Question

Let S be the surface of the cube bounded by x \(=\) 0, x \(=\) 1, y \(=\) 0, y \(=\) 1, z \(=\) 0, z \(=\) 1. The value of the surface integral \(\iint_S \vec{F} \cdot d\vec{s}\) of a vector field \(\vec{F} = 4xz \hat{i} - y^2 \hat{j} + yz \hat{k}\) over the entire surface S of the cube, is _____

• A. 5/2
• B. 1
• C. 1/2
• D. 3/2

Hint:

Whenever you see a closed surface integral over a simple geometric volume (like a cube, sphere, or cylinder), always use the Gauss Divergence Theorem.
It converts complex 2D surface patch integrations into a much simpler 1D/3D volume integration.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question asks for the value of a closed surface integral of a vector field over the boundaries of a unit cube.

Step 2: Key Formula or Approach:
According to the Gauss Divergence Theorem, the flux of a vector field \( \vec{F} \) through a closed surface \( S \) is equal to the volume integral of the divergence of \( \vec{F} \) over the volume \( V \) enclosed by \( S \):
\[ \iint_S \vec{F} \cdot d\vec{s} = \iiint_V (\nabla \cdot \vec{F}) dV \]
where:
\[ \nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \]

Step 3: Detailed Explanation:

• Calculate the Divergence of \( \vec{F} \):
- The given vector field is:
\[ \vec{F} = 4xz \hat{i} - y^2 \hat{j} + yz \hat{k} \]
- Compute the partial derivatives:
\[ \frac{\partial F_x}{\partial x} = \frac{\partial}{\partial x}(4xz) = 4z \]
\[ \frac{\partial F_y}{\partial y} = \frac{\partial}{\partial y}(-y^2) = -2y \]
\[ \frac{\partial F_z}{\partial z} = \frac{\partial}{\partial z}(yz) = y \]
- Sum these terms to find the divergence:
\[ \nabla \cdot \vec{F} = 4z - 2y + y = 4z - y \]

• Evaluate the Volume Integral over the Unit Cube:
- The bounds for the cube are \( x \in [0, 1] \), \( y \in [0, 1] \), and \( z \in [0, 1] \).
\[ \iiint_V (\nabla \cdot \vec{F}) dV = \int_0^1 \int_0^1 \int_0^1 (4z - y) \, dx \, dy \, dz \]
- Since the integrand does not depend on \( x \), integrating with respect to \( x \) simply yields a factor of 1:
\[ \int_0^1 \int_0^1 (4z - y) \, dy \, dz \]
- Integrate with respect to \( y \):
\[ \int_0^1 \left[ 4zy - \frac{y^2}{2} \right]_0^1 dz = \int_0^1 \left( 4z - \frac{1}{2} \right) dz \]
- Integrate with respect to \( z \):
\[ \left[ 2z^2 - \frac{z}{2} \right]_0^1 = \left(2(1)^2 - \frac{1}{2}(1)\right) - 0 = 2 - \frac{1}{2} = \frac{3}{2} \]


Step 4: Final Answer:
The value of the surface integral is \( 3/2 \).
Therefore, the correct choice is option (D).