field at given points.

Understanding vector fields is crucial, as they are involved in modeling physical phenomena such as electromagnetic fields, gravitational fields, and fluid flow. A vector field can be used to calculate quantities such as flux, which is evaluated using methods like the divergence theorem.

Surface Integral

Surface integrals allow us to calculate the flux across surfaces. In physical contexts, they often represent things like the flow rate of a fluid across a boundary or the electric flux through a surface.
For a given surface \( S \) and a vector field \( \mathbf{F} \), the surface integral \( \int_{S} \mathbf{F} \cdot \mathbf{n} \, ds \) calculates how \( \mathbf{F} \) behaves over \( S \).

• Here, \( \mathbf{n} \) is the unit normal vector to the surface, pointing outward if the surface encloses a volume.
• The dot product \( \mathbf{F} \cdot \mathbf{n} \) captures the component of \( \mathbf{F} \) that passes through \( S \).

The divergence theorem, also known as Gauss's theorem, offers a way to transform a difficult surface integral into a straightforward volume integral, facilitating computation by evaluating over the volume enclosed by the surface instead.

Cube Volume Integration

Volume integration over a cube involves evaluating a triple integral within defined bounds, specifically focusing on the volume \( V \) enclosed by the cube.
Here's how to evaluate the triple integral: \( \iiint_{V} f(x, y, z) \, dV \).

• This problem uses a cube defined by \( 0 \leq x \leq 1 \), \( 0 \leq y \leq 1 \), and \( 0 < z \leq 1 \).
• The integral \( \iiint_{V} (x^2 + 3)\, dV \) is computed in three steps:
  • First, integrate with respect to \( z \): \( \int_{0}^{1} (x^2 + 3) \, dz \), generating \( x^2 + 3 \).
  • Next, integrate with respect to \( y \): \( \int_{0}^{1} (x^2 + 3) \, dy \), yielding \( x^2 + 3 \).
  • Finally, integrate with respect to \( x \): \( \int_{0}^{1} (x^2 + 3) \, dx \), culminating in \( \frac{10}{3} \).

Volume integrals are the backbone of various applications in physics and engineering, enabling the calculation of mass, charge, energy, or fluid passing through a given region over space.