# Vector Calculus

Math 20E Requirement Fulfillment Exam Practice Exam Note: This is a practice exam. The actual exam may differ in terms of difficulty and the topics covered. Instructions: Please read the instructions carefully. • You have only one chance to take the Math 20E Requirement Fulfillment Exam! Do not take this exam if you are not prepared! • Write your name above. • Please write your answers in this booklet. • You must show your work to get full credit. • No calculators or notes are allowed. • You have 40 minutes to complete the exam. • Each problem is worth 10 points. 1 1. Consider the vector field F = 2xy i + (x2 + z) j + y k. (a) Find a function f = f (x, y, z) such that F = gradf . (b) Find ∇ × F (i.e. the curl of F.) (c) Using your answer to (a), calculate the line integral Z F · dR, C π where C is the curve given by R = 2t i − 3 sin t j + t2 k, for 2 0 ≤ t ≤ 1. 2 2. Let F be the vector field F = 3x i + xz 3 j − ey k. Let B be the solid unit ball x2 + y 2 + z 2 ≤ 1, and let S be the boundary of B. If n is the outward facing unit normal to S then use the Divergence Theorem to calculate the surface integral ZZ F · n dS. S Hint: you may use the fact that the volume of B is 3 4π . 3 3. Let S be the portion of the paraboloid z = 4 − x2 − y 2 that lies above the plane z = 0 and let F be the vector field F = (z − y)i + (x + z)j − (exyz cos y)k. Using Stoke’s Theorem, find the surface integral ZZ (∇ × F) · n dS, S where n is the unit normal to S (the one that points in the direction of positive z.) 4 Math 20E Requirement Fulfillment Exam Suggested Problems Here are some suggested review problems for the Math 20E Requirement Fulfillment Exam taken from Vector Calculus, 6th Edition, by Jerrold E. Marsden and Anthony J. Tromba. Note that you should not expect the problems on the actual exam to be similar to those listed here. This list is intended to be representative rather than exhaustive, and may not include examples of all possible problems that could be encountered on the exam. • Section 2.4. Problems: 13, 19 • Section 2.6. Problems: 1, 9(c), 13 • Section 3.1. Problems: 15 • Section 3.3. Problems: 11 • Section 4.2. Problems: 13 • Section 4.4. Problems: 3, 7, 15, 33 • Section 5.2. Problems: 7 • Section 5.3. Problems: 3 • Section 5.4. Problems: 3 • Section 5.5. Problems: 11, 13 • Section 6.2. Problems: 3, 5, 7, 15, 23 • Section 7.1. Problems: 10 • Section 7.2. Problems: 3, 17 • Section 7.3. Problems: 3, 15 • Section 7.4. Problems: 1, 17, 19 • Section 7.5. Problems: 13 • Section 7.6. Problems: 9, 13 • Section 8.1. Problems: 11, 13, 21 • Section 8.2. Problems: 11, 13, 15, 17, 19 • Section 8.3. Problems: 1, 7, 11, 13, 19 • Section 8.4. Problems: 5, 7, 9, 11, 15 Math 20E Requirement Fulfillment Exam Suggested Problems Here are some suggested review problems for the Math 20E Requirement Fulfillment Exam taken from Vector Calculus, 5th Edition, by Jerrold E. Marsden and Anthony J. Tromba. Note that you should not expect the problems on the actual exam to be similar to those listed here. This list is intended to be representative rather than exhaustive, and may not include examples of all possible problems that could be encountered on the exam. • Section 2.4. Problems: 11, 17 • Section 2.6. Problems: 1, 5(c), 9 • Section 3.1. Problems: 9 • Section 3.3. Problems: 11 • Section 4.2. Problems: 9 • Section 4.4. Problems: 3, 7, 15, 25 • Section 5.2. Problems: 3 • Section 5.3. Problems: 1 • Section 5.4. Problems: 1 • Section 5.5. Problems: 9, 11 • Section 6.2. Problems: 1, 3, 5, 13, 21 • Section 7.1. Problems: 3 • Section 7.2. Problems: 1, 15 • Section 7.3. Problems: 3, 11 • Section 7.4. Problems: 1, 13, 15 • Section 7.5. Problems: 7 • Section 7.6. Problems: 5, 9 • Section 8.1. Problems: 3, 5, 13 • Section 8.2. Problems: 3, 5, 7, 9, 11 • Section 8.3. Problems: 3, 7, 9, 13, 15 • Section 8.4. Problems: 1, 3, 5, 7, 9 4:06 4 ASSO < … Hello, If you are receiving this email, you are registered for the remote Math 20E Requirement Fulfillment Exam on Monday, March 29, 2021. You should have already received an email that you were added to the Gradescope account for the exam. This is where the exam will be made available and where you will submit the exam. On Monday morning (10 AM), the exam will be made available on Gradescope. You will have approximately one hour to complete the exam. You will be required to submit the exam via Gradescope before the submission deadline on Monday morning. You will have from 10:00 AM to 11:00 AM to complete and upload the exam. Detailed instructions about Gradescope submission can be found in the attached document below. If you experience any technical difficulties with uploading to Gradescope, you should email the exam files to frdesk@math.ucsd.edu before the submission deadline in order for your exam to be considered. Please continue to check your UCSD email address, as any future correspondence about the exam will be sent to that email address. Lastly, if you have not already done so, please submit the petition to us via email (mathadvising@math.ucsd.edu) before the exam. If you have any questions that were not answered in the below document, please let us know. Sincerely, Front Desk Staff UCSD Department of TVIACHEMatts 4:079 X Math 20E Exam Instructio… 4 Û Please read the following instructions carefully. The Math 20E Fulfillment Examination will be administered as follows: You should receive an email notification from Gradescope that you have been added to the Gradescope account named “Math20E FulfillmentExam.” If you have not been added to this by Monday morning, please let us know via email to . A copy of the exam will be uploaded to Gradescope by 10:00 am on Monday, March 29th, 2021. You will write your solutions on your own paper or iPad/tablet in an organized way. If you are able to write your solution on the blank exam pdf using your iPad/tablet or on a printed copy of the blank exam, you are encouraged to do so, but it is not required. When you are done, you will scan or take pictures of (or export to pdf from your iPad/tablet) your completed exam and upload it to the “Fulfillment Exam” assignment on Gradescope. Instructions for uploading an assignment (exam) to Gradescope can be found at https://www.gradescope.com/help#help-center-item-student-scanning and https://gradescope-static-assets.s3.amazonaws.com/help/submitting_hw_guide.pdf. Please make sure your files are legible before submitting and carefully follow the Gradescope instructions to match every question to the appropriate pages of your submission. Unreadable solutions will not be graded. You will be given until 11:00 am Monday to upload your exam solution, with a 15-minute late window (with no penalty). If your exam solution is not uploaded to the Fulfillment Exam assignment on Gradescope by 11:00am on Monday, your exam will not be accepted. If you are experiencing technical difficulties with uploading your exam to Gradescope by the above deadline, send us your pictures or scan via email to by the deadline and, if necessary, we will upload them for you. 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