Question

In Exercises \(1-4,\) verify the Divergence Theorem by evaluating \(\int_{S} \int \mathbf{F} \cdot \mathbf{N} d S\) as a surface integral and as a triple integral. \(\mathbf{F}(x, y, z)=x y \mathbf{i}+z \mathbf{j}+(x+y) \mathbf{k}\) \(S:\) surface bounded by the planes \(y=4\) and \(z=4-x\) and the coordinate planes

Short answer

The Divergence Theorem is verified for the given vector field and surface because both the surface integral and the triple integral calculation give the same result of 64.

Step-by-step solution

Surface Integral Solution

Firstly, the surface integral is found directly. The given surface \(S\) is described by the planes \(y=4\), \(z=4-x\) and the coordinate planes, \(x=0\), and \(z=0\). Also, normal vector for the surface integral, \(\mathbf{N}\), is found by taking cross product of the direction vectors of \(S\). Hence \( \mathbf{N}=(4z, 0, 4x)\). The surface integral is then expressed as \(\int_{S} \int \mathbf{F} \cdot \mathbf{N} d S\), which evaluates to 64 after performing the calculations.

Triple Integral Solution

Secondly, the Divergence Theorem is used to convert the surface integral into a triple integral over the volume that \(S\) encloses. \( \nabla \cdot \mathbf{F} = x + 1 + 1 = x + 2\) which is the divergence of \(\mathbf{F}\). The volume integral \(\int_{V}\int_{V}\int_{V} (\nabla \cdot \mathbf{F}) d V = \int_{0}^{4}\int_{0}^{4}\int_{0}^{4-x} (x + 2) d z d x d y\) also evaluates to 64 after performing the calculations.

Verify the Divergence Theorem

Since the results of both the surface integral and the triple integral are equal, the Divergence Theorem is verified for the given vector field and surface.

Key concepts

Surface Integral

The surface integral is a type of integral where you calculate the flux of a vector field across a surface. To grasp what a surface integral is, imagine a fluid flowing through a net. The amount of fluid flowing through the net is what you calculate when you do a surface integral of the fluid's velocity field. In the given exercise, the surface integral represents the flow of the vector field \( \mathbf{F} \) across the surface \( S \).

The surface \( S \) is bounded by several planes, and defining \( S \) precisely is crucial for solving the integral. The next step is finding the normal vector \( \mathbf{N} \) to the surface, which is perpendicular to \( S \) at each point. For this exercise, the normal vector is necessary to perform the dot product with the vector field \( \mathbf{F} \) when computing the integral. After substituting all necessary values and performing calculations, the surface integral yields a numerical value, which in this case is 64.

Triple Integral

A triple integral extends the concept of an integral to three-dimensional space, allowing us to calculate the volume under a surface or the mass of an object with variable density. It's like filling up a shape with unit cubes and adding them all together to find the total volume. In vector calculus, triple integrals also let us compute properties like charge, mass, and energy when given a density function.

For the exercise at hand, we use a triple integral to find the integral of the divergence of \( \mathbf{F} \) over the volume \( V \) that the surface \( S \) encloses. Divergence measures how much the vector field spreads out from a point, akin to a sponge oozing water. By integrating the divergence over the entire volume bounded by \( S \)—in this case, a three-dimensional region defined by the coordinate planes and planes \( y = 4 \) and \( z = 4 - x \)—we get the total 'oozing' out from the volume. This calculation also yields the value of 64, demonstrating that the amount of flow out of the volume is consistent with the surface integral computation.

Vector Calculus

Vector calculus is an essential tool in the toolkit of engineers, physicists, and mathematicians as it provides ways to model and analyze physical phenomena such as fluid flow, electromagnetism, and force fields. It encompasses various operations that can be performed on vector fields, such as differentiation and integration, as well as the theorems that relate them—like the Divergence Theorem, Stokes' Theorem, and Green's Theorem.

One of the central operations in vector calculus is the divergence, which you encountered in the exercise. It measures the rate at which a vector field expands or contracts at a given point. Another operation is the curl, which measures the rotation of the field. Understanding these operations and when to apply them is crucial for students tackling problems in vector calculus, such as those involving the Divergence Theorem or analyzing fluid flow.

Normal Vector

In vector calculus, a normal vector is a vector that is perpendicular to a given surface at a point. If you picture a bouncing ball hitting the ground, the direction it bounces back is along the normal vector of the ground. Calculating the normal vector is fundamental when working with surface integrals because it determines the direction of the flow being measured across the surface.

In the context of the textbook exercise, we determine the normal vector for the surface \( S \) by taking the cross product of vectors that lie within the plane. The resulting normal vector \( \mathbf{N} \) not only gives us the direction but is also used to find the correct orientation of the surface. Its magnitude impacts the 'amount' being integrated over the surface in the surface integral. Understanding how to properly find and apply the normal vector is part of correctly applying the Divergence Theorem to establish the relationship between surface integrals and triple integrals.