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The Divergence Theorem is a powerful tool for simplifying the calculation of flux in certain problems. It allows us to convert a surface integral into a volume integral, which is often easier to compute. The theorem states that if you have a vector field \( \vec{F} \) and a closed surface \( S \) enveloping a volume \( V \), the flux through \( S \) is equal to the integral of the divergence of \( \vec{F} \) over \( V \). This is represented mathematically as:
\[ \iint_S \vec{F} \cdot d\vec{S} = \iiint_V (abla \cdot \vec{F}) \, dV \]
The divergence measures how much the vector field spreads out from a given point. If a vector field has a positive divergence, it indicates that more of the field is "flowing out" than "flowing in." Conversely, a negative divergence signals more inflows toward a point.
Understanding and applying the Divergence Theorem is crucial when dealing with closed surfaces like spheres, as demonstrated in the problem, turning surface computations into more manageable volume computations.

A vector field assigns a vector to every point in space. It is a way of specifying a direction and magnitude for regions in mathematical spaces. In our exercise, the vector field is defined as \( \vec{F} = -5 \vec{r} \), where \( \vec{r} \) is the position vector \( (x, y, z) \). Therefore, \( \vec{F} \) explicitly becomes \( (-5x, -5y, -5z) \).
Key characteristics of vector fields include:

• Directionality: The vector points in a specific direction at each point.
• Magnitude: This is represented by the length of the arrow at each point.

Vector fields are often used in physics to represent forces like magnetic, gravitational, or electric fields. By understanding the nature of the vector field, we can better analyze behaviors such as flux or movement through a region, giving insight into the field's effects within the volume or across the surface.

A surface integral extends the concept of integrating over a curve to integrating over a surface. Surface integrals are used when you want to calculate the flux of a vector field through a surface. In our task, the surface is a sphere centered at the origin.
The equation of the sphere is \( x^2 + y^2 + z^2 = 4 \), which describes all points equidistant from a central point (the origin, in this case) by a radius of 2. Surface integrals involve summing up vector field values over each differential element of the surface area, represented as \( \iint_S \vec{F} \cdot d\vec{S} \).
When using the Divergence Theorem, we convert this surface integral into a volume integral, making the calculation process much simpler. This is especially useful for surfaces that enclose a volume, like a sphere, where checking each differential surface element can be quite complex.

Volume integrals allow us to calculate values over three-dimensional spaces. In our problem, after establishing the divergence of the vector field, we proceed with a volume integral to simplify the task at hand.
The divergence calculated was \(-15\) for the vector field \( \vec{F} \). Thus, the volume integral equation becomes:
\[ \iiint_V (-15) \, dV \]The sphere's volume helps determine the bounds of the integral, calculated using its volume formula \( \frac{4}{3} \pi r^3 \), resulting in:
\[ \frac{4}{3} \pi (2)^3 = \frac{32}{3} \pi \]Multiplying the divergence by this volume gives the flux through the sphere's surface. This straightforward calculation simplifies what would otherwise be a cumbersome surface computation. Understanding volume integrals is essential when applying them in contexts like calculating mass, charge, or any property distributed throughout a region.