Flux across curves in a vector field Consider the vector Field F = 〈 y, x 〉 shown in the figure. a. Compute the outward flux across the quarter circle C : r (t) = 〈2 cos t , 2 sin t ), for 0 ≤ t ≤ π /2 . b. Compute the outward flux across the quarter circle C : r (t) = 〈2 cos t , 2 sin t ), for π /2 ≤ t ≤ π. c. Explain why the flux across the quarter circle in the third quadrant equals the flux computed in part (a). d. Explain why the flux across the quarter circle in the fourth quadrant equals the flux computed in part (b). e. What is the outward flux across the full circle?
Flux across curves in a vector field Consider the vector Field F = 〈 y, x 〉 shown in the figure. a. Compute the outward flux across the quarter circle C : r (t) = 〈2 cos t , 2 sin t ), for 0 ≤ t ≤ π /2 . b. Compute the outward flux across the quarter circle C : r (t) = 〈2 cos t , 2 sin t ), for π /2 ≤ t ≤ π. c. Explain why the flux across the quarter circle in the third quadrant equals the flux computed in part (a). d. Explain why the flux across the quarter circle in the fourth quadrant equals the flux computed in part (b). e. What is the outward flux across the full circle?
Flux across curves in a vector field Consider the vector Field F = 〈y, x〉 shown in the figure.
a. Compute the outward flux across the quarter circle C: r(t) = 〈2 cos t, 2 sin t), for 0 ≤ t ≤ π/2.
b. Compute the outward flux across the quarter circle C: r(t) = 〈2 cos t, 2 sin t), for π/2 ≤ t ≤ π.
c. Explain why the flux across the quarter circle in the third quadrant equals the flux computed in part (a).
d. Explain why the flux across the quarter circle in the fourth quadrant equals the flux computed in part (b).
e. What is the outward flux across the full circle?
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.
Flux across curves in a vector field Consider the vector fieldF = ⟨y, x⟩ shown in the figure.a. Compute the outward flux across the quarter-circleC: r(t) = ⟨2 cos t, 2 sin t⟩ , for 0 ≤ t ≤ π/2.b. Compute the outward flux across the quarter-circleC: r(t) = ⟨2 cos t, 2 sin t⟩ , for π/2 ≤ t ≤ π.c. Explain why the flux across the quarter-circle in the third quadrant equals the flux computed in part (a). d. Explain why the flux across the quarter-circle in the fourth quadrant equals the flux computed in part (b).e. What is the outward flux across the full circle?
use Green’s Theorem to find the counterclock-wise circulation and outward flux for the field F and curve C.
F = (x + y)i - (x2 + y2 )j
C: The triangle bounded by y = 0, x = 1, and y = x
use Green’s Theorem to find the counterclock-wise circulation and outward flux for the field F and curve C.
F = (y2 - x2 )i + (x2 + y2 )j
C: The triangle bounded by y = 0, x = 3, and y =x.
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