Tilted disks Let S be the disk enclosed by the curve C: r ( t ) = 〈cos ϕ cos t, sin t, sin ϕ cos t 〉, for 0 ≤ t ≤ 2 p, where 0 ≤ ϕ ≤ p /2 is a fixed angle. 32. Use Stokes’ Theorem and a surface integral to find the circulation on C of the vector field F = 〈 –y, x , 0〉 as a function of ϕ. For what value of ϕ is the circulation a maximum ?
Tilted disks Let S be the disk enclosed by the curve C: r ( t ) = 〈cos ϕ cos t, sin t, sin ϕ cos t 〉, for 0 ≤ t ≤ 2 p, where 0 ≤ ϕ ≤ p /2 is a fixed angle. 32. Use Stokes’ Theorem and a surface integral to find the circulation on C of the vector field F = 〈 –y, x , 0〉 as a function of ϕ. For what value of ϕ is the circulation a maximum ?
Tilted disks Let S be the disk enclosed by the curve C:r(t) = 〈cos ϕ cos t, sin t, sin ϕ cos t〉, for 0 ≤ t ≤ 2p, where 0 ≤ ϕ ≤ p/2 is a fixed angle.
32. Use Stokes’ Theorem and a surface integral to find the circulation on C of the vector field F = 〈–y, x, 0〉 as a function of ϕ. For what value of ϕ is the circulation a maximum ?
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.
Vector F is mathematically defined as F = M x N, where M = p 2p² cos + 2p2 sind while N
is a vector normal to the surface S. Determine F as well as the area of the plane perpendicular
to F if surface S = 2xy + 3z.
a) A three dimensional motion of an object is given by the vector function
r(t) = 4 cos t i+ 4 sin tj+5 k.
Sketch the motion of the object when 0
=
Use Stokes' Theorem to find the circulation of F 5y + 5zj+2ak around the triangle obtained by tracing out the path
(6,0,0) to (6,0, 6), to (6, 3, 6) back to (6,0,0).
Circulation =
- dr = -45
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