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. $$ \begin{aligned} &\mathbf{F}(x, y, z)=2 x \mathbf{i}-2 y \mathbf{j}+z^{2} \mathbf{k}\\\ &S: \text { cube bounded by the planes } x=0, x=a, y=0, y=a,\\\ &z=0, z=a \end{aligned} $$

Short answer

The Divergence Theorem is verified for this vector field and surface if the computed surface integral and the triple integral yield the same result.

Step-by-step solution

Calculate Surface Integral

The surface of the cube consists of six faces. We can compute the surface integral on each of these faces individually and then add the results together. First, find the normal vector for each face. Then, substitute the function values into the given vector field \( \mathbf{F} \). Finally, use these to calculate the surface integral. The surface integral of \(\mathbf{F}\cdot \mathbf{N}\) over the cube is the sum of surface integrals over each face.

Calculate the Triple Integral

The Divergence Theorem relates surface integrals of vector fields to triple integrals over the volume enclosed by the surface. Hence, we need to calculate the triple integral of the divergence of the vector field over the volume of the cube. First, calculate the divergence of the vector field \( \mathbf{F} \). Then, substitute this into the triple integral and calculate the integral over the volume of the cube.

Verifying the Divergence Theorem

After both integrals are computed, compare them to see if they are equal. If they are equal, then the Divergence Theorem is verified for this particular vector field and surface. Make sure to calculate correctly and check your steps.

Key concepts

Surface Integral

Understanding the surface integral involves calculating the flux of a vector field through a given surface. Imagine a vector field representing, for instance, the flow of a fluid or the force exerted by a magnetic field. Now, visualize a surface, like the skin of a drum, placed within this field. The surface integral essentially measures how much of the field passes through that drum's skin.

Regarding our example with the cube, we'd calculate the surface integral for each face separately. A face parallel to the yz-plane, for instance, would have a normal vector pointing in the x-direction. The surface integral over this face would involve the component of the vector field pointing the same way. Considering all six faces and their respective normal vectors, the sum of their surface integrals gives the total flux out of the cube.

Triple Integral

The triple integral extends the idea of a single integral, which calculates the area under a curve, to three dimensions, where it calculates the volume under a surface within a spatial region. In our cube example, we apply the triple integral to the divergence of the vector field over the cube's volume. Conceptually, it's like filling the cube with infinitesimally small building blocks (volume elements), finding the value of the divergence at each block, and summing all these values up to get a total. This gives us a measurement of how much the vector field is 'spreading out' from each point within the cube.

Vector Field

A vector field is a map that assigns a vector to every point in space. Think about placing a compass at every location in a room; the needle of each compass gives you a magnetic field vector. Similarly, in our textbook problem, the vector field \(\mathbf{F}(x, y, z)=2 x \mathbf{i}-2 y \mathbf{j}+z^{2} \mathbf{k}\) assigns a vector to each point \( (x, y, z) \) in the space inside and around the cube. This field might represent something real, like the velocity field of a flowing fluid.

Normal Vector

The normal vector to a surface at a given point is a perpendicular vector pointing outward. For simple geometries like the cube in our problem, we determine the normal vector direction based on which plane the face lies in relation to the coordinate system. For example, if we're looking at a face in the xy-plane, its normal vector points either straight up or straight down along the z-axis, depending on the direction we consider as 'outward'. The significance of the normal vector is that it dictates how we gauge the flow or flux across a particular surface.

Divergence of a Vector Field

The divergence of a vector field is a scalar quantity that represents the rate at which the vector field is spreading out from a given point. It's calculated using the partial derivatives of the field's components. For the vector field in our example, the divergence would be found by calculating \( abla \cdot \mathbf{F} \), where \( abla \) represents the gradient operator. Intuitively, if you imagine the vector field as representing air flow, the divergence at a point would tell you whether air is being blown out (positive divergence) or sucked in (negative divergence) at that location.

Volume Integral

A volume integral refers to the integration over a three-dimensional region. It's somewhat like adding up the contributions of a quantity within every infinitesimally small piece of the volume. When we're using the Divergence Theorem, we compute a volume integral of the divergence of our vector field over the region bounded by the surface. Going back to our cube example, we are essentially adding up how much the vector field spreads out or converges inside the entire cube. This integral tells us the total 'outflow' minus the 'inflow' of the field throughout the volume of the cube.