Complete this assignment on paper but make sure you send clear photos so I can read the solutions and work on each question.
Course: Multivariable Calculus
MATH2004C Assignment 3 Last Name: First Name: Student ID: • You may either write your answers on a copy of this assignment, or on your own paper or on your electronic devices (you do not need to copy the questions). • Due date: Before 11:59 pm on Tuesday, April 5th. • The assignment has 10 questions and is out of 40 points. Each question is out of 4. • No email submission will be accepted. • No excuses regarding technical issues will be accepted. It is your responsibility to double check that you submitted the right file. Don’t wait until too close to the deadline to start working. • Submission Requirements: Submit your work as only one .pdf document. Files in format different then .pdf or that are not in one document will not be marked and you will obtain 0. Submit your file at the appropriate link on Brightspace. • If you have changes after submission, you can resubmit before the deadline. Only the last submitted file before the deadline will be marked. • In the next page, you must certified that this is your own work by signing at the different places. • You .pdf must be legible. The questions must be in the right order and the files should have the correct orientation • Your file must be the following format: LastName,FirstName/Name of file. For example, if my name is Matt Lemire, I would name my file as: Lemire, Matt Assignment 3.pdf • Show your work: Means that you must show all your steps with justification. • No decimal answers will be accepted. We only want exact and simplified answers in the form of fractions. For example, 0.125 is not accepted but 1 8 would be. An expression of the form 3π − 1 2 would an example of an exact answer. • You can use the Discord forum to write privately to other people in the class regarding answers and work but please do not post any kind of solutions or major hints on the forum. The goal is for you to learn as much as you can your own. It is okay to get help from others as long you understand it yourself eventually. Question 0a. This assignment is open book. I would kindly ask you to do this assignment without just copying down other people answers. I would kindly ask you to promise (code of honour) that you accept the following: I promise not to have someone else doing my assignment. I am allowed to consult textbooks, notes, the internet, some classmates, but I will only do so to help my understanding and not for others to do my work. Signature: (For students who do not write on a printed version of the exam, simply write 0a: and then put your signature.) Question 0b: By signing here, I hereby certify that I have read all the instruc- tions and conditions on the first page and that I will follow them. Signature: (For students who do not write on a printed version of the exam, simply write 0b: and then put your signature.) Question 0c: By signing here, I hereby certify that I understand that I must submit all my work no later than Monday April 4th, no later than 23:59 at the appropriate link on Brightspace. I also know that my work will not be accepted passed that day and time. Signature: (For students who do not write on a printed version of the exam, simply write 0c: and then put your signature.) Important Trigonometric Identities: sin2(x) + cos2(x) = 1 cos2(x) = 1 + cos(2x) 2 sin2(x) = 1− cos(2x) 2 sin(x) sin(y) = 1 2 cos(x−y)− 1 2 cos(x+y) = 1 2 cos(y−x)− 1 2 cos(x+y) = sin(y) sin(x) cos(x) cos(y) = 1 2 cos(x−y)+ 1 2 cos(x+y) = 1 2 cos(y−x)+ 1 2 cos(x+y) = cos(y) sin(x) sin(2x) = 2 sin(x)cos(x) cos(2x) = cos2(x)− sin2(x) 1. Evaluate the surface integral ∫∫ S F · dS in a vector field for the vector field F (x, y, z) = (x2, xy, z) and S is the part of the plane x+ y + z = 2 above the region in the xy-plane given by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 in the xy-plane. Calculate that sur- face integral directly without the use of Stokes’ Theorem. Show all your work. 2. Compute the flux of the vector field F (x, y, z) = (−x,−y, z) across the part of the cone z = √ x2 + y2 between the planes z = 1 and z = 3 (these are just bounds) whose orientation is pointing toward the positive z direction. The surface is not closed since the two planes are not part of the surface. (Do not use Stokes’ Theorem). 3. Use Stokes’ Theorem (and only Stokes’ Theorem) to evaluate ∫∫ S curlF · dS for the vector field F (x, y, z) = (z2, 2x,−y3) and the surface S that is the upper half of the unit sphere x2 + y2 + z2 = 1. (So to be clear, if you want to evaluate this and use Stokes’ Theorem then you must be calculating a certain line integral in a vector field). Show all your work. 4. Let F (x, y, z) = (zey, x cos y, xz). Let S be the hemisphere x2+y2+z2 = 16, y ≥ 0 oriented in the direction of the positive y-axis. Use Stokes’ Theorem (and only that theorem) to compute ∫∫ S curlF · dS. Show all your work. 5. Use Stokes’ Theorem (and only Stokes’ Theorem) to evaluate ∫ C F · dr, where F (x, y, z) = (3y,−2x, 3y) and C is the curve given by x2 + y2 = 9, z = 2. (So to be clear, if you want to evaluate this and use Stokes’ Theorem then you must be calculating the surface integral of the curl of F of a certain surface S.) 6. Use the Divergence Theorem (and only that theorem) to evaluate ∫∫ S F · dS if F (x, y, z) = (xy, yz,−yz) and S is the closed surface given by z = √ 16− x2 − y2 and z = 0. Show all your work. 7. Let S be the surface consisting of the part of the cylinder x2 + y2 = 9 between the planes z = 0 and z = 2. Use the Divergence Theorem (and only that theorem) to compute the flux of the vector field F (x, y, z) = (x, y2, 5z) across the surface S. Show all your work. 8. Evaluate the line integral ∫ C (x+4 √ y) ds, where C is the path going counterclock- wise around the square with vertices (0, 0), (2, 0), (2, 2) and (0, 2). Show all your work. Important: Make sure to realize that we are not talking of a line integral in a vector field here. Also, A regular line integral share the property that line integral in a vector have when C is the union of curves. 9. Find the surface area of the surface S that is the portion of the graph z = x+ y2 where 0 ≤ x ≤ y, 0 ≤ y ≤ 1. Show all your work. 10. Compute the surface integral of f(x, y, z) = x + y + z along the surface S parametrized by r(u, v) = (u+ v, u− v, 1 + 2u+ v), for 0 ≤ u ≤ 2, 0 ≤ v ≤ 1. Show all your work.