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23S-MATH-32B-LEC-2 Final Exam version B DAWEI ZHANG TOTAL POINTS 175 / 190 QUESTION 1 25 pts 1.1 10 / 10 ✓ + 10 pts Correct + 4 pts $$r=2\cos\theta$$ drawn correctly + 4 pts $$r=2\sin\theta$$ drawn correctly + 1 pts circles labelled correctly + 1 pts axes labelled correctly + 0 pts incorrect or no progress 1.2 15 / 15 ✓ - 0 pts Correct - 2 pts One incorrect bound for $$r$$ OR correct bounds, but swapped - 4 pts Both incorrect bounds for $$r$$ - 2 pts One incorrect bound for $$\theta$$ - 4 pts Both incorrect bounds for $$\theta$$ - 2 pts Didn't multiply integrand by $$r$$ OR incorrect integrand - 1 pts Computational/algebraic/transcription error while integrating - 2 pts Conceptual/calculus error while integrating OR while using a geometric argument to get the final answer - 3 pts made partial progress calculating the integral but didn't arrive at a final answer - 13 pts Some correct stuff written, but little correct progress - 15 pts no progress - 5 pts set up but didn't solve integral QUESTION 2 15 pts 2.1 0 / 0 ✓ + 10 pts curved - "no harm" question - 0 pts Correct - 2 pts Mostly correct - 8 pts Mostly incorrect ✓ - 10 pts Incorrect - 1 pts Minor error - 4 pts Missing bounds - 3 pts Incorrect illustration - 5 pts Half correct 2.2 15 / 15 ✓ - 0 pts Correct - 4 pts Incorrect determinant - 3 pts Errors in integral calculation - 11 pts Correct Jacobian area formula, but otherwise incorrect - 13.5 pts Assumes the transformation is linear - 15 pts Incorrect or missing - 3 pts Incorrect matrix of partials - 1.5 pts Minor computational error - 4 pts Correct given (a), but problem has been

significantly simplified due to errors in (a) - 1.5 pts A bit more needed - 6 pts Missing integration - 11 pts Mostly incorrect QUESTION 3 25 pts 3.1 10 / 10 ✓ - 0 pts Correct - 1 pts Incorrect or missing axis labels - 1.5 pts Missing or incorrect shading - 1.5 pts Incorrect boundary points/lines - 6 pts Incorrect region - 2 pts $$x=\sqrt{y}$$ curve a little inaccurate 3.2 15 / 15 ✓ - 0 pts Correct - 1 pts Minor computational error - 5 pts Incorrect $$u$$-substitution - 15 pts Incorrect - 10 pts Major computational errors - 3 pts Incorrect bounds - 4 pts Several small errors - 1 pts More simplifying needed QUESTION 4 25 pts 4.1 15 / 15 ✓ - 0 pts Correct - 2 pts Minor error in parametrization - 4 pts Major error in parametrization - 5 pts Set up but didn't solve integral - 2 pts Computational error during integration or computing $$\mathbf{F}(\mathbf{r}(t))\cdot \mathbf{N}(t)$$ - 5 pts Found the work of $$\mathbf{F}$$ along $$C$$ instead of the flux - 12 pts Some relevant work, but little correct progress - 1 pts Minor error in formula for normal vector - 2 pts Major error in formula for normal vector - 4 pts Major error in formula for flux 4.2 10 / 10 ✓ - 0 pts Correct - 4 pts Asserted $$\mathbf{F}$$ is not conservative by showing (incorrectly) that $$\frac{\partial F_1}{\partial y} = \frac{\partial F_2}{\partial x}$$ or by trying to show that $$\mathbf{F}$$ is not path-independent - 6 pts Asserted $$\mathbf{F}$$ is conservative, but didn't find a potential function or had some conceptual misunderstanding of what a potential function is - 2 pts computational error in potential function - 10 pts no progress - 8 pts Asserted $$\mathbf{F}$$ is not conservative with an incorrect justification, e.g. by showing that the curl is zero QUESTION 5 25 pts 5.1 9 / 9 ✓ + 9 pts Correct ($$\bm{F}$$ is not conservative, with correct justification) + 7 pts Claimed that the work to move from P to Q is different along different paths, but did not

