Concordia University
Faculty of Engineering and Computer Science
Applied Advanced Calculus – ENGR233/4-J
2020 Final Examination
• Professor: Dmitry Korotkin
• Start: 14-00. Finish: 17-00
April 17, 2020
Important points
• Total marks: 118. Evaluation out of 100. The exam implies 18% bonus. All questions should be answered.
• There are 14 problems. Some problems consist of several parts; mark related to each part is given in brackets.
• There is an answer sheet attached to this exam. Please write your answer to each question in the answer sheet.
• You can also provide your detailed solutions of up to three questions as optional supple- mentary attachments.
• If you choose to submit optional solutions to three questions, solution to each question should be submitted as one separate file
• Your mandatory answer sheet and your optional supplementary attachments should be sent as one email with up to four attachments to
and copy to
engr233.j@gmail.com
dmitry.korotkin@concordia.ca
The answer sheet can be filled and sent in any electronic format. Alternatively, if can be printed, filled and photographed/scanned.
• Students are given an additional 20 minutes administration time from 17:00 to 17:20 to prepare their email attachments and submit their exam.
• Exams that are received after 17:21 PM will not be marked.
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Problems
1. The line l1 has the direction vector ⟨1, 0, −1⟩ and passes through the point (0, −1, −1). The line l2 passes through the points (1, 2, 3) and (1, 3, 2).
a. [2] What is the angle between l1 and l2 in radians? The answer should lie between 0 and π/2. b. [6] What is the distance between l1 and l2?
2. Suppose r(t) = t2i + (t3 − t)j + (t − 1)2k is the position vector of a moving particle. a. [2] At what point does the particle pass through the xy-plane?
b. [2] What is its velocity vector at this point?
c. [4] What is the radius of curvature of the trajectory at this point?
3. ([8]) Find the point on the surface z = x2 + 2y2 where the tangent plane is orthogonal to the line connecting the points (3, 0, 1) and (1, 4, 0).
4. Considertheplanecurvey= x3 for0≤x<∞. 45
a. [4] Find the x-coordinate of the point where the curvature of the curve is minimal.
b. [4] Find the x-coordinate of the point where the curvature of the curve is maximal.
Useful formula: The curvature of the plane curve y = f(x) is given by κ(x) = |f′′|(1 + f′2)−3/2.
5. The temperature T at a point (x,y,z) in space is inversely proportional to the square of the distance from (x, y, z) to the origin. It is known that T (0, 0, 1) = 500.
a. [2] Compute T (2, 0, 0).
b. [3] Find the rate of change of T at the point (2, 3, 3) in the direction of the point (3, 1, 1). c. [3] What is the maximal rate of change of T at the point (2, 3, 3)?
6. ([8])FindtheworkdonebytheforcefieldF(x,y,z)=yzi+xzj+xykactingalongthecurve given by r(t) = t3 i + t2 j + t k from the point (1, 1, 1) to the point (8, 4, 2).
7. Consider the sum of double integrals
1 x
√ √ xydydx+
√√
√
2x 1 0
coordinates. Give the range for the radius r. b. [4] Compute the integral.
8. Consider the surface which is the part of the cone z = x2 + y2 that lies between the plane y = x and the parabolic cylinder y = x2.
a. [2] Represent the scalar surface element dS in the form a dx dy and find a. b. [6] Compute the area of the surface.
2 4−x2 xydydx+ √ xydydx.
1/2 1−x2
a. [4] Combine into one integral and describe the domain of integration in terms of polar
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9. Consider the integral xy dx + x2 dy where C is the boundary of the region bounded by x = 0, C
x=y,x2+y2 =64,x,y≥0,takeninclockwisedirection.
a. [3] Apply the Green’s theorem and rewrite the obtained double integral in polar coordinates.
Give the range for the polar angle θ.
b. [6] Compute the original line integral using Green’s theorem.
F · dr where F = yi − xj + z3k where C is the triangle with vertices a. [2] Represent the equation of the plane containing the triangle in the form z = ax + by + c
10. Consider the integral
(0, 0, 3), (1, 1, 4), (2, 0, 0), oriented counterclockwise if viewed from above.
C
and find a, b and c.
b. [2] Compute curl F.
c. [4] Compute the original line integral using Stokes’ theorem.
11. ([9]) Use spherical coordinates to find the volume of the solid situated below x2 + y2 + z2 = 1 and above z = x2 + y2 and lying in the first octant.
12. Consider the inward flux (F·n)dS of the vector field F = y2i+xz3j+z2k where S is the S
surface of the region D bounded by the cylinder x2 +y2 = 16 and the planes z = 1, z = 5, √
x= 3y,y=0,x,y≥0.
a. [2] Compute the divergence of the vector field F at the point (1, 1, −1).
b. [7] Transform the surface integral into the triple integral using the divergence theorem and evaluate.
13. Consider the following line integral of the conservative vector field:
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(y sinz−z)dx+2xysinzdy+(xy cosz−x)dz
C
where C is the contour given by r(t) = ⟨t3, 2t2 − 1, πt⟩, 0 ≤ t ≤ 1/2.
a. [4] Find the potential f of the vector field satisfying the condition f (1, 1, 0) = 0. b. [5] Compute the line integral.
14. a. [2] Compute the divergence of vector field F = x3y2i + yj − 3zx2y2k
b. [7] Use divergence theorem to compute the outward flux of the vector field F through the
surface of the solid bounded by the surfaces z = x2 + y2 and z = 2y.
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