91Ó°ÊÓ

Surface integrals are used to integrate vector fields over surfaces. In this problem, we are working over the surface of a cube. The cube is bounded by planes such as \( x = 0 \), \( x = 2 \), \( y = 0 \), and so on. The key thing with surface integrals is to understand the role played by the surface's orientation and normal vectors. The direction of the normal vector determines which part of the vector field contributes to the integral. Here, normal vectors are picked, for instance, as \(-\mathbf{j}\) or \(\mathbf{j}\) depending on which face of the cube we evaluate.### Calculating the Surface IntegralEach side of the cube gives a contribution based on the vector field and the orientation of the surface:- The face at \( y = 0 \) contributes nothing since \( yz \) when \( y = 0 \) gives zero.- Conversely, the face at \( y = 2 \) leads to contributions since the vector field there becomes \( yz \). This needs careful computation of \( \int_0^2 \int_0^2 2z \, dx \, dz \), yielding \( 8 \).The procedure involves analyzing contributions from each face and summing them. This requires understanding normals and the behavior of the vector field on those surfaces, which together inform what part of the vector field affects the integral.

Triple integrals extend the idea of double integrals from two-dimensions to three-dimensional spaces. They allow us to integrate functions over a volume. Here, we apply the triple integral to measure the divergence of our vector field across the cube's volume.### Setting up the Triple IntegralA triple integral requires setting up the bounds for integration correctly, typically reflecting the shapes' geometric features—in this case, our cube.The integral that corresponds to the volume is:\[\int_0^2 \int_0^2 \int_0^2 (y + z + x) \, dx \, dy \, dz\]Breaking it down step by step, we:- First integrate with respect to \( x \), treating \( y \) and \( z \) as constants, giving us \( 2(y + z) + 2 \).- Then, integrate the result with respect to \( y \), giving \( 8 + 4z \).- Finally, integrate with respect to \( z \), culminating in a result of \( 24 \).Through these incremental steps, triple integrals encapsulate the summation of the divergence over the entire volume of the cube.

A vector field assigns a vector to every point in a space. In this exercise, our vector field is defined as \( \mathbf{F}(x, y, z) = xy \mathbf{i} + yz \mathbf{j} + xz \mathbf{k} \). Vector fields have direction and magnitude at each position in space, often visualized as arrows.### Understanding the Vector Field- The component \( xy \mathbf{i} \) varies with the product of \( x \) and \( y \), showing dependency on these variables in the \( x \)-direction.- The component \( yz \mathbf{j} \) indicates influence in the \( y \)-direction based on both \( y \) and \( z \).- Lastly, \( xz \mathbf{k} \) suggests contribution in the \( z \)-direction, reliant on \( x \) and \( z \).### Role in the Divergence TheoremThe divergence of this vector field, \( abla \cdot \mathbf{F} = y + z + x \), encapsulates how the field interacts over a region, leading us to explore how it "expands" or "contracts" within the cube. Specifically, calculating this allows us to evaluate both terms of the Divergence Theorem, comparing the internal volume expansion (triple integral) and the flux through the boundary (surface integral). Grasping how each component varies helps in understanding how the Divergence Theorem verifies that the volume-based and surface-based assessments align—highlighting consistency and conservation within the vector field over the region.