Problem 387 - 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. Use the divergence theorem to calculate surface integral \(\quad \iint_{S} \mathbf{F} \cdot d \mathbf{S}\) when \(\mathbf{F}(x, y, z)=x^{2} z^{3} \mathbf{i}+2 x y z^{3} \mathbf{j}+x z^{4} \mathbf{k}\) and \(S\) is the surface of the box with vertices (±1,±2,±3) .

Step-by-Step Solution

Answer

The value of the surface integral is 0.

Step 1: Understand the Divergence Theorem

The divergence theorem relates a surface integral over a closed surface to a volume integral over the region bounded by the surface. It states that \( \iint_{S} \mathbf{F} \cdot d \mathbf{S} = \iiint_{V} (abla \cdot \mathbf{F}) \, dV \), where \( V \) is the volume enclosed by the surface \( S \), and \( abla \cdot \mathbf{F} \) is the divergence of \( \mathbf{F} \).

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

Given \( \mathbf{F}(x, y, z) = x^{2} z^{3} \mathbf{i} + 2 x y z^{3} \mathbf{j} + x z^{4} \mathbf{k} \), find the divergence \( abla \cdot \mathbf{F} \). This is computed as: \[ abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x^{2} z^{3}) + \frac{\partial}{\partial y}(2 x y z^{3}) + \frac{\partial}{\partial z}(x z^{4}) \].

Step 3: Differentiate \(\mathbf{F}\)

Calculate the derivatives: \(\frac{\partial}{\partial x}(x^{2} z^{3}) = 2xz^{3}\), \(\frac{\partial}{\partial y}(2 x y z^{3}) = 2xz^{3}\), and \(\frac{\partial}{\partial z}(x z^{4}) = 4xz^{3}\).

Step 4: Sum the Divergence Components

Add up the components to get the divergence: \[ abla \cdot \mathbf{F} = 2xz^{3} + 2xz^{3} + 4xz^{3} = 8xz^{3} \].

Step 5: Set Up the Volume Integral

The region is bounded by a box defined with vertices at \((\pm 1, \pm 2, \pm 3)\). Integrate \( 8xz^{3} \) over each dimension of this box: \(-1 \leq x \leq 1\), \(-2 \leq y \leq 2\), and \(-3 \leq z \leq 3\).

Step 6: Evaluate the Integral

The integral can be evaluated as \[ \int_{-3}^{3} \int_{-2}^{2} \int_{-1}^{1} 8xz^{3} \, dx \, dy \, dz \]. First, integrating with respect to \( x \), we have:\[ \int_{-1}^{1} 8xz^{3} \, dx = \left[ 4x^{2}z^{3} \right]_{-1}^{1} = 8z^{3} \]. Next, integrate with respect to \( y \):\[ \int_{-2}^{2} 8z^{3} \, dy = 8z^{3} \times 4 = 32z^{3} \]. Finally, integrate with respect to \( z \):\[ \int_{-3}^{3} 32z^{3} \, dz = 32 \left[ \frac{z^{4}}{4} \right]_{-3}^{3} = 32 \times 0 = 0 \].

Step 7: Conclusion

The evaluation of the surface integral, using the divergence theorem and a CAS, results in \( 0 \). The symmetry and cancellation of positive and negative values in the region result in a net zero integral over the surface.

Key Concepts

Surface Integral

A surface integral is a way to calculate the total flow of a vector field across a surface. Imagine a gentle breeze moving across a sheet of paper. The surface integral effectively measures how much of this breeze passes through the paper at every point.
Calculating the surface integral can seem daunting, but with the Divergence Theorem, it becomes manageable, especially for closed surfaces like the one described in the exercise. Here, the surface was a box defined by vertices. Instead of computing directly over this complex surface, the Divergence Theorem lets us convert it to a simpler volume integral.
Using this technique, we evaluated \( \iint_{S} \mathbf{F} \cdot d\mathbf{S} \) through the surface \( S \) by finding the divergence and computing the volume integral of the field \( \mathbf{F} \). This was crucial, as it leveraged symmetry and eased calculation.

Volume Integral

A volume integral extends the idea of integrals to three-dimensional spaces, allowing us to sum a quantity over a volume. In simpler terms, think about filling a box with sand efficiently, ensuring you account for every grain from bottom to top.
The Divergence Theorem relates this concept directly to surface integrals: it states that the total outflow through a closed surface equals the volume integral of the divergence over the volume the surface encloses. In shape terms, the surface's box acts as a boundary, and our task was to evaluate \( \iiint_{V} (abla \cdot \mathbf{F}) \, dV \).
For the exercise, this involved calculating the integral of the divergence, \( 8xz^{3} \), over a box bound by \(-1 \leq x \leq 1\), \(-2 \leq y \leq 2\), and \(-3 \leq z \leq 3\). This conversion from a surface to a volume integral demonstrates how spatial understanding of the problem helps reach a solution more efficiently.

Vector Field Divergence

Vector field divergence indicates how much a fluid expands or compresses at a point, acting like a scanner highlighting areas where the field density changes. It’s calculated by extracting partial derivatives of the vector components.
In our problem, we needed \( abla \cdot \mathbf{F} \), the divergence of \( \mathbf{F}(x, y, z) = x^{2} z^{3} \mathbf{i} + 2 x y z^{3} \mathbf{j} + x z^{4} \mathbf{k} \). By differentiating each part with respect to its variable, we found the divergence was \( 8xz^{3} \).
Why is this useful? Well, because it simplifies the surface integral using the Divergence Theorem. It connects the surface's external behavior to what's happening inside, offering a computational shortcut. This divergence effectively "summarizes" what's happening to the vector field through the box, allowing us to proceed with the simpler volume integral instead of the directly complex surface integral.

Computer Algebra System

A Computer Algebra System (CAS) is like a digital assistant for math, handling complex calculations and symbol manipulations that would otherwise require tedious manual work. In calculus, where precision and complexity grow quickly, a CAS becomes invaluable.
With powerful software, operations such as differentiating \( \mathbf{F} \), computing \( abla \cdot \mathbf{F} \), and evaluating multiple integrals become less prone to human error. This allows students and professionals to focus on understanding the underlying math concepts rather than bogging down in arithmetic.
For the exercise, having a CAS expedited finding derivatives and evaluating integrals. This emphasis on the CAS isn't just about ease—it's about leveraging technology for deeper understanding and accurate results. By automating laborious calculations, we're free to explore the rich implications of the Divergence Theorem and other calculus concepts without getting lost in the nitty-gritty.