School of Mathematics and Statistics MAST20009 Vector Calculus, Semester 2 2019 Assignment 3 and Cover Sheet Student Name Student Number Tutor’s Name Tutorial Day/Time Submit your assignment to your tutor’s MAST20009 assignment box before 11am on Tuesday 8th of October. • This assignment is worth 5% of your final MAST20009 mark. • Assignments must be neatly handwritten in blue or black pen on A4 paper or typed using Lateχ. For Lateχ assignments, also email a copy of the source code to [email protected]. • Diagrams can be drawn in colour on grid paper (use ruler and compass). Tikz pictures are acceptable as long as they include a grid. • You must complete the plagiarism declaration on the LMS before submitting your assignment. • Full working must be shown in your solutions. • Marks will be deducted for incomplete working, insufficient justification of steps, incorrect mathematical notation and for messy presentation of solutions. The skill practiced in this Assignment is the mathematical modelling of a problem. Do not be dis- heartened if at a first read the questions seem vague and weird. You are the one who will make sense of them. This involves very carefully reading the questions, asking which coordinate system offers itself for which question, possibly sorting out some irrelevant information, if necessary, breaking your task into simpler steps. Explain your thought process very carefully. Since we are missing our Friday class this week and with it the best opportunity to ask questions, I will accept email questions until Friday 5pm and put my responses on the LMS for all to see. It is fine to use Wikipedia and Wookiepedia, where required. 1. The death star is being audited by the occupational health and safety team. After extensive mea- surements, the team becomes concerned, because they detect a Force field emanating from the 120 km diameter battle station. Strangely the Force is strongest in the trash compactor, on the sixth subfloor. The Empirial Auditors determine ~F (x, y, z) = 1 x2 + y2 + (z + 3)2 11 1 . They have parametrized the star such that the origin is in the center and scaled their picture such that one unit length in the picture corresponds to 10km. Viewing the Force as the velocity of a cosmic energy flow, the question is now what is the net quantity of cosmic energy to flow accross the surface of the star per unit time in the direction of the outward normal. 2. The rebel force is pouring over the intelligence they have received about the death star and are trying to figure out its orbit around Endor. They have drawn a graph of the orbital plane. They know that the orbit is an ellipse of width 100,000 km and hight 60,000 km. (a) Using the scale where 1 space unit is 10,000 km, draw this orbit on grid paper, putting the centre at the origin, the major axis along the x-axis and the minor axis along the y-axis. Compute the focal points and mark them. Let’s say that Endor sits on the focal point on the negative x-axis. (b) Write down the anti-clockwise parametrization of this ellipse that we found in the notes, using the specific values for a and b that you can figure out from the information above. To fix notation, let’s say the result is a path ~c(θ) in R2. To avoid giving the markers a headache, please do set this up exactly like in the notes. (c) Parametrize the surface swept out by the ray from Endor to the death star in the time interval [θ0, θ1] and compute its surface area A. A Hint: The answer to the first question should be a function Φ : [0, 1]× [θ0, θ1] −→ R2[ r θ ] 7−→ [ ? ? ] . You will likely want to find Φ in several steps: (i) Consider the easier scenario where the ray starts at the origin. Call the resulting parametrization Ψ. (ii) Draw the grid you obtain for Ψ when fixing values of r or θ. (iii) Play around with the equation for the rays Ψ(r, θfixed) for fixed θ and figure out what you need to adjust to turn these into rays starting at Endor. (iv) Now you should be in good shape to figure out how to adjust Ψ to obtain Φ. After this, you still need to compute the area. For this you will need to find the Jacobian of the relavant derivative matrix. (d) Reparametrize your path ~c such that at any time t the ray from the previous question has swept out a surface of area t. To keep things simple, let us ignore the question of what our time unit is. To fix notation, let us call the new path ~γ(t). (e) Using whichever method you like, determine the point of time at which ~γ is closest to Endor.
