Flux Consider the vector fields and curves in Exercises 57–58. a. Based on the picture, make a conjecture about whether the outward flux of F across C is positive, negative, or zero. b. Compute the flux for the vector fields and curves. 59. F and C given in Exercise 57 57. F = 〈 y − x , x 〉 ; C : r ( t ) = 〈 2 cos t , 2 sin t 〉 , for 0 ≤ t ≤ 2 π
Flux Consider the vector fields and curves in Exercises 57–58. a. Based on the picture, make a conjecture about whether the outward flux of F across C is positive, negative, or zero. b. Compute the flux for the vector fields and curves. 59. F and C given in Exercise 57 57. F = 〈 y − x , x 〉 ; C : r ( t ) = 〈 2 cos t , 2 sin t 〉 , for 0 ≤ t ≤ 2 π
Solution Summary: The flow of F on C is negative. The vector field F is directed inwards, but the flow is opposite to the orientation of the curve.
Flux Consider the vector fields and curves in Exercises 57–58.
a. Based on the picture, make a conjecture about whether the outward flux of F across C is positive, negative, or zero.
b. Compute the flux for the vector fields and curves.
59. F and C given in Exercise 57
57.
F
=
〈
y
−
x
,
x
〉
;
C :
r
(
t
)
=
〈
2
cos
t
,
2
sin
t
〉
,
for 0 ≤ t ≤ 2π
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.
a. Outward flux and area Show that the outward flux of theposition vector field F = xi + yj across any closed curve towhich Green’s Theorem applies is twice the area of the regionenclosed by the curve.b. Let n be the outward unit normal vector to a closed curve towhich Green’s Theorem applies. Show that it is not possiblefor F = x i + y j to be orthogonal to n at every point of C.
Rain on a roof Consider the vertical vector field F = ⟨0, 0, -1⟩, correspondingto a constant downward flow. Find the flux in the downward direction acrossthe surface S, which is the plane z = 4 - 2x - y in the first octant.
Explain why or why not Determine whether the following statementsare true and give an explanation or counterexample.a. A paddle wheel with its axis in the direction ⟨0, 1, -1⟩ would not spin when put in the vector field F = ⟨1, 1, 2⟩ x ⟨x, y, z⟩.b. Stokes’ Theorem relates the flux of a vector field F across a surface to values of F on the boundary of the surface.c. A vector field of the form F = ⟨a + ƒ(x), b + g(y), c + h(z)⟩, where a, b, and c are constants, has zero circulation on a closed curve.d. If a vector field has zero circulation on all simple closed smooth curves C in a region D, then F is conservative on D.
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