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. 41. F = ∇ ( x 2 + y 2 ) , where R is the half annulus { ( r , θ ) : 1 ≤ r ≤ 3 , 0 ≤ θ ≤ π }
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. 41. F = ∇ ( x 2 + y 2 ) , where R is the half annulus { ( r , θ ) : 1 ≤ r ≤ 3 , 0 ≤ θ ≤ π }
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.
41.
F
=
∇
(
x
2
+
y
2
)
, where R is the half annulus
{
(
r
,
θ
)
:
1
≤
r
≤
3
,
0
≤
θ
≤
π
}
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.
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 = ⟨-y, x⟩; R is the annulus {(r, θ); 1 ≤ r ≤ 3, 0 ≤ θ ≤ π}.
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