(a) Consider a cylinder of height H and radius R centred around the positive z-axis whose...

 

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(a) Consider a cylinder of height H and radius R centred around the positive z-axis whose base is a disk of radius R lying in the x-y plane, with its centre at the origin. Let V denote the three-dimensional region (i.e., volume) represented by this cylinder. (i) Define the vector field F = 3xi + 3yj + 2zk. Working in an appropriate coordinate system, Ishow that the volume integral [][V] where a is a formula in terms of R and H that you should determine. V.FdV=aT [5 marks] (ii) Sketch V, label each of the three parts of the surface that encloses V and write down a unit vector in that points in the outward normal direction to each one. [3 marks] (iii) Verify that the Divergence Theorem is satisfied for the chosen cylinder V and the vector field F defined in part (i). [6 marks] (a) Consider a cylinder of height H and radius R centred around the positive z-axis whose base is a disk of radius R lying in the x-y plane, with its centre at the origin. Let V denote the three-dimensional region (i.e., volume) represented by this cylinder. (i) Define the vector field F = 3xi + 3yj + 2zk. Working in an appropriate coordinate system, Ishow that the volume integral [][V] where a is a formula in terms of R and H that you should determine. V.FdV=aT [5 marks] (ii) Sketch V, label each of the three parts of the surface that encloses V and write down a unit vector in that points in the outward normal direction to each one. [3 marks] (iii) Verify that the Divergence Theorem is satisfied for the chosen cylinder V and the vector field F defined in part (i). [6 marks]