(20 points) Parts (a) and (b) are distinct from each other….
(20 points) Parts (a) and (b) are distinct from each other. (a) Use Stokes’ Theorem to evaluate \(\displaystyle \iint_S \text{curl} \vec{G} \cdot d\vec{S}\), where \(\vec{G}(x,y,z)= \langle yz, xy, xz \rangle\) and \(S\) is the part of the sphere \(x^2+y^2+z^2=4\) with \(x \geq 0\) oriented in the direction of the positive x-axis that lies inside the cylinder \(y^2+z^2=1\). (b) Consider the solid region E bounded below by the cone \(z=\sqrt{x^2+y^2}\) and above by the paraboloid \(z=2-(x^2+y^2)\). Use the Divergence Theorem and cylindrical coordinates to evaluate \(\displaystyle \iint_S \vec{F} \cdot d\vec{S}\), where \(\vec{F} (x,y,z)= (5x\sin^2(z)+\arctan(y))\vec{i}+(5y\cos^2(z)-\ln(1+x^2))\vec{j}+(7z-e^{x-y})\vec{k}\), and S is the boundary of the region E. PLEASE READ THE INSTRUCTIONS BELOW CAREFULLY: When you have completed your entire exam do not hit “Submit” to submit the exam on Canvas yet, get your phone and scan your written work for questions 6, 7 and 8 (only) to upload to Gradescope www.gradescope.com or use the gradescope app. Make sure that all pages have been uploaded to Gradescope. Once you’re done scanning and uploading to Gradescope, then submit the exam on Canvas. If you experience any difficulties uploading your solutions to Gradescope, only then submit a single .pdf file in the space provided below. Do not close the current Honorlock session until you have submitted your solutions to Gradescope and then submitted the exam on Canvas. Submitting any solutions to Gradescope or Canvas after the Honorlock proctoring session has ended constitutes a violation of the UMass Academic Honesty policy and will be dealt with accordingly.
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