School of Mathematics and Statistics MAST20009 Vector Calculus, Semester 2 2019 Assignment 4 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 22nd 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. 1. Consider the domain D ⊂ R2 obtained by removing the unit circle from the square centred at the origin with side length 4. Equip D with the positive orientation. (a) Draw D and indicate clearly in your drawing the orientation of D and the compatible orientation on each of the boundary components. (b) Let now ~F be the vector field ~F [ x y ] = [ x− y xy ] on R2. Verify Green’s theorem for ~F on D. 2. Let T be the torus, parametrized as in the notes, with R = 3. Consider the function f(x, y, z) = 2x− y on R3. (a) Write down the parametrization of T from the notes. (b) Produce a high quality drawing (following specifications) of the torus T and the grid you obtain when fixing values of ϕ or θ, depicting at least five lines in either direction of the grid. (c) Compute the tangent vectors ~Tϕ and ~Tθ and the outward normal vector ~n. (d) Compute the surface integral ∫∫ T f dS. 3. Let ~F be the vector field ~F [ x y ] = [−y y ] on R2, and let c be the semi-circle of radius 1 in the x − y-plane centred at (2, 0) and moving in anti- clockwise direction from (2,−1) to (2, 1). (a) Determine the work done by ~F to move a particle along c. (b) Compute the curl and the divergence of ~F .