provide justification (_e.g, curves not oriented; did not explain conclusions about line integrals_). + 4 pts Described a way to determine if a vector field is conservative (level curves) + 3 pts Described a way to determine if a vector field is conservative (path independence) + 3 pts Described a way to determine if a vector field is conservative (curl of F = 0 AND simply connected) + 0 pts Incorrect or invalid explanation 5.2 3 / 8 + 8 pts Correct (zero, with correct reasoning about flux line integrals) + 6 pts Described geometric interpretation of div, but did not justify conclusions stated (_e.g, curves not oriented; did not explain conclusions about flux line integrals_). ✓ + 3 pts Described geometric interpretation of div, but incorrect justifications/conclusions + 3 pts Described a formula for div, but incorrect justifications/conclusions + 0 pts Incorrect or invalid reasoning 5.3 8 / 8 ✓ + 8 pts Correct (Positive, with correct reasoning about vector line integrals) + 6 pts Described geometric interpretation of curl, but did not justify conclusions stated (_e.g, curves not oriented; did not explain conclusions about vector line integrals_). + 4 pts Used fundamental theorem of conservative vector fields, consistent with answer in part (A) + 3 pts Described geometric interpretation of curl, but incorrect justifications/conclusions + 3 pts Described a formula for curl, but incorrect justifications/conclusions + 0 pts Incorrect or invalid reasoning QUESTION 6 25 pts 6.1 11 / 15 - 0 pts Correct - 2 pts Minor mistake in parametrizing the hemisphere (including bounds of integration) - 4 pts Error in parametrizing the hemisphere (e.g. found the wrong hemisphere, as indicated by the bounds of integration, or parametrization does not trace out a part of a sphere). If using the divergence theorem, it does not matter which hemisphere one integrates over as the result only depends on the volume of the hemisphere, but in order to receive credit there must be an indication of this observation. ✓ - 2 pts Error in finding normal vector - 4 pts Major error in computing normal vector to the hemisphere or normal vector is missing altogether. ✓ - 2 pts Mistake in computing integral - 4 pts Set up the integral for flux calculation / divergence but did not compute or computed incorrectly. - 6 pts Divergence theorem: did not use closed surface - 4 pts Divergence theorem: did not subtract the flux across the disk / other chosen surface

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bounding the hemisphere (this flux turns out to be zero, but awareness of the fact that this flux also had to be calculated to correctly apply the divergence theorem must be demonstrated for a fully correct approach using this theorem). - 15 pts Incorrect or no attempt made. - 2 pts The divergence of $$\mathbf{F}$$ was computed incorrectly 6.2 10 / 10 ✓ - 0 pts Correct - 3 pts Divergence theorem: did not use closed surface. - 2 pts Divergence theorem: added a disk to the cone to close the surface, but computed the flux across the disk incorrectly - 1 pts Incorrect orientation (normals need to point outward, i.e. in the negative $$y$$ direction for the cone, so if using the divergence theorem, for example, it is not necessary to multiply the triple integral obtained through the divergence theorem by $$-1$$. - 1 pts Divergence theorem: incorrect divergence for $$\mathbf{F}$$ (no double jeopardy from part a), if used) - 1 pts Minor mistake in parametrizing / describing the cone. - 3 pts Incorrect parametrization of cone (or description of cone, if using divergence theorem) - 2 pts Calculating flux directly: mistake in normal vector or normal vector missing - 2 pts Mistake in integral calculation or calculation incomplete - 10 pts Incorrect or no attempt made. - 1 pts Computational error in finding volume of cone. QUESTION 7 25 pts 7.1 10 / 10 ✓ - 0 pts Correct - 2 pts Minor mistake in $$x$$-component of the curl - 2 pts Minor mistake in $$y$$-component of the curl - 2 pts Minor mistake in $$z$$-component of the curl - 3 pts Major mistake in $$x$$-component of the curl or missing - 3 pts Major mistake in $$y$$-component of the curl or missing - 3 pts Major mistake in $$z$$-component of the curl or missing - 10 pts Incorrect or no attempt made. 7.2 15 / 15 ✓ - 0 pts Correct. Full credit even if the orientation on $$C$$ was not specified (it's not necessary to get the correct answer but to correctly apply Stokes' theorem it is necessary to be clear about which orientation one is choosing). - 1 pts Correctly applied Stokes' theorem but mistake from part a) made the answer ambiguous (the integrand is zero) - 3 pts Applied Stokes' theorem but made minor mistake in computing normal vector to the plane - 5 pts Applied Stokes' theorem but made major mistake in obtaining normal vector to the plane

- 2 pts Applied Stokes' theorem, set up the surface integral, but did not compute or computed incorrectly. - 12 pts Did not apply Stokes' theorem, but used a specific curve $$C$$ (the reasoning should apply to any simple closed curve $$C$$ in the plane) - 12 pts Used Green's theorem instead of Stokes': this reasoning is incorrect because the curve is not in the xy-plane. The _idea_ behind Green's theorem should lead to a solution, since the curve is contained in a plane, but to make the necessary modifications to the original form of the theorem is the same as applying Stokes' theorem. - 5 pts Applied Stokes' theorem incorrectly by finding the flux of $$\mathbf{F}$$ across the part of the plane bounded by $$C$$. - 15 pts Incorrect or no attempt made QUESTION 8 25 pts 8.1 5 / 5 ✓ + 5 pts Correct (E) + 0 pts Incorrect 8.2 5 / 5 ✓ + 5 pts Correct (A) + 0 pts Incorrect 8.3 5 / 5 ✓ + 5 pts Correct (C) + 2 pts Chose "Not sure" + 0 pts Incorrect 8.4 4 / 10 + 10 pts Correct (2x-3y+z=0) + 8 pts Identified that curl(F) should be the normal vector ✓ + 4 pts identified that $$curl(F) \cdot n$$ should be maximized + 0 pts Incorrect - 1 pts did not give equation of a plane - 1 pts plane does not pass through origin + 0 pts incorrect work or no work shown Page 5

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