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. 47. F = 〈 x + y 2 , x 2 − y 〉 ; R = { ( x , y ) : y 2 ≤ x ≤ 2 − y 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 are oriented counterclockwise. 47. F = 〈 x + y 2 , x 2 − y 〉 ; R = { ( x , y ) : y 2 ≤ x ≤ 2 − y 2 } .
Solution Summary: The author evaluates the circulation of F on the simple closed curve C is equal to the double integral of the curl throughout the region R.
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.
47.
F
=
〈
x
+
y
2
,
x
2
−
y
〉
;
R
=
{
(
x
,
y
)
:
y
2
≤
x
≤
2
−
y
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.
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}.
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.
calculate div(F) and curl(F).
F = (xy, yz, y² – x³)
Calculus, Single Variable: Early Transcendentals (3rd Edition)
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