Answer: $8\pi$

Set up the integral: For the plane $z = 3$, we can parametrize as $\mathbf{r}(x,y) = \langle x, y, 3\rangle$. Then $\mathbf{r}_x \times \mathbf{r}_y = \langle 0,0,1\rangle$, so $|\mathbf{r}_x \times \mathbf{r}_y| = 1$ and $dS = dA$. Evaluate using polar coordinates: $$\iint_D (x^2+y^2)\,dA = \int_0^{2\pi}\int_0^{2} r^2 \cdot r\,dr\,d\theta = 2\pi \int_0^2 r^3\,dr = 2\pi \cdot \frac{r^4}{4}\Big|_0^2 = 2\pi \cdot 4 = 8\pi.$$ Why distractors fail: Option A ($4\pi$) results from forgetting the extra $r$ in the polar area element. Option C ($16\pi$) comes from evaluating $\frac{r^4}{4}$ as $\frac{16}{1}$ instead of $4$. Option D ($12\pi$) is an arithmetic error.

Answer: Swap the order and use $\mathbf{r}_v \times \mathbf{r}_u$ to reverse the normal direction.

Cross product and orientation: The cross product $\mathbf{r}_u \times \mathbf{r}_v$ determines the direction of the normal. Reversing the order gives $\mathbf{r}_v \times \mathbf{r}_u = -(\mathbf{r}_u \times \mathbf{r}_v)$. Why the correct answer works: Since $\langle -2,1,-3\rangle = -\langle 2,-1,3\rangle$, swapping the cross product order yields the desired outward normal direction. Why distractors fail: Option A is wrong because the sign of the normal directly determines the sign of the flux. Option C is wrong because the magnitude is a scalar and cannot have a negative direction. Option D is an unnecessary and unrelated change — any parametrization can have its normal reversed by swapping the cross product order.

Answer: The integral sums the normal component of $\mathbf{F}$ over the surface, giving the net flow through $S$ in the direction of the chosen orientation.

Key idea: flux measures the normal component: $\mathbf{F} \cdot d\mathbf{S} = \mathbf{F} \cdot \hat{\mathbf{n}}\, dS$, so the integrand picks out the component of $\mathbf{F}$ perpendicular to the surface. Why the correct answer works: Option B correctly states that the flux integral sums the normal component over the surface, and that the sign depends on the chosen orientation. Why distractors fail: Option A confuses flux with a line integral that uses the tangential component. Option C describes work (a line integral concept). Option D ignores the dot product — angle matters, so $|\mathbf{F}|\,dS$ would overcount.

Answer: $24\pi$

Set up with the graph-surface formula: For $z = g(x,y) = 4-x^2-y^2$ with upward normal: $\iint_S \mathbf{F}\cdot d\mathbf{S} = \iint_D (-F_1 g_x - F_2 g_y + F_3)\,dA$. Here $g_x = -2x$, $g_y = -2y$, so the integrand is $-x(-2x) - y(-2y) + (4-x^2-y^2) = 2x^2 + 2y^2 + 4 - x^2 - y^2 = x^2 + y^2 + 4$. Evaluate in polar coordinates: Domain: $x^2+y^2 \le 4$ (where the paraboloid meets $z=0$). In polar: $$\int_0^{2\pi}\int_0^2 (r^2+4)\,r\,dr\,d\theta = 2\pi\int_0^2 (r^3+4r)\,dr = 2\pi\left[\frac{r^4}{4}+2r^2\right]_0^2 = 2\pi(4+8) = 24\pi.$$ Verify with the Divergence Theorem: Close the surface by adding the disk $D$ at $z=0$ with downward normal $\hat{\mathbf{n}}=\langle 0,0,-1\rangle$. Then $\nabla\cdot\mathbf{F}=3$. The volume under the paraboloid is $V = \int_0^{2\pi}\int_0^2\int_0^{4-r^2} r\,dz\,dr\,d\theta = 2\pi\int_0^2 r(4-r^2)\,dr = 2\pi[2r^2 - r^4/4]_0^2 = 8\pi$. So $\iiint 3\,dV = 24\pi$. Flux through the disk: $\iint_D \langle x,y,0\rangle\cdot\langle 0,0,-1\rangle\,dA = 0$. Therefore flux through the paraboloid $= 24\pi - 0 = 24\pi$. ✓ Why distractors fail: Option A ($8\pi$) equals the volume under the paraboloid, not the flux. Option B ($16\pi$) arises from omitting one of the $x^2$ or $y^2$ terms in the integrand. Option D ($4\pi$) results from using the wrong domain radius or dropping terms.

