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.
Find the maximum rate of change of f(x, y, z) = x + y/z at the point (1, 1, –5) and the direction in which it occurs.
Maximum rate of change:
Direction (unit vector) in which it occurs:
Suppose that over a certain region of space the electrical potential V is given by the following equation.
V(x, y, z) 4x2 - 2xy xyz
(a) Find the rate of change of the potential at P(2, 4, 4) in the direction of the vector v = i + j - k.
(b) In which direction does V change most rapidly at P?
(c) What is the maximum rate of change at P?
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 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}.
Chapter 17 Solutions
Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
01 - What Is an Integral in Calculus? Learn Calculus Integration and how to Solve Integrals.; Author: Math and Science;https://www.youtube.com/watch?v=BHRWArTFgTs;License: Standard YouTube License, CC-BY