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.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 30E
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There was an error in the question, the 'delta T' should not be bold, the equation should read-
F(x,y,z)=−k∇T
The temperature is a scalar function, with that taken into consideration, could you make any necessary changes in the working and answer?
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