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.
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}.
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:
X
Suppose f(x,y) = 4₁ P= (-4₁3) and v= 2i + 4;
y
)
A. Find the gradient of f
)
=
V f
i +
B. Find the gradient off at the point P.
(f)(-4₁3)=
i +-
D
Find the directional derivative of fat P
in the direction of v, where the
direction u of a vector v is the unit
vector obtained by normalizing that
vector, ie; u=
(Duf) -4,3)=
Find the maximum rate of change
of
f at P.
.
u= Tlvll
E. Find the (unit) direction vector in
which the maximum rate of change
occurs at P.
i + s
Chapter 17 Solutions
Calculus, Early Transcendentals, Single Variable Loose-Leaf Edition Plus MyLab Math with Pearson eText - 18-Week Access Card Package
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