Let S = [0, 1] x [0, 1] be the unit square in R2, and C...

   

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Let S = [0, 1] x [0, 1] be the unit square in R2, and C [0, 1] x [0, 1] x [0, 1] be the unit cube in R'. For each map below, do the following: (i) Draw the image of the map, either in the ry-plane or in ryz-space; (ii) Calculate the Jacobian determinant of the map; (iii) Integrate to calculate the volume of the image of each map. (a) TpSR², where T,(r, 0) = (r cos 270, r sin 270). (b) TC R, where T.(r, 0, 2) = (rcos 270, r sin 27e, z). (c) T:C + R3, where T,(r, 0, 4) = (r cos 2n0 sin rp, r sin 2O sin np, r cos TY). (d) T:S - R?, where Te(u, v) = (2u – v, u - 3u). (e) T;:S + R', where T,(u, v) = (u? – v2, 2uv). (Change of variables, multiple integration, geometry of planar maps) Let S = [0, 1] x [0, 1] be the unit square in R2, and C [0, 1] x [0, 1] x [0, 1] be the unit cube in R'. For each map below, do the following: (i) Draw the image of the map, either in the ry-plane or in ryz-space; (ii) Calculate the Jacobian determinant of the map; (iii) Integrate to calculate the volume of the image of each map. (a) TpSR², where T,(r, 0) = (r cos 270, r sin 270). (b) TC R, where T.(r, 0, 2) = (rcos 270, r sin 27e, z). (c) T:C + R3, where T,(r, 0, 4) = (r cos 2n0 sin rp, r sin 2O sin np, r cos TY). (d) T:S - R?, where Te(u, v) = (2u – v, u - 3u). (e) T;:S + R', where T,(u, v) = (u? – v2, 2uv). (Change of variables, multiple integration, geometry of planar maps)