Circulation and flux For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region, Assume boundary curves have counterclockwise orientation. 42. F = ( y cos x , − sin x ) , where R is the square { ( x , y ) : 0 ≤ x ≤ π / 2 , 0 ≤ y ≤ π / 2 }
Circulation and flux For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region, Assume boundary curves have counterclockwise orientation. 42. F = ( y cos x , − sin x ) , where R is the square { ( x , y ) : 0 ≤ x ≤ π / 2 , 0 ≤ y ≤ π / 2 }
Solution Summary: The author explains how to compute the circulation of the vector field F=langle ymathrmcosx,-
Circulation and fluxFor the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region, Assume boundary curves have counterclockwise orientation.
42.
F
=
(
y
cos
x
,
−
sin
x
)
, where R is the square
{
(
x
,
y
)
:
0
≤
x
≤
π
/
2
,
0
≤
y
≤
π
/
2
}
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.
Circulation and flux For the following vector fields, compute (a) the circulation on, and (b) the outward flux across, the boundary of the given region. Assume boundary curves are oriented counterclockwise.
F = ⟨x + y2, x2 - y⟩; R = {(x, y): y2 ≤ x ≤ 2 - y2}.
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}.
Heat flux The heat flow vector field for conducting objects is F = -k∇T, where T(x, y, z) is the temperature in the object and k > 0 is a constant that depends on the material. Compute the outward flux of F across the following surfaces S for the given temperature distributions. Assume k = 1.
T(x, y, z) = -ln (x2 + y2 + z2); S is the sphere x2 + y2 + z2 = a2.
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
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