Answer: A cone with vertex at the origin and height 2

Interpret the parametrization: We have $x = u\cos v$, $y = u\sin v$, $z = u$. Therefore $x^2 + y^2 = u^2 = z^2$, giving $z = \sqrt{x^2+y^2}$. Identify the surface: The equation $z = \sqrt{x^2+y^2}$ is the upper half of a cone with vertex at the origin, half-angle $45°$. With $u \in [0,2]$, the cone extends to height 2. Why distractors fail: Option A (hemisphere) would require $x^2+y^2+z^2 = \text{const}$. Option C (cylinder) would require $x^2+y^2 = \text{const}$, but here the radius grows with $z$. Option D (paraboloid) would give $z = x^2+y^2$, not $z = \sqrt{x^2+y^2}$.

Answer: It converts the parameter-domain area element $dA$ into the surface area element $dS$.

Definition of $dS$: By definition, $dS = |\mathbf{r}_u \times \mathbf{r}_v|\, dA$, where $dA = du\, dv$ is the area element in parameter space. Why the correct answer works: The magnitude of the cross product accounts for how the parametrization stretches or compresses area, converting a flat $du\,dv$ element into the true surface area element. Why distractors fail: Option A confuses the magnitude with the unit normal; the unit normal is $\frac{\mathbf{r}_u \times \mathbf{r}_v}{|\mathbf{r}_u \times \mathbf{r}_v|}$. Option C describes flux integrals, which use $(\mathbf{r}_u \times \mathbf{r}_v)$ as a vector, not its magnitude. Option D describes a Jacobian for volume integrals, a different context.

Answer: $\pi$

Use the graph-surface flux formula: For $z = g(x,y)$ with upward orientation, $\iint_S \mathbf{F}\cdot d\mathbf{S} = \iint_D (-F_1 g_x - F_2 g_y + F_3)\,dA$. With $\mathbf{F}=\langle 0,0,1\rangle$, this becomes $\iint_D 1\,dA$. Evaluate the integral: The domain $D$ is the disk $x^2+y^2 \le 1$, whose area is $\pi(1)^2 = \pi$. Therefore the flux is $\pi$. Why distractors fail: Option A ($\pi/2$) might arise from incorrectly including a factor of $1/2$. Option C ($2\pi$) could result from confusing the area of a disk of radius 1 with the circumference. Option D ($3\pi/2$) has no valid derivation for this problem.

Answer: Both formulas are correct and give the same result.

Derive the equivalence: With $\mathbf{r}(u,v) = \langle u, v, f(u,v)\rangle$, we get $\mathbf{r}_u = \langle 1,0,f_u\rangle$ and $\mathbf{r}_v = \langle 0,1,f_v\rangle$. Then $\mathbf{r}_u \times \mathbf{r}_v = \langle -f_u, -f_v, 1\rangle$, and $|\mathbf{r}_u \times \mathbf{r}_v| = \sqrt{f_u^2 + f_v^2 + 1}$. Why the correct answer works: The cross-product magnitude $\sqrt{1+f_x^2+f_y^2}$ is exactly the integrand in the first formula, so both expressions are identical. Why distractors fail: Option A is wrong because the cross product approach with the given parametrization reproduces the same formula. Option B is wrong because the square root formula is exact, not approximate. Option D is wrong because surface area is a scalar integral, not a flux integral.

Answer: The Möbius strip is non-orientable, so a continuous unit normal cannot be defined over the entire surface.

Flux requires orientation: A flux integral $\iint_S \mathbf{F}\cdot d\mathbf{S}$ requires choosing a consistent normal direction over the entire surface, i.e., the surface must be orientable. Why the correct answer works: The Möbius strip is the classic non-orientable surface: traversing a loop on it reverses the normal direction. Since no continuous choice of $\hat{\mathbf{n}}$ exists globally, flux is undefined. Why distractors fail: Option A is false; a Möbius strip of finite dimensions has finite area. Option C is false; the cross product is generally nonzero on smooth parametrizations of the strip. Option D is false; flux integrals can be defined on open, oriented surfaces (e.g., a hemisphere).

