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Problem 0.8 Any object conducts heat if we heat up one part of the object (i.e. increase its temperature), then eventually that heat will spread throughout the object. Fourier's law of heat conduction states that if T(x, y, z) gives the temperature of the object at the location (x, y, z), then heat flows in the direction q = -√T. In this problem we will consider a 2D metal sheet with temperature function T(x,y) = xy - x. (1) At the point (1, 1), in which direction does the temperature increase the fastest (per unit distance)? (2) How fast does the temperature increase (per unit distance) in the direction of the vector 31+ 4ĵ? (3) Sketch the isothermal curves T = -2, -1,0, 1, 2. At several points along each isothermal curve, including the point (1, 1), draw a vector which indicates the direction that heat would flow. (4) Sketch the curves along which heat would flow (i.e. curves tangent to the direction of heat flow)

Problem 0.8 Any object conducts heat if we heat up one part of the object (i.e. increase its temperature), then eventually that heat will spread throughout the object. Fourier's law of heat conduction states that if T(x, y, z) gives the temperature of the object at the location (x, y, z), then heat flows in the direction q = -√T. In this problem we will consider a 2D metal sheet with temperature function T(x,y) = xy - x. (1) At the point (1, 1), in which direction does the temperature increase the fastest (per unit distance)? (2) How fast does the temperature increase (per unit distance) in the direction of the vector 31+ 4ĵ? (3) Sketch the isothermal curves T = -2, -1,0, 1, 2. At several points along each isothermal curve, including the point (1, 1), draw a vector which indicates the direction that heat would flow. (4) Sketch the curves along which heat would flow (i.e. curves tangent to the direction of heat flow)

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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Problem 0.8 Any object conducts heat - if we heat up one part of the object (i.e. increase its temperature), then eventually that heat will spread throughout the object. Fourier's law of heat conduction states that if T(x, y, z) gives the temperature of the object at the location (x, y, z), then heat flows in the direction q=-VT. In this problem we will consider a 2D metal sheet with temperature function T(x, y) =xy - x. (1) At the point (1, 1), in which direction does the temperature increase the fastest (per unit distance)? How fast does the temperature increase (per unit distance) in the direction of the vector 31+ 4j? (3) Sketch the isothermal curves T = -2, -1,0, 1,2. At several points along each isothermal curve, including the point (1, 1), draw a vector which indicates the direction that heat would flow. (4) Sketch the curves along which heat would flow (i.e. curves tangent to the direction of heat flow)
Transcribed Image Text:Problem 0.8 Any object conducts heat - if we heat up one part of the object (i.e. increase its temperature), then eventually that heat will spread throughout the object. Fourier's law of heat conduction states that if T(x, y, z) gives the temperature of the object at the location (x, y, z), then heat flows in the direction q=-VT. In this problem we will consider a 2D metal sheet with temperature function T(x, y) =xy - x. (1) At the point (1, 1), in which direction does the temperature increase the fastest (per unit distance)? How fast does the temperature increase (per unit distance) in the direction of the vector 31+ 4j? (3) Sketch the isothermal curves T = -2, -1,0, 1,2. At several points along each isothermal curve, including the point (1, 1), draw a vector which indicates the direction that heat would flow. (4) Sketch the curves along which heat would flow (i.e. curves tangent to the direction of heat flow)
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