Problem 376 - 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. [T] \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} ; S\) is the surface of cube \(0 \leq x \leq 1,0 \leq y \leq 1,0

Step-by-Step Solution

Answer

The value of the surface integral is 3.

Step 1: Understanding the Divergence Theorem

The Divergence Theorem states that \( \int_{S} \mathbf{F} \cdot \mathbf{n} \, ds = \int_{V} abla \cdot \mathbf{F} \, dV \), where \( S \) is a closed surface and \( V \) is the volume inside the surface. We will use this theorem to convert the surface integral into a volume integral.

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

Given \( \mathbf{F}(x, y, z) = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \), we calculate its divergence: \[ abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x) + \frac{\partial}{\partial y}(y) + \frac{\partial}{\partial z}(z) = 1 + 1 + 1 = 3 \]

Step 3: Set Up the Volume Integral

With \( abla \cdot \mathbf{F} = 3 \), substitute into the volume integral: \[ \int_{V} 3 \, dV \] The volume \( V \) is bounded by the cube, so we need to integrate over each dimension from 0 to 1.

Step 4: Evaluate the Volume Integral

Calculate the volume integral:\[ \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} 3 \, dx \, dy \, dz = 3 \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \, dx \, dy \, dz \]This results in:\( \int_{0}^{1} \int_{0}^{1} 3 \cdot 1 \, dy \, dz = 3 \int_{0}^{1} 1 \, dz = 3 \cdot 1 = 3 \)

Key Concepts

Surface Integral

Surface integrals are a type of integral that allow us to calculate the flux of a vector field across a given surface. The surface integral is expressed as \( \int_{S} \mathbf{F} \cdot \mathbf{n} \, ds \), where \( \mathbf{F} \) is a vector field, \( \mathbf{n} \) is the unit normal vector to the surface \( S \), and \( ds \) represents an infinitesimal element of the surface. This type of integral is used to measure how much of a vector field passes through a surface.

• **Flux**: Represents the quantity of the field flowing through the surface.
• **Surface**: Can be any shape, such as a sphere, cube, or even a more complex form.
• **Vector Field (\( \mathbf{F} \))**: Describes the flow or direction of points in a space, like the movement of air or water.

In the problem above, \( \mathbf{F} \) is defined as \( x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \), and \( S \) is the surface of a cube. Using the Divergence Theorem transforms this complex surface calculation into a simpler volume calculation, simplifying the problem.

Volume Integral

The volume integral is a key concept in vector calculus that helps in calculating the total value of some quantity over a 3D region. When we talk about the volume integral, we often refer to integrating a function throughout a volume. It is expressed as \( \int_{V} abla \cdot \mathbf{F} \, dV \), where \( abla \cdot \mathbf{F} \) is the divergence of the vector field and \( dV \) represents a volume element.

• **Volume Element (\( dV \))**: Usually a small cube or rectangular box used to sum over the volume.
• **Divergence**: Measures the magnitude of a source or sink at a point in a vector field.

In the given exercise, we computed the divergence of \( \mathbf{F} \) as 3, simplifying the volume integral to \( \int_{V} 3 \, dV \). Given the cube boundary, the limits of integration are from 0 to 1 in all three dimensions. This results in multiplying the constant divergence through the unit cube, leading to a straightforward calculation of the integral.

Vector Calculus

Vector calculus is a branch of mathematics focused on vector fields and is fundamental to fields such as physics and engineering. It deals with both differentiation and integration of vector fields. Key concepts include gradient, divergence, curl, and theorems like the Divergence Theorem and Stokes' Theorem.

• **Gradient**: Measures how a scalar field changes in space.
• **Divergence**: Indicates how much a vector field spreads out from a point.
• **Curl**: Describes the rotation of a vector field around a point.
• **Theorems**: Such as the Divergence Theorem, help bridge the gap between surface and volume integrals.

In the problem we addressed, vector calculus simplifies solving complex integrals by converting them from surface to volume, using the Divergence Theorem. Instead of painstakingly evaluating the vector field's flux across each face of the cube, we compute the divergence over the volume inside the cube. This approach not only saves time but also provides deeper insight into the behavior of the vector field within the region.