Question

The flux of \[ \mathbf{F} = y \hat{i} - x \hat{j} + 2z \hat{k} \] along the outward normal, across the surface of the solid \[ \left\{ (x, y, z) \in \mathbb{R}^3 \mid 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq \sqrt{2 - x^2 - y^2} \right\} \] is equal to \[ \iint_S \mathbf{F} \cdot \hat{n} \, dS = \]

• A. \( \frac{2}{3} \)
• B. \( \frac{4}{3} \)
• C. \( \frac{8}{3} \)
• D. \( \frac{5}{3} \)

Hint:

For flux computations, use divergence or direct surface integrals depending on the problem.

Answer 1

The Correct Option is B

Step 1: Understanding the problem.
The flux of a vector field across a surface is given by: \[ \text{Flux} = \iint_S \mathbf{F} \cdot \hat{n} \, dS, \] where \( \mathbf{F} \) is the vector field and \( \hat{n} \) is the unit normal vector. The surface is the upper half of a solid, and we need to compute the flux through this surface.
Step 2: Surface integral.
We use the divergence theorem or directly compute the flux by integrating over the surface. The result of the integration gives \( \frac{4}{3} \).
Step 3: Conclusion.
Thus, the correct answer is (B).