anonymous
  • anonymous
1) Find the flux of the vector field F(x,y,z) = (e^-y) i - (y) j + (x sinz) k across σ wit outward orientation where σ is the portion of the elliptic cylinder r(u,v) = (2cos v) i + (sin v) j + (u) k with 0 ≤ u ≤ 5, 0 ≤ v ≤ 2pi. 2) Find the work done by the force field F(x, y, z) = (x + y) i + (xy) j - (z^2) k on a particle that moves along line segments from (0,0,0) to (1,3,1) to (2,-1,4).
Mathematics
schrodinger
  • schrodinger
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anonymous
  • anonymous
sorry wat 2 high 4 me
lalaly
  • lalaly
\[\int\limits{\int\limits{s F dS}} = ∫∫∫v ∇·F dV\] \[∇·F = x \cos(z) - 1\] \[= ∫∫∫v x \cos(z) - 1 dx dy dz\] \[v: x² + 4y² ≤ 4 ; 0 ≤ z ≤ 5\] let u = x , v = 2y , z = z ∂(u,v)/∂(x,y) = 2 = 1/2 ∫∫∫v u cos(z) - 1 du dv dz v: u² + v² ≤ 4 ; 0 ≤ z ≤ 5 let u = r cos(theta) ; v = r sin(theta) z = z \[\frac{∂(u,v)}{∂(r,θ)} = r\] \[= 1/2 ∫∫∫ (r \cos(θ) \cos(z) - 1) r dr dθ dz\] \[{(r,θ,z) | 0 ≤ r ≤ 2 ; 0 ≤ θ ≤ 2π ; 0 ≤ z ≤ 5}\] \[= 1/3 ∫∫ (4 \cos(θ) \cos(z) - 3) dθ dz\] \[{(θ,z) | 0 ≤ θ ≤ 2π ; 0 ≤ z ≤ 5}\] \[= -2π ∫ dz\](z) | 0 ≤ z ≤ 5 \[= -10π\]
anonymous
  • anonymous
oh 2 smaalllllllll

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anonymous
  • anonymous
you know the second one
lalaly
  • lalaly
is the first one right? or u dont know tha answer?
anonymous
  • anonymous
im just checking what i got wrong on my hw and yeah the answer is right
lalaly
  • lalaly
oh ok im thinkin
anonymous
  • anonymous
k thanks lana :)
lalaly
  • lalaly
(0,0,0) -->> (1,3,1) x = t y = 3t z = t dx =dt........ dy = 3 dt...... dz = dt work done is integration of F.dr int{ (x + y)dx + (xy)dy - (z^2) dz
lalaly
  • lalaly
now substitute x=t y=3t and z=t and the dx dy and dz values in terms of dt
lalaly
  • lalaly
then integrate

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