91Ó°ÊÓ

The flux calculation involves determining how much of a vector field passes through a given surface. In this exercise, we use the Divergence Theorem to calculate the flux of a vector field \( \mathbf{F} \) across the boundary of a region. The theorem helps to simplify the problem by converting a complex surface integral into a more manageable volume integral. Here's how it works:

• The flux of \( \mathbf{F} \) through a closed surface \( S \) is equal to the triple integral of the divergence of \( \mathbf{F} \) within the volume \( V \) enclosed by \( S \).
• This means instead of computing a detailed surface integral, we look at how much the vector field spreads out or diverges within the volume.

This understanding is essential because it moves from a geometrically complex task to an algebraically approachable problem using the divergence formula \( \iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{V} abla \cdot \mathbf{F} \, dV \). This formula links the surface and volume integrals in a helpful way.

The divergence of a vector field illustrates how the field behaves at any given point. It essentially measures the "outflowing" nature of the field. For the vector field given by \( \mathbf{F} = 2x \mathbf{i} - xy \mathbf{j} - z^2 \mathbf{k} \), we calculate its divergence using the formula:\[ abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(2x) + \frac{\partial}{\partial y}(-xy) + \frac{\partial}{\partial z}(-z^2) \].This results in \( abla \cdot \mathbf{F} = 2 - x - 2z \). Breaking this calculation down makes the concept clearer:

• The term \( \frac{\partial}{\partial x}(2x) \) evaluates to 2, indicating flux along the x-axis.
• The term \( \frac{\partial}{\partial y}(-xy) \) results in \(-x\), representing a sink effect as y increases.
• The term \( \frac{\partial}{\partial z}(-z^2) \) gives \(-2z\), signifying that the field decreases as z increases.

This measure of divergence is a core aspect of the problem and drives the integral setup and evaluation.

Setting the correct integration limits is crucial when solving any integral, especially in three dimensions. For the region \( D \), bounded by given planes and surfaces, the limits determine how we integrate over the volume. For this exercise:

• The plane \( y + z = 4 \) and the elliptical cylinder \( 4x^2 + y^2 = 16 \) bound the region.
• Additional bounds include \( y = 0 \) and \( z = 0 \).

Projection of the region onto the \( xy \)-plane provides us with new limits:

• For \( x \), the limits are from \(-2\) to \(2\).
• For \( y \), given specific \( x \), it ranges from \(0\) to \(\sqrt{16 - 4x^2}\).
• For \( z \), given \( y \), it ranges from \(0\) to \(4 - y\).

These bounds define the region where the divergence will be integrated and translate the geometric constraints into a workable form for integration.

Evaluating the triple integral is the final step required to find the flux using the divergence theorem. It computes the total divergence over the volume. For the case of:\[ \iiint_{V}(2 - x - 2z) \, dV = \int_{-2}^{2}\int_{0}^{\sqrt{16-4x^2}}\int_{0}^{4-y}(2-x-2z)dz\,dy\,dx \], follow this sequence:1. **Innermost Integral**: Integrate with respect to \( z \); evaluate the expression \((2 - x - 2z)\) from \(0\) to \(4-y\). This gives \[ (2 - x)(4-y) - (4-y)^2 \].2. **Middle Integral**: Integrate the result with respect to \( y \). This step computes the output from the previous evaluation across the range given.3. **Outermost Integral**: Finally, integrate this outcome with respect to \( x \), within its specific bounds. This collects the entire contribution of divergence across the volume.By solving each step systematically, we find the total flux. Properly handling each integral section ensures that the physical boundaries and mathematical calculations fit together correctly.