In Exercises 5-16, use the Divergence Theorem to find the...

In Exercises $5-16,$ use the Divergence Theorem to find the outward flux of $\mathbf{F}$ across the boundary of the region $D .$ $\mathbf{F}=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k}$ a. Cube $D :$ The cube cut from the first octant by the planes $$x=1, y=1,$ and $z=1$$ b. Cube $\quad D :$ The cube bounded by the planes $x=\pm 1$ $$y=\pm 1, \text { and } z=\pm 1$$ c. Cylindrical can $D :$ The region cut from the solid cylinder $$\begin{array}{l}{x^{2}+y^{2} \leq 4 \text { by the planes } z=0 \text { and }} \\ {z=1}\end{array}$$

In Exercises $5-16,$ use the Divergence Theorem to find the outward flux of $\mathbf{F}$ across the boundary of the region $D .$
$\mathbf{F}=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k}$
a. Cube $D :$ The cube cut from the first octant by the planes
$$x=1, y=1,$ and $z=1$$
b. Cube $\quad D :$ The cube bounded by the planes $x=\pm 1$
$$y=\pm 1, \text { and } z=\pm 1$$
c. Cylindrical can $D :$ The region cut from the solid cylinder
$$\begin{array}{l}{x^{2}+y^{2} \leq 4 \text { by the planes } z=0 \text { and }} \\ {z=1}\end{array}$$

Key Concepts

Divergence Theorem

The Divergence Theorem is a fundamental theorem in vector calculus that relates the flux of a vector field across a closed surface to the divergence of the field integrated over the volume bounded by the surface. It allows one to convert a surface integral into a volume integral, thereby simplifying the computation of flux when the divergence is easier to integrate over the given region.

Divergence of a Vector Field

The divergence of a vector field is a scalar function that measures the net rate at which the vector field 'spreads out' from a point. It is computed as the sum of the partial derivatives of the components of the vector field with respect to their corresponding coordinates. Calculating divergence is crucial when applying the Divergence Theorem because it determines the integrand of the resulting volume integral.

Volume Integration

Volume integration involves integrating a function over a three-dimensional region. In the context of the Divergence Theorem, it is used to integrate the divergence of the vector field over the volume enclosed by the surface. Setting up these integrals properly and choosing the appropriate coordinates for the region is key to simplifying the calculations.

Coordinate System Selection

Choosing an appropriate coordinate system, such as Cartesian or cylindrical coordinates, simplifies the process of evaluating volume integrals. The decision depends on the symmetry and boundaries of the region being integrated over; for example, cylindrical coordinates are often more suitable for regions with circular symmetry, while Cartesian coordinates may be more convenient for box-like or rectangular regions.

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