Answer: A surface in three-dimensional space

Recall the definition of a parametric surface: A vector-valued function of two parameters $\mathbf{r}(u,v)$ maps a 2D domain $D$ into $\mathbb{R}^3$, sweeping out a surface. Why the correct answer works: Because $(u,v)$ varies over a two-dimensional region, the image of $\mathbf{r}$ is a two-dimensional object in 3-space — a surface. Why distractors fail: Option A describes a vector-valued function of one parameter $\mathbf{r}(t)$, which gives a curve. Option C confuses a parametrization with a vector field $\mathbf{F}(x,y,z)$. Option D describes an implicit surface, not a parametric one.

Answer: No — if $\mathbf{F}$ points predominantly inward across $S$, the dot product $\mathbf{F}\cdot\hat{\mathbf{n}}$ is negative, yielding negative flux.

Understand the sign of flux: Flux $\iint_S \mathbf{F}\cdot \hat{\mathbf{n}}\,dS$ depends on the angle between $\mathbf{F}$ and $\hat{\mathbf{n}}$. When $\mathbf{F}$ points opposite to $\hat{\mathbf{n}}$, the dot product is negative. Why the correct answer works: If the field flows into the enclosed region (against the outward normal), $\mathbf{F}\cdot\hat{\mathbf{n}} < 0$ at those points, and the overall flux can be negative. Orientation fixes which direction is 'positive,' but does not force the integral's value to be positive. Why distractors fail: Option A incorrectly equates a sign convention with the value of the integral. Option C is wrong because the Divergence Theorem gives $\iint_S \mathbf{F}\cdot d\mathbf{S} = \iiint_V \nabla\cdot\mathbf{F}\,dV$, which is zero only if $\nabla\cdot\mathbf{F}=0$ everywhere. Option D confuses $dS$ (a scalar area element) with the vector $d\mathbf{S}$; the dot product can still be negative.

Answer: Use $\mathbf{r}_\theta \times \mathbf{r}_\phi$ instead, so the normal points outward.

Determine the normal direction: The standard spherical parametrization with the ordering $(\phi,\theta)$ can yield $\mathbf{r}_\phi \times \mathbf{r}_\theta$ pointing inward (toward the origin). To reverse it, swap the cross product order. Why the correct answer works: By anti-commutativity, $\mathbf{r}_\theta \times \mathbf{r}_\phi = -(\mathbf{r}_\phi \times \mathbf{r}_\theta)$, which reverses the normal to point outward. Why distractors fail: Option A (negate the integrand) would produce the same numerical result, but the standard correct procedure is to swap the cross product order — negating $\mathbf{F}\cdot(\mathbf{r}_\phi \times \mathbf{r}_\theta)$ is algebraically equivalent but is not standard practice and obscures the geometric reasoning. Option C changes the surface itself, not the orientation. Option D is false; orientation depends on the parametrization order and must be checked.

Answer: A continuous choice of unit normal vector has been made at every point of the surface.

Definition of surface orientation: An oriented surface is one on which a continuous unit normal vector field $\hat{\mathbf{n}}$ can be (and has been) chosen. This distinguishes one 'side' of the surface from the other. Why the correct answer works: Orientation requires a consistent, continuous normal direction everywhere on the surface — exactly what Option B states. Why distractors fail: Option A conflates parametrization alignment with orientation; how $u$ maps to coordinates does not define orientation. Option C describes a meshing technique, not orientation. Option D describes a closed surface, but orientation is defined for open surfaces as well.

Answer: $u\sqrt{2}$

Compute the partial derivatives: $\mathbf{r}_u = \langle \cos v, \sin v, 1\rangle$ and $\mathbf{r}_v = \langle -u\sin v, u\cos v, 0\rangle$. Compute the cross product: $\mathbf{r}_u \times \mathbf{r}_v = \langle -u\cos v,\, -u\sin v,\, u\cos^2 v + u\sin^2 v\rangle = \langle -u\cos v,\, -u\sin v,\, u\rangle$. Find the magnitude: $|\mathbf{r}_u \times \mathbf{r}_v| = \sqrt{u^2\cos^2 v + u^2\sin^2 v + u^2} = \sqrt{u^2 + u^2} = u\sqrt{2}$. Why distractors fail: Option A ($u$) omits the contribution from the $z$-component, essentially missing that the cone is tilted. Option C ($\sqrt{2}$) drops the factor of $u$ from the cross product. Option D ($2u$) incorrectly simplifies $\sqrt{2u^2}$.