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i) Using the change of coordinat.es from r, y, z to u, v, uw cxpress the volume of the tetrahedron T as a triple integral in the coordinates u, v, w. Evaluate this integral and verify that the volume of T is ii) The tetrahedron T has a density function 8(r, y, 2) same coordinate transformation, fnd the z-coordinate of the contre of mass of the tetrahedron T (by symmetry of the shape and the density, the r and y coordinates of the centre of mass are zero: you do not need to prove this). iii) Find the outward flux of the vector ficld 2/2 31 #+ Using the F(r, y, 2)- ( 2y, 6 sin z + y, 7r – 2rz) 2y, 6 sin 2 + y, 7r 2r2) across the surface S of the tetrahedron T. In other words, evaluate the integral F dS.

i) Using the change of coordinat.es from r, y, z to u, v, uw cxpress the volume of the tetrahedron T as a triple integral in the coordinates u, v, w. Evaluate this integral and verify that the volume of T is ii) The tetrahedron T has a density function 8(r, y, 2) same coordinate transformation, fnd the z-coordinate of the contre of mass of the tetrahedron T (by symmetry of the shape and the density, the r and y coordinates of the centre of mass are zero: you do not need to prove this). iii) Find the outward flux of the vector ficld 2/2 31 #+ Using the F(r, y, 2)- ( 2y, 6 sin z + y, 7r – 2rz) 2y, 6 sin 2 + y, 7r 2r2) across the surface S of the tetrahedron T. In other words, evaluate the integral F dS.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Needed to be solved part b only in 30 minutes and get the thumbs up please show neat and clean work. Please provide correct solution
a)A triple integral /ia written using spherical mordinates an follows, D Sketch the region of integration i) Rewrite the integral / in cartesian coordinatea, uaing the atandard coordi- nate tranaformalion, 2-pandcond, -paine -pcoad. You do not need to evalunte this integral. b) Consider a regular tetrahedron T with vertiecs 1,0, 1,0, 1, The coordinae transformation ((4, 1, w) where (7(4,4, u), (4, v, u), 2(4, u, w)), * w, z(4, , w) bapetively mapa the telrahedron U with the vertices (0,0,0). (1,0,0), (), 1, 0), (0, 0, 1) to the original tetrahedron I (you do not need to prove Lhis). The tetrahedron 7 s shown in (he image below on the keft, and the tetrahedron Us shown n the imge on the right. 集 10.
Transcribed Image Text:a)A triple integral /ia written using spherical mordinates an follows, D Sketch the region of integration i) Rewrite the integral / in cartesian coordinatea, uaing the atandard coordi- nate tranaformalion, 2-pandcond, -paine -pcoad. You do not need to evalunte this integral. b) Consider a regular tetrahedron T with vertiecs 1,0, 1,0, 1, The coordinae transformation ((4, 1, w) where (7(4,4, u), (4, v, u), 2(4, u, w)), * w, z(4, , w) bapetively mapa the telrahedron U with the vertices (0,0,0). (1,0,0), (), 1, 0), (0, 0, 1) to the original tetrahedron I (you do not need to prove Lhis). The tetrahedron 7 s shown in (he image below on the keft, and the tetrahedron Us shown n the imge on the right. 集 10.
i) Using the change of coordinat.es from r, y, z to u, v, w cxpress the volume of the tetrahedron T as a triple integral in the coordinates u, v, w. Evaluate this integral and verify that the volume of T is ii) The tetrahedron T has a density function 8(r, y, 2) same coordinate transformation, fnd the z-coordinate of the contre of mass of the tetrahedron T (by symmetry of the shape and the density, the r and y coordinates of the centre of mass are zero: you do not need to prove this). iii) Find the outward fux of the vector ficld 2/2 #+ Using the F(r, y, 2)- ( – 2rz) 2y, 6 sin z + y, 7r across the surface S of the tetrahedron T. In other words, cvaluate the integral F dS.
Transcribed Image Text:i) Using the change of coordinat.es from r, y, z to u, v, w cxpress the volume of the tetrahedron T as a triple integral in the coordinates u, v, w. Evaluate this integral and verify that the volume of T is ii) The tetrahedron T has a density function 8(r, y, 2) same coordinate transformation, fnd the z-coordinate of the contre of mass of the tetrahedron T (by symmetry of the shape and the density, the r and y coordinates of the centre of mass are zero: you do not need to prove this). iii) Find the outward fux of the vector ficld 2/2 #+ Using the F(r, y, 2)- ( – 2rz) 2y, 6 sin z + y, 7r across the surface S of the tetrahedron T. In other words, cvaluate the integral F dS.
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