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MATH 2413 Midterm 2 solutions.: Columbia University Department of Mathematics Calculus IV – MIDTERM #1

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Columbia University Department of Mathematics Calculus IV – MIDTERM #1 Instructor: Mike Miller 10/04/2020 Name: No calculator is allowed. Please give exact answers, including ⇡ and p as appr ... opriate; do not write answers with decimals. Show your work; partial credit will be given as appropriate. Distribution of Points Question Points Score Calculus IV MIDTERM #1 10/04/2020 Exam time and submission This exam is available between 9AM and 9PM EST on October 4. You will submit your exam on Courseworks as you would a homework assignment. Submissions will no longer be available after 9PM, and no exceptions will be made to this. You should, therefore, plan to start submitting plenty of time in advance of 9PM, so that you can contact me if there are any unexpected technical issues. I suggest that you stop working by 8PM and start submitting then. Honor code This is an open book, open note exam; the book and the notes on Courseworks are the only authorized material for this exam. In particular, you may not refer to posted homework or practice midterm solutions; you may not use a calculator; you may not use other online references or tools; you may not communicate about the problems with other students. You must sign and submit the following statement1 when submitting your exam. (If you don’t have a printer, copy the statement onto a piece of paper, sign below it, and submit that.) I a!rm that I have not plagiarized nor used unauthorized materials in taking this exam, nor have I given or received illegitimate help with this exam. Signature: 1This statement is modeled on the Columbia College code of honor. Calculus IV MIDTERM #1 10/04/2020 2. (20 points) Compute the line integral RC F • d~r, where C is the the displayed curve (which is made up of semicircles of radius 1 and line segments), oriented counterclockwise, and F = Calculus IV MIDTERM #1 10/04/2020 3. (15 points) Determine whether or not the vector field F = h2xyex2, ex2 % cos z, y sin zi is conservative. If it is not, explain why not; if it is, find a function f(x, y, z) so that F = rf. Page 5 of 9 Calculus IV MIDTERM #1 10/04/2020 4. Let S be the portion of the cylinder x2 + y2 = 1 in the first octant and below the plane z = 1, oriented outwards (in the positive x-direction, or equivalently here, the positive y-direction). (a) (5 points) Parameterize S. Is your parameterization oriented or not? (b) (10 points) If F = h2x + z, 2y % z, tan(z)i, compute RRS F • dA. ~ Page 6 of 9 Calculus IV MIDTERM #1 10/04/2020 5. (15 points) Consider the ‘spherical cap’ S = {(x, y, z) | x2 + y2 + z2 = 2, x ' 1}, oriented in the positive x-direction. If F = h2z2, tan(x2 % x), 2xy2, compute ZZS F • dA. ~ Page 7 of 9 Calculus IV MIDTERM #1 10/04/2020 6. (20 points) Let S be the top hemisphere of the sphere of radius 1, S = {(x, y, z) | x2 + y2 + z2 = 1, z ' 0}, oriented outwards/upwards. If F = curlh2x + z, 2y % z, y2 tan zi, compute ZZS curl F • dA. ~ Calculus IV MIDTERM #1 10/04/2020 [Show More]

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