3 Parametrization of surfaces and surface integrals The Sydney Opera house is a domed-shape building with many triangular-shaped segments, similar to a series of sweeping curved shells, making up it's unique structure. An engineering maths student, would like to model these segments by designing different shapes for the roof. The Sydney Opera house roof can be model by three surfaces: 01, $2 and ?s (see below). For rainy days, if the wind is given by a velocity vector field F = ri + yj + 5k, then the total flux of water through the "roofs", the surfaces $1, $2 and $3, can be calculated and compared. Note: Flux is the quantity of a "fluid" passing through a surface. "Fluid" is used for lack of a better word because, it can mean a gas, electric field, water or just about anything that flows and can be represented by a vector field. Surfaces to analyse the roof : 1. 0, is defined by the boundary of the region enclosed by the surface a? + y' + 2? = 1 and the planes = = 0, y = 0 and = = 0. 2. 0, is defined by the boundary of the region enclosed by the surface = = 1 - 12 - yz and the plane = = 0, y = 0 and = = 0. 3. ng is defined by the boundary of the region enclosed by the surface = = 1 - vx2 + y' and the plane = = 0, y = 0 and = = 0. In this worksheet you will : a) analyse the parametric representations of the three surfaces and fill the attached table (use may need to show your work on more pages, the table should present only a summary, include all work for full marks). b) evaluate the flux ff F . ndS of the vector field F across the surfaces $21, , and ng from part a). n c) decide which geometry gives the largest flux of the vector field F. d) compare with the true geometry chosen by the architect. On 29 January 1957, Jorn Utzon's sail-like drawings were announced as the winning design for the Sydney Opera House. The construct tion began on 2 March 1959.Surfaces and Sketch of sur- Parametrization of surfaces r(?, ?) N = or du X I F . nds faces 01: 2 +y? + 2? = 1, x 2 0, y 2 0, z 2 0 (use spher- ical coordinates for the parametrization) 02: 2 =1 - (x2 +y'), x 2 0, y 2 0, = 2 0 (use cylin- drical coordinates for the parametrization) 13: 2=-Vx2+y2+1, x 2 0, y 2 0, 2 2 0. (use cylindrical coordi- nates for the parametrica- tion)