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MAT 267 – Calculus for Engineers III - Spring 2020 Due Date: Format: Note: Exam must be submitted by Wednesday, May 6, 2020 at 11:59pm. Please note: If you need more time, I can grant time up until Friday, May 8, 2020 at 5:00pm. One single PDF file Fine details listed in canvas. Show work on all free response questions below. Showing your work is important to receive credit. Any problems email me immediately. Good luck! 1. 2. Match the vector fields with the plots labeled A - D. Field A–D Field 𝐅(π‘₯, 𝑦) = 〈0, 𝑦βŒͺ 𝐅(π‘₯, 𝑦) = βŒ©π‘¦ 3 , βˆ’π‘₯βŒͺ 𝐅(π‘₯, 𝑦) = βŒ©βˆ’π‘₯, 𝑦 3 βŒͺ 𝐅(π‘₯, 𝑦) = 〈π‘₯, βˆ’π‘¦ 3 βŒͺ Consider the given vector field: A–D 𝐅(π‘₯, 𝑦, 𝑧) = 〈π‘₯ βˆ’ 𝑦𝑧, 𝑦 + π‘₯𝑧, 𝑒 π‘₯𝑦 + 𝑧 2 βŒͺ (a) Find the divergence of the vector field. (b) Find the curl of the vector field. (c) Multiple Choice: Does there exist a function 𝑓 with 𝐅 = βˆ‡π‘“? Circle your answer. A. Yes B. No C. We do not have enough information. 3. Here are plots of five vector fields on the box where 0 ≀ π‘₯ ≀ 1, 0 ≀ 𝑦 ≀ 1 and 0 ≀ 𝑧 ≀ 1 . For each part, circle the best answer. (a) Circle the letter of the vector field that is 𝐅(x, y, z) = βŒ©βˆ’π‘§, 0, 1 + π‘₯βŒͺ: A B C D E (b) Exactly one of these vector fields has nonzero divergence. Circle it: A B C D E (c) The vector field C is conservative: 4. TRUE or FALSE Define 𝐹⃑ (π‘₯, 𝑦, 𝑧) as a vector field and 𝑓(π‘₯, 𝑦, 𝑧) as a scalar field in space. Determine if the expression is defined, a scalar field or vector field. Check with a β€œX” mark to which applies. Formula curl(div(grad(f))) βƒ— ))) grad(div(curl(F βƒ— ))) grad(curl(div(F div(grad(curl(f))) Not Defined Scalar Vector 5. Consider the function 𝑓 (π‘₯, 𝑦) on the rectangle 𝐷 = {0 ≀ π‘₯ ≀ 4 π‘Žπ‘›π‘‘ 0 ≀ 𝑦 ≀ 2} whose graph is shown below. Think about what the contour map looks like for this surface when answering the following questions. Multiple Choice: For each part, circle the best answer. (a) πœ•π‘“ At the point 𝑃 = (1, 0.5) is πœ•π‘¦: A. negative (b) B. zero C. positive How many critical points does 𝑓 have in the interior of D? 0 (c) 1 2 B. zero C. positive For the curve C shown, the line integral ∫𝐢 𝛁𝑓 βˆ™ 𝑑𝒓 is : A. βˆ’3 (e) 4 The integral ∬𝐷 𝑓(π‘₯, 𝑦)𝑑𝐴 is: A. negative (d) 3 B. βˆ’1.5 C. 0 D. 1.5 E. 3 Mark the plot below of the gradient vector field 𝛁𝑓. 6. Consider a conservative vector field 𝐅(π‘₯, 𝑦, 𝑧) = (3𝑦 2 )𝐒 + (6π‘₯𝑦 + 3𝑒 3𝑧 )𝐣 + (9𝑦𝑒 3𝑧 )𝐀 a) How would you verify that 𝐅 is a conservative vector field? Explanation only. No need for calculations. b) Find a potential function for 𝐅 such that 𝐅 = βˆ‡π‘“. c) Find the work done by the force field 𝐅 in moving the object from point (0,0,0) to point (1,1,1) along each of the following paths. Provide setup and solve for EXACT ANSWER. i. Path 1 is a straight-line segment from (0,0,0) to (1,1,1). ii. (extra credit) Path 2 consists of three-line segments, the first from (0,0,0) to (1,0,0), the second from (1,0,0) to (1,1,0), and the third from (1,1,0) to (1,1,1). iii. (extra credit) Path 3 is the curve βŒ©π‘‘, 𝑑 2 . 𝑑 3 βŒͺ over the time interval 0 ≀ 𝑑 ≀ 1. 7. Convert the following triple integral to spherical coordinates and solve. Show work for each integration step. Exact answer only. 4 √16βˆ’π‘₯ 2 ∫ ∫ βˆ’4 0 √16βˆ’π‘₯ 2 βˆ’π‘¦ 2 ∫ 0 3 (π‘₯ 2 + 𝑦 2 + 𝑧 2 )2 𝑑𝑧 𝑑𝑦 𝑑π‘₯ ...
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