Question

Consider a vector function \( \vec{F} = (y-z+2)\hat{i} + (yz+8)\hat{j} - xz\hat{k} \), on the surface (S) in the XY plane at \( z = 0 \) (as shown in the figure).
Considering Stokes theorem for space, the ABSOLUTE value of the surface integral \[ \iint_S (\nabla \times \vec{F}) \cdot \hat{n}\, dS \] is __________. (Answer in integer)

(Consider \( \hat{i}, \hat{j}, \hat{k} \), and \( \hat{n} \) as unit vectors)

Hint:

Compute curl F, dot it with the unit normal on the flat square, and integrate over the 4 by 4 region, or equivalently evaluate the line integral around T to U to V to W to T.

Answer 1

Correct Answer: 16

Step 1: Set up Stokes theorem as a line integral: 
Stokes theorem states that $\iint_S (\nabla \times \vec{F}) \cdot \hat{n}\, dS = \oint_C \vec{F} \cdot d\vec{r}$, where C is the closed boundary of S traversed in the direction consistent with the chosen normal. From the figure, C is the path T(0,0) to U(4,0) to V(4,4) to W(0,4) back to T(0,0), all of which lies in the plane z = 0.

Step 2: Simplify F on the boundary: 
Since the entire path lies at z = 0, substitute z = 0 into F. This gives $\vec{F} = (y + 2)\hat{i} + 8\hat{j} + 0\hat{k}$. Also, since the path stays in the XY plane, $d\vec{r} = dx\,\hat{i} + dy\,\hat{j}$, so $\vec{F} \cdot d\vec{r} = (y+2)\,dx + 8\,dy$.

Step 3: Integrate along T to U: 
On this segment y = 0 and dy = 0, with x going from 0 to 4. The integrand is $(0+2)\,dx = 2\,dx$, so the integral is $2 \times (4 - 0) = 8$.

Step 4: Integrate along U to V: 
On this segment x = 4 and dx = 0, with y going from 0 to 4. The integrand is $8\,dy$, so the integral is $8 \times (4 - 0) = 32$.

Step 5: Integrate along V to W: 
On this segment y = 4 and dy = 0, with x going from 4 to 0. The integrand is $(4+2)\,dx = 6\,dx$, so the integral is $6 \times (0 - 4) = -24$.

Step 6: Integrate along W to T: 
On this segment x = 0 and dx = 0, with y going from 4 to 0. The integrand is $8\,dy$, so the integral is $8 \times (0 - 4) = -32$.

Step 7: Add the four segments and take the absolute value: 
Adding all four parts gives $8 + 32 - 24 - 32 = -16$. This matches the surface integral found from the curl, confirming the calculation. The absolute value requested by the question is 16.

Final Answer: 
\[ \boxed{16} \]