Question

Q4 only:

Question 3. Consider the region of R3 given by V is bounded by three surfaces. Si is a disc of radius 1 in the plane z -0. S3 is a disc of radius 2 in the plane z 3 and a) Make a clear sketch of V. (Hint: You could consider the cross-section of S2 with y-0, and then use the circular symmetry. (b) Express V in cylindrical coordinates. (c) Calculate the volume of V, working in cylindrical coordinates Question 4. Consider the vector field F(r) - -yi+rj+zk. We still consider V and its bounding surfaces, as defined in Question 3 above (a) What is the flux of F through S1? (No integration is required.) (b) What is the flux of F through S3? (No integration is required.) (c) Find the divergence of F, and then find . .V.FdV. (No new integration is required.) (d) Use the divergence theorem and your answers to (a), (b) and (c) to calculate the flux of F through S2 (e) Parametrise the surface S2 by using u θ and v = z as parameters, where θ and are standard cylindrical coordinates. That is, give r(u,v) as done for various examples in Section 4.3.1 of the subject noes. (f) Verify your answer to (d) above by explicitly calculating the flux of F through the surface S2, using the parametrisation of part (e)

Answer 1

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