3. A greenhouse has a glass dome in the shape of the paraboloid z=8-2x²-2y2 and a flat wooden floor at z = 0. Let S be the closed surface formed by the dome and the floor, oriented with outward unit normal. Suppose that the temperature in the greenhouse is given by T(x, y, z) = x²+ y²+3(z− 2)². The temperature gives rise to a heat flux density field F(x, y, z)-kVT - where k is a positive constant that depends on the insulating properties of the medium. Assume that k = 1 on the glass dome and k = 3 on the wooden floor of the greenhouse. (a) Sketch S, clearly labelling any intercepts and the direction of the normal vector. (b) Write down an expression in terms of x, y and z for the vector field F on the greenhouse. (c) By direct calculation (do not use any integral theorems), find the total heat flux F.ndS across the greenhouse in the direction of the outward unit normal.

3. A greenhouse has a glass dome in the shape of the paraboloid z=8-2x²-2y2 and a flat wooden floor at z = 0. Let S be the closed surface formed by the dome and the floor, oriented with outward unit normal. Suppose that the temperature in the greenhouse is given by T(x, y, z) = x²+ y²+3(z− 2)². The temperature gives rise to a heat flux density field F(x, y, z)-kVT - where k is a positive constant that depends on the insulating properties of the medium. Assume that k = 1 on the glass dome and k = 3 on the wooden floor of the greenhouse. (a) Sketch S, clearly labelling any intercepts and the direction of the normal vector. (b) Write down an expression in terms of x, y and z for the vector field F on the greenhouse. (c) By direct calculation (do not use any integral theorems), find the total heat flux F.ndS across the greenhouse in the direction of the outward unit normal.

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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Question

3. A greenhouse has a glass dome in the shape of the paraboloid
z=8-2x²-2y2
and a flat wooden floor at z = 0. Let S be the closed surface formed by the dome and the floor,
oriented with outward unit normal.
Suppose that the temperature in the greenhouse is given by
T(x, y, z) = x²+ y²+3(z − 2)².
The temperature gives rise to a heat flux density field
F(x, y, z) -kVT
=
where k is a positive constant that depends on the insulating properties of the medium. Assume
that k = 1 on the glass dome and k = 3 on the wooden floor of the greenhouse.
(a) Sketch S, clearly labelling any intercepts and the direction of the normal vector.
(b) Write down an expression in terms of x, y and z for the vector field F on the greenhouse.
(c) By direct calculation (do not use any integral theorems), find the total heat flux
F.ndS
across the greenhouse in the direction of the outward unit normal.

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F(x,y,z)=−k∇T

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