Heat flux in a plate A square plate R = {( x , y ): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} has a temperature distribution T ( x , y ) = 100 – 50 x – 25 y . a. Sketch two level curves of the temperature in the plate. b. Find the gradient of the temperature ▿ T ( x, y ) . c. Assume that the flow of heat is given by the vector field F = –▿ T ( x , y ). Compute F . d. Find the outward heat flux across the boundary {( x , y ): x = 1, 0 ≤ y ≤ 1}. e. Find the outward heat flux across the boundary {( x , y ): 0 ≤ x ≤ 1, y = 1}.
Heat flux in a plate A square plate R = {( x , y ): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} has a temperature distribution T ( x , y ) = 100 – 50 x – 25 y . a. Sketch two level curves of the temperature in the plate. b. Find the gradient of the temperature ▿ T ( x, y ) . c. Assume that the flow of heat is given by the vector field F = –▿ T ( x , y ). Compute F . d. Find the outward heat flux across the boundary {( x , y ): x = 1, 0 ≤ y ≤ 1}. e. Find the outward heat flux across the boundary {( x , y ): 0 ≤ x ≤ 1, y = 1}.
Solution Summary: The author illustrates the two level curves of the temperature in the square plate.
Heat flux in a plate A square plate R = {(x, y): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} has a temperature distribution T(x, y) = 100 – 50x – 25y.
a. Sketch two level curves of the temperature in the plate.
b. Find the gradient of the temperature ▿ T(x, y).
c. Assume that the flow of heat is given by the vector field F = –▿ T(x, y). Compute F.
d. Find the outward heat flux across the boundary {(x, y): x= 1, 0 ≤ y ≤ 1}.
e. Find the outward heat flux across the boundary {(x, y): 0 ≤ x ≤ 1, y = 1}.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Elementary Statistics: Picturing the World (7th Edition)
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